CINXE.COM

<!DOCTYPE html> <html lang="en" dir="ltr"> <head>  <script async src="https://www.googletagmanager.com/gtag/js?id=G-P63WKM1TM1"></script> <script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-P63WKM1TM1'); </script>  <script type="text/javascript" > (function(m,e,t,r,i,k,a){m[i]=m[i]||function(){(m[i].a=m[i].a||[]).push(arguments)}; m[i].l=1*new Date(); for (var j = 0; j < document.scripts.length; j++) {if (document.scripts[j].src === r) { return; }} k=e.createElement(t),a=e.getElementsByTagName(t)[0],k.async=1,k.src=r,a.parentNode.insertBefore(k,a)}) (window, document, "script", "https://mc.yandex.ru/metrika/tag.js", "ym"); ym(55165297, "init", { clickmap:false, trackLinks:true, accurateTrackBounce:true, webvisor:false }); </script> <noscript><div><img src="https://mc.yandex.ru/watch/55165297" style="position:absolute; left:-9999px;" alt="" /></div></noscript>    <title>Search results for: exact solutions of the Einstein equations</title> <meta name="description" content="Search results for: exact solutions of the Einstein equations"> <meta name="keywords" content="exact solutions of the Einstein equations"> <meta name="viewport" content="width=device-width, initial-scale=1, minimum-scale=1, maximum-scale=1, user-scalable=no"> <meta charset="utf-8"> <link href="https://cdn.waset.org/favicon.ico" type="image/x-icon" rel="shortcut icon"> <link href="https://cdn.waset.org/static/plugins/bootstrap-4.2.1/css/bootstrap.min.css" rel="stylesheet"> <link href="https://cdn.waset.org/static/plugins/fontawesome/css/all.min.css" rel="stylesheet"> <link href="https://cdn.waset.org/static/css/site.css?v=150220211555" rel="stylesheet"> </head> <body> <header> <div class="container"> <nav class="navbar navbar-expand-lg navbar-light"> <a class="navbar-brand" href="https://waset.org"> <img src="https://cdn.waset.org/static/images/wasetc.png" alt="Open Science Research Excellence" title="Open Science Research Excellence" /> </a> <button class="d-block d-lg-none navbar-toggler ml-auto" type="button" data-toggle="collapse" data-target="#navbarMenu" aria-controls="navbarMenu" aria-expanded="false" aria-label="Toggle navigation"> <span class="navbar-toggler-icon"></span> </button> <div class="w-100"> <div class="d-none d-lg-flex flex-row-reverse"> <form method="get" action="https://waset.org/search" class="form-inline my-2 my-lg-0"> <input class="form-control mr-sm-2" type="search" placeholder="Search Conferences" value="exact solutions of the Einstein equations" name="q" aria-label="Search"> <button class="btn btn-light my-2 my-sm-0" type="submit"><i class="fas fa-search"></i></button> </form> </div> <div class="collapse navbar-collapse mt-1" id="navbarMenu"> <ul class="navbar-nav ml-auto align-items-center" id="mainNavMenu"> <li class="nav-item"> <a class="nav-link" href="https://waset.org/conferences" title="Conferences in 2024/2025/2026">Conferences</a> </li> <li class="nav-item"> <a class="nav-link" href="https://waset.org/disciplines" title="Disciplines">Disciplines</a> </li> <li class="nav-item"> <a class="nav-link" href="https://waset.org/committees" rel="nofollow">Committees</a> </li> <li class="nav-item dropdown"> <a class="nav-link dropdown-toggle" href="#" id="navbarDropdownPublications" role="button" data-toggle="dropdown" aria-haspopup="true" aria-expanded="false"> Publications </a> <div class="dropdown-menu" aria-labelledby="navbarDropdownPublications"> <a class="dropdown-item" href="https://publications.waset.org/abstracts">Abstracts</a> <a class="dropdown-item" href="https://publications.waset.org">Periodicals</a> <a class="dropdown-item" href="https://publications.waset.org/archive">Archive</a> </div> </li> <li class="nav-item"> <a class="nav-link" href="https://waset.org/page/support" title="Support">Support</a> </li> </ul> </div> </div> </nav> </div> </header> <main> <div class="container mt-4"> <div class="row"> <div class="col-md-9 mx-auto"> <form method="get" action="https://publications.waset.org/abstracts/search"> <div id="custom-search-input"> <div class="input-group"> <i class="fas fa-search"></i> <input type="text" class="search-query" name="q" placeholder="Author, Title, Abstract, Keywords" value="exact solutions of the Einstein equations"> <input type="submit" class="btn_search" value="Search"> </div> </div> </form> </div> </div> <div class="row mt-3"> <div class="col-sm-3"> <div class="card"> <div class="card-body"><strong>Commenced</strong> in January 2007</div> </div> </div> <div class="col-sm-3"> <div class="card"> <div class="card-body"><strong>Frequency:</strong> Monthly</div> </div> </div> <div class="col-sm-3"> <div class="card"> <div class="card-body"><strong>Edition:</strong> International</div> </div> </div> <div class="col-sm-3"> <div class="card"> <div class="card-body"><strong>Paper Count:</strong> 6015</div> </div> </div> </div> <h1 class="mt-3 mb-3 text-center" style="font-size:1.6rem;">Search results for: exact solutions of the Einstein equations</h1> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">6015</span> The Generalized Lemaitre-Tolman-Bondi Solutions in Modeling the Cosmological Black Holes </h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Elena%20M.%20Kopteva">Elena M. Kopteva</a>, <a href="https://publications.waset.org/abstracts/search?q=Pavlina%20Jaluvkova"> Pavlina Jaluvkova</a>, <a href="https://publications.waset.org/abstracts/search?q=Zdenek%20Stuchlik"> Zdenek Stuchlik</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In spite of the numerous attempts to close the discussion about the influence of cosmological expansion on local gravitationally bounded systems, this question arises in literature again and again and remains still far from its final resolution. Here one of the main problems is the problem of obtaining a physically adequate model of strongly gravitating object immersed in non-static cosmological background. Such objects are usually called ‘cosmological’ black holes and are of great interest in wide set of cosmological and astrophysical areas. In this work the set of new exact solutions of the Einstein equations is derived for the flat space that generalizes the known Lemaitre-Tolman-Bondi solution for the case of nonzero pressure. The solutions obtained are pretending to describe the black hole immersed in nonstatic cosmological background and give a possibility to investigate the hot problems concerning the effects of the cosmological expansion in gravitationally bounded systems, the structure formation in the early universe, black hole thermodynamics and other related problems. It is shown that each of the solutions obtained contains either the Reissner-Nordstrom or the Schwarzschild black hole in the central region of the space. It is demonstrated that the approach of the mass function use in solving of the Einstein equations allows clear physical interpretation of the resulting solutions, that is of much benefit to any their concrete application. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=exact%20solutions%20of%20the%20Einstein%20equations" title="exact solutions of the Einstein equations">exact solutions of the Einstein equations</a>, <a href="https://publications.waset.org/abstracts/search?q=cosmological%20black%20holes" title=" cosmological black holes"> cosmological black holes</a>, <a href="https://publications.waset.org/abstracts/search?q=generalized%20Lemaitre-Tolman-Bondi%20solutions" title=" generalized Lemaitre-Tolman-Bondi solutions"> generalized Lemaitre-Tolman-Bondi solutions</a>, <a href="https://publications.waset.org/abstracts/search?q=nonzero%20pressure" title=" nonzero pressure"> nonzero pressure</a> </p> <a href="https://publications.waset.org/abstracts/63654/the-generalized-lemaitre-tolman-bondi-solutions-in-modeling-the-cosmological-black-holes" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/63654.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">423</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">6014</span> Bianchi Type- I Viscous Fluid Cosmological Models with Stiff Matter and Time Dependent Λ- Term</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Rajendra%20Kumar%20Dubey">Rajendra Kumar Dubey</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Einstein’s field equations with variable cosmological term Λ are considered in the presence of viscous fluid for Bianchi type I space time. Exact solutions of Einstein’s field equations are obtained by assuming cosmological term Λ Proportional to (R is a scale factor and m is constant). We observed that the shear viscosity is found to be responsible for faster removal of initial anisotropy in the universe. The physical significance of the cosmological models has also been discussed. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=bianchi%20type" title="bianchi type">bianchi type</a>, <a href="https://publications.waset.org/abstracts/search?q=I%20cosmological%20model" title=" I cosmological model"> I cosmological model</a>, <a href="https://publications.waset.org/abstracts/search?q=viscous%20fluid" title=" viscous fluid"> viscous fluid</a>, <a href="https://publications.waset.org/abstracts/search?q=cosmological%20constant%20%CE%9B" title=" cosmological constant Λ"> cosmological constant Λ</a> </p> <a href="https://publications.waset.org/abstracts/28301/bianchi-type-i-viscous-fluid-cosmological-models-with-stiff-matter-and-time-dependent-l-term" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/28301.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">528</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">6013</span> Exploring Solutions in Extended Horava-Lifshitz Gravity</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Aziza%20Altaibayeva">Aziza Altaibayeva</a>, <a href="https://publications.waset.org/abstracts/search?q=Ertan%20G%C3%BCdekli"> Ertan Güdekli</a>, <a href="https://publications.waset.org/abstracts/search?q=Ratbay%20Myrzakulov"> Ratbay Myrzakulov</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this letter, we explore exact solutions for the Horava-Lifshitz gravity. We use of an extension of this theory with first order dynamical lapse function. The equations of motion have been derived in a fully consistent scenario. We assume that there are some spherically symmetric families of exact solutions of this extended theory of gravity. We obtain exact solutions and investigate the singularity structures of these solutions. Specially, an exact solution with the regular horizon is found. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=quantum%20gravity" title="quantum gravity">quantum gravity</a>, <a href="https://publications.waset.org/abstracts/search?q=Horava-Lifshitz%20gravity" title=" Horava-Lifshitz gravity"> Horava-Lifshitz gravity</a>, <a href="https://publications.waset.org/abstracts/search?q=black%20hole" title=" black hole"> black hole</a>, <a href="https://publications.waset.org/abstracts/search?q=spherically%20symmetric%20space%20times" title=" spherically symmetric space times "> spherically symmetric space times </a> </p> <a href="https://publications.waset.org/abstracts/18654/exploring-solutions-in-extended-horava-lifshitz-gravity" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/18654.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">581</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">6012</span> Poisson Type Spherically Symmetric Spacetimes</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Gonzalo%20Garc%C3%ADa-Reyes">Gonzalo García-Reyes</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Conformastat spherically symmetric exact solutions of Einstein's field equations representing matter distributions made of fluid both perfect and anisotropic from given solutions of Poisson's equation of Newtonian gravity are investigated. The approach is used in the construction of new relativistic models of thick spherical shells and three-component models of galaxies (bulge, disk, and dark matter halo), writing, in this case, the metric in cylindrical coordinates. In addition, the circular motion of test particles (rotation curves) along geodesics on the equatorial plane of matter configurations and the stability of the orbits against radial perturbations are studied. The models constructed satisfy all the energy conditions. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=general%20relativity" title="general relativity">general relativity</a>, <a href="https://publications.waset.org/abstracts/search?q=exact%20solutions" title=" exact solutions"> exact solutions</a>, <a href="https://publications.waset.org/abstracts/search?q=spherical%20symmetry" title=" spherical symmetry"> spherical symmetry</a>, <a href="https://publications.waset.org/abstracts/search?q=galaxy" title=" galaxy"> galaxy</a>, <a href="https://publications.waset.org/abstracts/search?q=kinematics%20and%20dynamics" title=" kinematics and dynamics"> kinematics and dynamics</a>, <a href="https://publications.waset.org/abstracts/search?q=dark%20matter" title=" dark matter"> dark matter</a> </p> <a href="https://publications.waset.org/abstracts/151913/poisson-type-spherically-symmetric-spacetimes" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/151913.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">87</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">6011</span> Classification of Cosmological Wormhole Solutions in the Framework of General Relativity</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Usamah%20Al-Ali">Usamah Al-Ali</a> </p> <p class="card-text"><strong>Abstract:</strong></p> We explore the effect of expanding space on the exoticity of the matter supporting a traversable Lorentzian wormhole of zero radial tide whose line element is given by ds2 = dt^2 − a^2(t)[ dr^2/(1 − kr2 −b(r)/r)+ r2dΩ^2 in the context of General Relativity. This task is achieved by deriving the Einstein field equations for anisotropic matter field corresponding to the considered cosmological wormhole metric and performing a classification of their solutions on the basis of a variable equations of state (EoS) of the form p = ω(r)ρ. Explicit forms of the shape function b(r) and the scale factor a(t) arising in the classification are utilized to construct the corresponding energy-momentum tensor where the energy conditions for each case is investigated. While the violation of energy conditions is inevitable in case of static wormholes, the classification we performed leads to interesting solutions in which this violation is either reduced or eliminated. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=general%20relativity" title="general relativity">general relativity</a>, <a href="https://publications.waset.org/abstracts/search?q=Einstein%20field%20equations" title=" Einstein field equations"> Einstein field equations</a>, <a href="https://publications.waset.org/abstracts/search?q=energy%20conditions" title=" energy conditions"> energy conditions</a>, <a href="https://publications.waset.org/abstracts/search?q=cosmological%20wormhole" title=" cosmological wormhole"> cosmological wormhole</a> </p> <a href="https://publications.waset.org/abstracts/150239/classification-of-cosmological-wormhole-solutions-in-the-framework-of-general-relativity" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/150239.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">63</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">6010</span> Particle and Photon Trajectories near the Black Hole Immersed in the Nonstatic Cosmological Background</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Elena%20M.%20Kopteva">Elena M. Kopteva</a>, <a href="https://publications.waset.org/abstracts/search?q=Pavlina%20Jaluvkova"> Pavlina Jaluvkova</a>, <a href="https://publications.waset.org/abstracts/search?q=Zdenek%20Stuchlik"> Zdenek Stuchlik</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The question of constructing a consistent model of the cosmological black hole remains to be unsolved and still attracts the interest of cosmologists as far as it is important in a wide set of research problems including the problem of the black hole horizon dynamics, the problem of interplay between cosmological expansion and local gravity, the problem of structure formation in the early universe etc. In this work, the model of the cosmological black hole is built on the basis of the exact solution of the Einstein equations for the spherically symmetric inhomogeneous dust distribution in the approach of the mass function use. Possible trajectories for massive particles and photons near the black hole immersed in the nonstatic dust cosmological background are investigated in frame of the obtained model. The reference system of distant galaxy comoving to cosmological expansion combined with curvature coordinates is used, so that the resulting metric becomes nondiagonal and involves both proper ‘cosmological’ time and curvature spatial coordinates. For this metric the geodesic equations are analyzed for the test particles and photons, and the respective trajectories are built. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=exact%20solutions%20for%20Einstein%20equations" title="exact solutions for Einstein equations">exact solutions for Einstein equations</a>, <a href="https://publications.waset.org/abstracts/search?q=Lemaitre-Tolman-Bondi%20solution" title=" Lemaitre-Tolman-Bondi solution"> Lemaitre-Tolman-Bondi solution</a>, <a href="https://publications.waset.org/abstracts/search?q=cosmological%20black%20holes" title=" cosmological black holes"> cosmological black holes</a>, <a href="https://publications.waset.org/abstracts/search?q=particle%20and%20photon%20trajectories" title=" particle and photon trajectories"> particle and photon trajectories</a> </p> <a href="https://publications.waset.org/abstracts/63353/particle-and-photon-trajectories-near-the-black-hole-immersed-in-the-nonstatic-cosmological-background" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/63353.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">339</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">6009</span> Exact Soliton Solutions of the Integrable (2+1)-Dimensional Fokas-Lenells Equation</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Meruyert%20Zhassybayeva">Meruyert Zhassybayeva</a>, <a href="https://publications.waset.org/abstracts/search?q=Kuralay%20Yesmukhanova"> Kuralay Yesmukhanova</a>, <a href="https://publications.waset.org/abstracts/search?q=Ratbay%20Myrzakulov"> Ratbay Myrzakulov</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Integrable nonlinear differential equations are an important class of nonlinear wave equations that admit exact soliton solutions. All these equations have an amazing property which is that their soliton waves collide elastically. One of such equations is the (1+1)-dimensional Fokas-Lenells equation. In this paper, we have constructed an integrable (2+1)-dimensional Fokas-Lenells equation. The integrability of this equation is ensured by the existence of a Lax representation for it. We obtained its bilinear form from the Hirota method. Using the Hirota method, exact one-soliton and two-soliton solutions of the (2 +1)-dimensional Fokas-Lenells equation were found. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=Fokas-Lenells%20equation" title="Fokas-Lenells equation">Fokas-Lenells equation</a>, <a href="https://publications.waset.org/abstracts/search?q=integrability" title=" integrability"> integrability</a>, <a href="https://publications.waset.org/abstracts/search?q=soliton" title=" soliton"> soliton</a>, <a href="https://publications.waset.org/abstracts/search?q=the%20Hirota%20bilinear%20method" title=" the Hirota bilinear method"> the Hirota bilinear method</a> </p> <a href="https://publications.waset.org/abstracts/99044/exact-soliton-solutions-of-the-integrable-21-dimensional-fokas-lenells-equation" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/99044.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">224</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">6008</span> Exact Solutions of K(N,N)-Type Equations Using Jacobi Elliptic Functions</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Edamana%20Krishnan">Edamana Krishnan</a>, <a href="https://publications.waset.org/abstracts/search?q=Khalil%20Al-Ghafri"> Khalil Al-Ghafri</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper, modified K(n,n) and K(n+1,n+1) equations have been solved using mapping methods which give a variety of solutions in terms of Jacobi elliptic functions. The solutions when m approaches 0 and 1, with m as the modulus of the JEFs have also been deduced. The role of constraint conditions has been discussed. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=travelling%20wave%20solutions" title="travelling wave solutions">travelling wave solutions</a>, <a href="https://publications.waset.org/abstracts/search?q=solitary%20wave%20solutions" title=" solitary wave solutions"> solitary wave solutions</a>, <a href="https://publications.waset.org/abstracts/search?q=compactons" title=" compactons"> compactons</a>, <a href="https://publications.waset.org/abstracts/search?q=Jacobi%20elliptic%20functions" title=" Jacobi elliptic functions"> Jacobi elliptic functions</a>, <a href="https://publications.waset.org/abstracts/search?q=mapping%20methods" title=" mapping methods"> mapping methods</a> </p> <a href="https://publications.waset.org/abstracts/59011/exact-solutions-of-knn-type-equations-using-jacobi-elliptic-functions" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/59011.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">305</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">6007</span> Analytical Solutions of Time Space Fractional, Advection-Dispersion and Whitham-Broer-Kaup Equations</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Muhammad%20Danish%20Khan">Muhammad Danish Khan</a>, <a href="https://publications.waset.org/abstracts/search?q=Imran%20Naeem"> Imran Naeem</a>, <a href="https://publications.waset.org/abstracts/search?q=Mudassar%20Imran"> Mudassar Imran</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this article, we study time-space Fractional Advection-Dispersion (FADE) equation and time-space Fractional Whitham-Broer-Kaup (FWBK) equation that have a significant role in hydrology. We introduce suitable transformations to convert fractional order derivatives to integer order derivatives and as a result these equations transform into Partial Differential Equations (PDEs). Then the Lie symmetries and corresponding optimal systems of the resulting PDEs are derived. The symmetry reductions and exact independent solutions based on optimal system are investigated which constitute the exact solutions of original fractional differential equations. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=modified%20Riemann-Liouville%20fractional%20derivative" title="modified Riemann-Liouville fractional derivative">modified Riemann-Liouville fractional derivative</a>, <a href="https://publications.waset.org/abstracts/search?q=lie-symmetries" title=" lie-symmetries"> lie-symmetries</a>, <a href="https://publications.waset.org/abstracts/search?q=optimal%20system" title=" optimal system"> optimal system</a>, <a href="https://publications.waset.org/abstracts/search?q=invariant%20solutions" title=" invariant solutions"> invariant solutions</a> </p> <a href="https://publications.waset.org/abstracts/2191/analytical-solutions-of-time-space-fractional-advection-dispersion-and-whitham-broer-kaup-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/2191.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">431</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">6006</span> The Application of Variable Coefficient Jacobian elliptic Function Method to Differential-Difference Equations</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Chao-Qing%20Dai">Chao-Qing Dai</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In modern nonlinear science and textile engineering, nonlinear differential-difference equations are often used to describe some nonlinear phenomena. In this paper, we extend the variable coefficient Jacobian elliptic function method, which was used to find new exact travelling wave solutions of nonlinear partial differential equations, to nonlinear differential-difference equations. As illustration, we derive two series of Jacobian elliptic function solutions of the discrete sine-Gordon equation. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=discrete%20sine-Gordon%20equation" title="discrete sine-Gordon equation">discrete sine-Gordon equation</a>, <a href="https://publications.waset.org/abstracts/search?q=variable%20coefficient%20Jacobian%20elliptic%20function%20method" title=" variable coefficient Jacobian elliptic function method"> variable coefficient Jacobian elliptic function method</a>, <a href="https://publications.waset.org/abstracts/search?q=exact%20solutions" title=" exact solutions"> exact solutions</a>, <a href="https://publications.waset.org/abstracts/search?q=equation" title=" equation"> equation</a> </p> <a href="https://publications.waset.org/abstracts/12987/the-application-of-variable-coefficient-jacobian-elliptic-function-method-to-differential-difference-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/12987.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">668</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">6005</span> Analytical Solution of Non–Autonomous Discrete Non-Linear Schrodinger Equation With Saturable Non-Linearity</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Mishu%20Gupta">Mishu Gupta</a>, <a href="https://publications.waset.org/abstracts/search?q=Rama%20Gupta"> Rama Gupta</a> </p> <p class="card-text"><strong>Abstract:</strong></p> It has been elucidated here that non- autonomous discrete non-linear Schrödinger equation is associated with saturable non-linearity through photo-refractive media. We have investigated the localized solution of non-autonomous saturable discrete non-linear Schrödinger equations. The similarity transformation has been involved in converting non-autonomous saturable discrete non-linear Schrödinger equation to constant-coefficient saturable discrete non-linear Schrödinger equation (SDNLSE), whose exact solution is already known. By back substitution, the solution of the non-autonomous version has been obtained. We have analysed our solution for the hyperbolic and periodic form of gain/loss term, and interesting results have been obtained. The most important characteristic role is that it helps us to analyse the propagation of electromagnetic waves in glass fibres and other optical wave mediums. Also, the usage of SDNLSE has been seen in tight binding for Bose-Einstein condensates in optical mediums. Even the solutions are interrelated, and its properties are prominently used in various physical aspects like optical waveguides, Bose-Einstein (B-E) condensates in optical mediums, Non-linear optics in photonic crystals, and non-linear kerr–type non-linearity effect and photo refracting medium. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=B-E-Bose-Einstein" title="B-E-Bose-Einstein">B-E-Bose-Einstein</a>, <a href="https://publications.waset.org/abstracts/search?q=DNLSE-Discrete%20non%20linear%20schrodinger%20equation" title=" DNLSE-Discrete non linear schrodinger equation"> DNLSE-Discrete non linear schrodinger equation</a>, <a href="https://publications.waset.org/abstracts/search?q=NLSE-non%20linear%20schrodinger%20equation" title=" NLSE-non linear schrodinger equation"> NLSE-non linear schrodinger equation</a>, <a href="https://publications.waset.org/abstracts/search?q=SDNLSE%20-%20saturable%20discrete%20non%20linear%20Schrodinger%20equation" title=" SDNLSE - saturable discrete non linear Schrodinger equation"> SDNLSE - saturable discrete non linear Schrodinger equation</a> </p> <a href="https://publications.waset.org/abstracts/121074/analytical-solution-of-non-autonomous-discrete-non-linear-schrodinger-equation-with-saturable-non-linearity" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/121074.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">155</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">6004</span> Foliation and the First Law of Thermodynamics for the Kerr Newman Black Hole</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Syed%20M.%20Jawwad%20Riaz">Syed M. Jawwad Riaz</a> </p> <p class="card-text"><strong>Abstract:</strong></p> There has been a lot of interest in exploring the thermodynamic properties at the horizon of a black hole geometry. Earlier, it has been shown, for different spacetimes, that the Einstein field equations at the horizon can be expressed as a first law of black hole thermodynamics. In this paper, considering r = constant slices, for the Kerr-Newman black hole, shown that the Einstein field equations for the induced 3-metric of the hypersurface is expressed in thermodynamic quantities under the virtual displacements of the hypersurfaces. As expected, it is found that the field equations of the induced metric corresponding to the horizon can only be written as a first law of black hole thermodynamics. It is to be mentioned here that the procedure adopted is much easier, to obtain such results, as here one has to essentially deal with (n - 1)-dimensional induced metric for an n-dimensional spacetime. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=black%20hole%20space-times" title="black hole space-times">black hole space-times</a>, <a href="https://publications.waset.org/abstracts/search?q=Einstein%27s%20field%20equation" title=" Einstein's field equation"> Einstein's field equation</a>, <a href="https://publications.waset.org/abstracts/search?q=foliation" title=" foliation"> foliation</a>, <a href="https://publications.waset.org/abstracts/search?q=hyper-surfaces" title=" hyper-surfaces"> hyper-surfaces</a> </p> <a href="https://publications.waset.org/abstracts/50127/foliation-and-the-first-law-of-thermodynamics-for-the-kerr-newman-black-hole" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/50127.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">346</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">6003</span> Exact Solutions of Discrete Sine-Gordon Equation</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Chao-Qing%20Dai">Chao-Qing Dai</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Two families of exact travelling solutions for the discrete sine-Gordon equation are constructed based on the variable-coefficient Jacobian elliptic function method and different transformations. When the modulus of Jacobian elliptic function solutions tends to 1, soliton solutions can be obtained. Some soliton solutions degenerate into the known solutions in literatures. Moreover, dynamical properties of exact solutions are investigated. Our analysis and results may have potential values for certain applications in modern nonlinear science and textile engineering. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=exact%20solutions" title="exact solutions">exact solutions</a>, <a href="https://publications.waset.org/abstracts/search?q=variable-coefficient%20Jacobian%20elliptic%20function%20method" title=" variable-coefficient Jacobian elliptic function method"> variable-coefficient Jacobian elliptic function method</a>, <a href="https://publications.waset.org/abstracts/search?q=discrete%20sine-Gordon%20equation" title=" discrete sine-Gordon equation"> discrete sine-Gordon equation</a>, <a href="https://publications.waset.org/abstracts/search?q=dynamical%20behaviors" title=" dynamical behaviors"> dynamical behaviors</a> </p> <a href="https://publications.waset.org/abstracts/48966/exact-solutions-of-discrete-sine-gordon-equation" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/48966.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">420</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">6002</span> Symbolic Computation for the Multi-Soliton Solutions of a Class of Fifth-Order Evolution Equations</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Rafat%20Alshorman">Rafat Alshorman</a>, <a href="https://publications.waset.org/abstracts/search?q=Fadi%20Awawdeh"> Fadi Awawdeh</a> </p> <p class="card-text"><strong>Abstract:</strong></p> By employing a simplified bilinear method, a class of generalized fifth-order KdV (gfKdV) equations which arise in nonlinear lattice, plasma physics and ocean dynamics are investigated. With the aid of symbolic computation, both solitary wave solutions and multiple-soliton solutions are obtained. These new exact solutions will extend previous results and help us explain the properties of nonlinear solitary waves in many physical models in shallow water. Parametric analysis is carried out in order to illustrate that the soliton amplitude, width and velocity are affected by the coefficient parameters in the equation. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=multiple%20soliton%20solutions" title="multiple soliton solutions">multiple soliton solutions</a>, <a href="https://publications.waset.org/abstracts/search?q=fifth-order%20evolution%20equations" title=" fifth-order evolution equations"> fifth-order evolution equations</a>, <a href="https://publications.waset.org/abstracts/search?q=Cole-Hopf%20transformation" title=" Cole-Hopf transformation"> Cole-Hopf transformation</a>, <a href="https://publications.waset.org/abstracts/search?q=Hirota%20bilinear%20method" title=" Hirota bilinear method"> Hirota bilinear method</a> </p> <a href="https://publications.waset.org/abstracts/9376/symbolic-computation-for-the-multi-soliton-solutions-of-a-class-of-fifth-order-evolution-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/9376.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">320</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">6001</span> On the Algorithmic Iterative Solutions of Conjugate Gradient, Gauss-Seidel and Jacobi Methods for Solving Systems of Linear Equations</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Hussaini%20Doko%20Ibrahim">Hussaini Doko Ibrahim</a>, <a href="https://publications.waset.org/abstracts/search?q=Hamilton%20Cyprian%20Chinwenyi"> Hamilton Cyprian Chinwenyi</a>, <a href="https://publications.waset.org/abstracts/search?q=Henrietta%20Nkem%20Ude"> Henrietta Nkem Ude</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper, efforts were made to examine and compare the algorithmic iterative solutions of the conjugate gradient method as against other methods such as Gauss-Seidel and Jacobi approaches for solving systems of linear equations of the form Ax=b, where A is a real n×n symmetric and positive definite matrix. We performed algorithmic iterative steps and obtained analytical solutions of a typical 3×3 symmetric and positive definite matrix using the three methods described in this paper (Gauss-Seidel, Jacobi, and conjugate gradient methods), respectively. From the results obtained, we discovered that the conjugate gradient method converges faster to exact solutions in fewer iterative steps than the two other methods, which took many iterations, much time, and kept tending to the exact solutions. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=conjugate%20gradient" title="conjugate gradient">conjugate gradient</a>, <a href="https://publications.waset.org/abstracts/search?q=linear%20equations" title=" linear equations"> linear equations</a>, <a href="https://publications.waset.org/abstracts/search?q=symmetric%20and%20positive%20definite%20matrix" title=" symmetric and positive definite matrix"> symmetric and positive definite matrix</a>, <a href="https://publications.waset.org/abstracts/search?q=gauss-seidel" title=" gauss-seidel"> gauss-seidel</a>, <a href="https://publications.waset.org/abstracts/search?q=Jacobi" title=" Jacobi"> Jacobi</a>, <a href="https://publications.waset.org/abstracts/search?q=algorithm" title=" algorithm"> algorithm</a> </p> <a href="https://publications.waset.org/abstracts/138341/on-the-algorithmic-iterative-solutions-of-conjugate-gradient-gauss-seidel-and-jacobi-methods-for-solving-systems-of-linear-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/138341.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">149</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">6000</span> Series Solutions to Boundary Value Differential Equations</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Armin%20Ardekani">Armin Ardekani</a>, <a href="https://publications.waset.org/abstracts/search?q=Mohammad%20Akbari"> Mohammad Akbari</a> </p> <p class="card-text"><strong>Abstract:</strong></p> We present a method of generating series solutions to large classes of nonlinear differential equations. The method is well suited to be adapted in mathematical software and unlike the available commercial solvers, we are capable of generating solutions to boundary value ODEs and PDEs. Many of the generated solutions converge to closed form solutions. Our method can also be applied to systems of ODEs or PDEs, providing all the solutions efficiently. As examples, we present results to many difficult differential equations in engineering fields. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=computational%20mathematics" title="computational mathematics">computational mathematics</a>, <a href="https://publications.waset.org/abstracts/search?q=differential%20equations" title=" differential equations"> differential equations</a>, <a href="https://publications.waset.org/abstracts/search?q=engineering" title=" engineering"> engineering</a>, <a href="https://publications.waset.org/abstracts/search?q=series" title=" series"> series</a> </p> <a href="https://publications.waset.org/abstracts/54764/series-solutions-to-boundary-value-differential-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/54764.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">336</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">5999</span> Comparing Numerical Accuracy of Solutions of Ordinary Differential Equations (ODE) Using Taylor's Series Method, Euler's Method and Runge-Kutta (RK) Method</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Palwinder%20Singh">Palwinder Singh</a>, <a href="https://publications.waset.org/abstracts/search?q=Munish%20Sandhir"> Munish Sandhir</a>, <a href="https://publications.waset.org/abstracts/search?q=Tejinder%20Singh"> Tejinder Singh</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The ordinary differential equations (ODE) represent a natural framework for mathematical modeling of many real-life situations in the field of engineering, control systems, physics, chemistry and astronomy etc. Such type of differential equations can be solved by analytical methods or by numerical methods. If the solution is calculated using analytical methods, it is done through calculus theories, and thus requires a longer time to solve. In this paper, we compare the numerical accuracy of the solutions given by the three main types of one-step initial value solvers: Taylor’s Series Method, Euler’s Method and Runge-Kutta Fourth Order Method (RK4). The comparison of accuracy is obtained through comparing the solutions of ordinary differential equation given by these three methods. Furthermore, to verify the accuracy; we compare these numerical solutions with the exact solutions. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=Ordinary%20differential%20equations%20%28ODE%29" title="Ordinary differential equations (ODE)">Ordinary differential equations (ODE)</a>, <a href="https://publications.waset.org/abstracts/search?q=Taylor%E2%80%99s%20Series%20Method" title=" Taylor’s Series Method"> Taylor’s Series Method</a>, <a href="https://publications.waset.org/abstracts/search?q=Euler%E2%80%99s%20Method" title=" Euler’s Method"> Euler’s Method</a>, <a href="https://publications.waset.org/abstracts/search?q=Runge-Kutta%20Fourth%20Order%20Method" title=" Runge-Kutta Fourth Order Method"> Runge-Kutta Fourth Order Method</a> </p> <a href="https://publications.waset.org/abstracts/56685/comparing-numerical-accuracy-of-solutions-of-ordinary-differential-equations-ode-using-taylors-series-method-eulers-method-and-runge-kutta-rk-method" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/56685.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">358</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">5998</span> Residual Power Series Method for System of Volterra Integro-Differential Equations</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Zuhier%20Altawallbeh">Zuhier Altawallbeh</a> </p> <p class="card-text"><strong>Abstract:</strong></p> This paper investigates the approximate analytical solutions of general form of Volterra integro-differential equations system by using the residual power series method (for short RPSM). The proposed method produces the solutions in terms of convergent series requires no linearization or small perturbation and reproduces the exact solution when the solution is polynomial. Some examples are given to demonstrate the simplicity and efficiency of the proposed method. Comparisons with the Laplace decomposition algorithm verify that the new method is very effective and convenient for solving system of pantograph equations. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=integro-differential%20equation" title="integro-differential equation">integro-differential equation</a>, <a href="https://publications.waset.org/abstracts/search?q=pantograph%20equations" title=" pantograph equations"> pantograph equations</a>, <a href="https://publications.waset.org/abstracts/search?q=system%20of%20initial%20value%20problems" title=" system of initial value problems"> system of initial value problems</a>, <a href="https://publications.waset.org/abstracts/search?q=residual%20power%20series%20method" title=" residual power series method"> residual power series method</a> </p> <a href="https://publications.waset.org/abstracts/35727/residual-power-series-method-for-system-of-volterra-integro-differential-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/35727.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">418</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">5997</span> Weak Solutions Of Stochastic Fractional Differential Equations</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Lev%20Idels">Lev Idels</a>, <a href="https://publications.waset.org/abstracts/search?q=Arcady%20Ponosov"> Arcady Ponosov</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Stochastic fractional differential equations have recently attracted considerable attention, as they have been used to model real-world processes, which are subject to natural memory effects and measurement uncertainties. Compared to conventional hereditary differential equations, one of the advantages of fractional differential equations is related to more realistic geometric properties of their trajectories that do not intersect in the phase space. In this report, a Peano-like existence theorem for nonlinear stochastic fractional differential equations is proven under very general hypotheses. Several specific classes of equations are checked to satisfy these hypotheses, including delay equations driven by the fractional Brownian motion, stochastic fractional neutral equations and many others. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=delay%20equations" title="delay equations">delay equations</a>, <a href="https://publications.waset.org/abstracts/search?q=operator%20methods" title=" operator methods"> operator methods</a>, <a href="https://publications.waset.org/abstracts/search?q=stochastic%20noise" title=" stochastic noise"> stochastic noise</a>, <a href="https://publications.waset.org/abstracts/search?q=weak%20solutions" title=" weak solutions"> weak solutions</a> </p> <a href="https://publications.waset.org/abstracts/146592/weak-solutions-of-stochastic-fractional-differential-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/146592.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">209</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">5996</span> Numerical Wave Solutions for Nonlinear Coupled Equations Using Sinc-Collocation Method</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Kamel%20Al-Khaled">Kamel Al-Khaled</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper, numerical solutions for the nonlinear coupled Korteweg-de Vries, (abbreviated as KdV) equations are calculated by Sinc-collocation method. This approach is based on a global collocation method using Sinc basis functions. First, discretizing time derivative of the KdV equations by a classic finite difference formula, while the space derivatives are approximated by a $\theta-$weighted scheme. Sinc functions are used to solve these two equations. Soliton solutions are constructed to show the nature of the solution. The numerical results are shown to demonstrate the efficiency of the newly proposed method. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=Nonlinear%20coupled%20KdV%20equations" title="Nonlinear coupled KdV equations">Nonlinear coupled KdV equations</a>, <a href="https://publications.waset.org/abstracts/search?q=Soliton%20solutions" title=" Soliton solutions"> Soliton solutions</a>, <a href="https://publications.waset.org/abstracts/search?q=Sinc-collocation%20method" title=" Sinc-collocation method"> Sinc-collocation method</a>, <a href="https://publications.waset.org/abstracts/search?q=Sinc%20functions" title=" Sinc functions"> Sinc functions</a> </p> <a href="https://publications.waset.org/abstracts/23564/numerical-wave-solutions-for-nonlinear-coupled-equations-using-sinc-collocation-method" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/23564.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">524</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">5995</span> Numerical Treatment of Block Method for the Solution of Ordinary Differential Equations</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=A.%20M.%20Sagir">A. M. Sagir</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Discrete linear multistep block method of uniform order for the solution of first order Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs) is presented in this paper. The approach of interpolation and collocation approximation are adopted in the derivation of the method which is then applied to first order ordinary differential equations with associated initial conditions. The continuous hybrid formulations enable us to differentiate and evaluate at some grids and off – grid points to obtain four discrete schemes, which were used in block form for parallel or sequential solutions of the problems. Furthermore, a stability analysis and efficiency of the block method are tested on ordinary differential equations, and the results obtained compared favorably with the exact solution. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=block%20method" title="block method">block method</a>, <a href="https://publications.waset.org/abstracts/search?q=first%20order%20ordinary%20differential%20equations" title=" first order ordinary differential equations"> first order ordinary differential equations</a>, <a href="https://publications.waset.org/abstracts/search?q=hybrid" title=" hybrid"> hybrid</a>, <a href="https://publications.waset.org/abstracts/search?q=self-starting" title=" self-starting "> self-starting </a> </p> <a href="https://publications.waset.org/abstracts/3426/numerical-treatment-of-block-method-for-the-solution-of-ordinary-differential-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/3426.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">482</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">5994</span> Quantum Mechanics as A Limiting Case of Relativistic Mechanics</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Ahmad%20Almajid">Ahmad Almajid</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The idea of unifying quantum mechanics with general relativity is still a dream for many researchers, as physics has only two paths, no more. Einstein's path, which is mainly based on particle mechanics, and the path of Paul Dirac and others, which is based on wave mechanics, the incompatibility of the two approaches is due to the radical difference in the initial assumptions and the mathematical nature of each approach. Logical thinking in modern physics leads us to two problems: - In quantum mechanics, despite its success, the problem of measurement and the problem of wave function interpretation is still obscure. - In special relativity, despite the success of the equivalence of rest-mass and energy, but at the speed of light, the fact that the energy becomes infinite is contrary to logic because the speed of light is not infinite, and the mass of the particle is not infinite too. These contradictions arise from the overlap of relativistic and quantum mechanics in the neighborhood of the speed of light, and in order to solve these problems, one must understand well how to move from relativistic mechanics to quantum mechanics, or rather, to unify them in a way different from Dirac's method, in order to go along with God or Nature, since, as Einstein said, "God doesn't play dice." From De Broglie's hypothesis about wave-particle duality, Léon Brillouin's definition of the new proper time was deduced, and thus the quantum Lorentz factor was obtained. Finally, using the Euler-Lagrange equation, we come up with new equations in quantum mechanics. In this paper, the two problems in modern physics mentioned above are solved; it can be said that this new approach to quantum mechanics will enable us to unify it with general relativity quite simply. If the experiments prove the validity of the results of this research, we will be able in the future to transport the matter at speed close to the speed of light. Finally, this research yielded three important results: 1- Lorentz quantum factor. 2- Planck energy is a limited case of Einstein energy. 3- Real quantum mechanics, in which new equations for quantum mechanics match and exceed Dirac's equations, these equations have been reached in a completely different way from Dirac's method. These equations show that quantum mechanics is a limited case of relativistic mechanics. At the Solvay Conference in 1927, the debate about quantum mechanics between Bohr, Einstein, and others reached its climax, while Bohr suggested that if particles are not observed, they are in a probabilistic state, then Einstein said his famous claim ("God does not play dice"). Thus, Einstein was right, especially when he didn't accept the principle of indeterminacy in quantum theory, although experiments support quantum mechanics. However, the results of our research indicate that God really does not play dice; when the electron disappears, it turns into amicable particles or an elastic medium, according to the above obvious equations. Likewise, Bohr was right also, when he indicated that there must be a science like quantum mechanics to monitor and study the motion of subatomic particles, but the picture in front of him was blurry and not clear, so he resorted to the probabilistic interpretation. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=lorentz%20quantum%20factor" title="lorentz quantum factor">lorentz quantum factor</a>, <a href="https://publications.waset.org/abstracts/search?q=new" title=" new"> new</a>, <a href="https://publications.waset.org/abstracts/search?q=planck%E2%80%99s%20energy%20as%20a%20limiting%20case%20of%20einstein%E2%80%99s%20energy" title=" planck’s energy as a limiting case of einstein’s energy"> planck’s energy as a limiting case of einstein’s energy</a>, <a href="https://publications.waset.org/abstracts/search?q=real%20quantum%20mechanics" title=" real quantum mechanics"> real quantum mechanics</a>, <a href="https://publications.waset.org/abstracts/search?q=new%20equations%20for%20quantum%20mechanics" title=" new equations for quantum mechanics"> new equations for quantum mechanics</a> </p> <a href="https://publications.waset.org/abstracts/159579/quantum-mechanics-as-a-limiting-case-of-relativistic-mechanics" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/159579.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">77</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">5993</span> Solution of Hybrid Fuzzy Differential Equations</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Mahmood%20Otadi">Mahmood Otadi</a>, <a href="https://publications.waset.org/abstracts/search?q=Maryam%20Mosleh"> Maryam Mosleh</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The hybrid differential equations have a wide range of applications in science and engineering. In this paper, the homotopy analysis method (HAM) is applied to obtain the series solution of the hybrid differential equations. Using the homotopy analysis method, it is possible to find the exact solution or an approximate solution of the problem. Comparisons are made between improved predictor-corrector method, homotopy analysis method and the exact solution. Finally, we illustrate our approach by some numerical example. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=fuzzy%20number" title="fuzzy number">fuzzy number</a>, <a href="https://publications.waset.org/abstracts/search?q=fuzzy%20ODE" title=" fuzzy ODE"> fuzzy ODE</a>, <a href="https://publications.waset.org/abstracts/search?q=HAM" title=" HAM"> HAM</a>, <a href="https://publications.waset.org/abstracts/search?q=approximate%20method" title=" approximate method"> approximate method</a> </p> <a href="https://publications.waset.org/abstracts/31754/solution-of-hybrid-fuzzy-differential-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/31754.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">511</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">5992</span> The Construction of Exact Solutions for the Nonlinear Lattice Equation via Coth and Csch Functions Method</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=A.%20Zerarka">A. Zerarka</a>, <a href="https://publications.waset.org/abstracts/search?q=W.%20Djoudi"> W. Djoudi</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The method developed in this work uses a generalised coth and csch funtions method to construct new exact travelling solutions to the nonlinear lattice equation. The technique of the homogeneous balance method is used to handle the appropriated solutions. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=coth%20functions" title="coth functions">coth functions</a>, <a href="https://publications.waset.org/abstracts/search?q=csch%20functions" title=" csch functions"> csch functions</a>, <a href="https://publications.waset.org/abstracts/search?q=nonlinear%20partial%20differential%20equation" title=" nonlinear partial differential equation"> nonlinear partial differential equation</a>, <a href="https://publications.waset.org/abstracts/search?q=travelling%20wave%20solutions" title=" travelling wave solutions"> travelling wave solutions</a> </p> <a href="https://publications.waset.org/abstracts/20374/the-construction-of-exact-solutions-for-the-nonlinear-lattice-equation-via-coth-and-csch-functions-method" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/20374.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">663</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">5991</span> Student Project on Using a Spreadsheet for Solving Differential Equations by Euler's Method</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Andriy%20Didenko">Andriy Didenko</a>, <a href="https://publications.waset.org/abstracts/search?q=Zanin%20Kavazovic"> Zanin Kavazovic</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Engineering students often have certain difficulties in mastering major theoretical concepts in mathematical courses such as differential equations. Student projects were proposed to motivate students’ learning and can be used as a tool to promote students’ interest in the material. Authors propose a student project that includes the use of Microsoft Excel. This instructional tool is often overlooked by both educators and students. An integral component of the experimental part of such a project is the exploration of an interactive spreadsheet. The aim is to assist engineering students in better understanding of Euler’s method. This method is employed to numerically solve first order differential equations. At first, students are invited to select classic equations from a list presented in a form of a drop-down menu. For each of these equations, students can select and modify certain key parameters and observe the influence of initial condition on the solution. This will give students an insight into the behavior of the method in different configurations as solutions to equations are given in numerical and graphical forms. Further, students could also create their own equations by providing functions of their own choice and a variety of initial conditions. Moreover, they can visualize and explore the impact of the length of the time step on the convergence of a sequence of numerical solutions to the exact solution of the equation. As a final stage of the project, students are encouraged to develop their own spreadsheets for other numerical methods and other types of equations. Such projects promote students’ interest in mathematical applications and further improve their mathematical and programming skills. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=student%20project" title="student project">student project</a>, <a href="https://publications.waset.org/abstracts/search?q=Euler%27s%20method" title=" Euler's method"> Euler's method</a>, <a href="https://publications.waset.org/abstracts/search?q=spreadsheet" title=" spreadsheet"> spreadsheet</a>, <a href="https://publications.waset.org/abstracts/search?q=engineering%20education" title=" engineering education"> engineering education</a> </p> <a href="https://publications.waset.org/abstracts/112422/student-project-on-using-a-spreadsheet-for-solving-differential-equations-by-eulers-method" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/112422.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">135</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">5990</span> Modified Newton's Iterative Method for Solving System of Nonlinear Equations in Two Variables</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Sara%20Mahesar">Sara Mahesar</a>, <a href="https://publications.waset.org/abstracts/search?q=Saleem%20M.%20Chandio"> Saleem M. Chandio</a>, <a href="https://publications.waset.org/abstracts/search?q=Hira%20Soomro"> Hira Soomro</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Nonlinear system of equations in two variables is a system which contains variables of degree greater or equal to two or that comprises of the transcendental functions. Mathematical modeling of numerous physical problems occurs as a system of nonlinear equations. In applied and pure mathematics it is the main dispute to solve a system of nonlinear equations. Numerical techniques mainly used for finding the solution to problems where analytical methods are failed, which leads to the inexact solutions. To find the exact roots or solutions in case of the system of non-linear equations there does not exist any analytical technique. Various methods have been proposed to solve such systems with an improved rate of convergence and accuracy. In this paper, a new scheme is developed for solving system of non-linear equation in two variables. The iterative scheme proposed here is modified form of the conventional Newton’s Method (CN) whose order of convergence is two whereas the order of convergence of the devised technique is three. Furthermore, the detailed error and convergence analysis of the proposed method is also examined. Additionally, various numerical test problems are compared with the results of its counterpart conventional Newton’s Method (CN) which confirms the theoretic consequences of the proposed method. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=conventional%20Newton%E2%80%99s%20method" title="conventional Newton’s method">conventional Newton’s method</a>, <a href="https://publications.waset.org/abstracts/search?q=modified%20Newton%E2%80%99s%20method" title=" modified Newton’s method"> modified Newton’s method</a>, <a href="https://publications.waset.org/abstracts/search?q=order%20of%20convergence" title=" order of convergence"> order of convergence</a>, <a href="https://publications.waset.org/abstracts/search?q=system%20of%20nonlinear%20equations" title=" system of nonlinear equations"> system of nonlinear equations</a> </p> <a href="https://publications.waset.org/abstracts/87602/modified-newtons-iterative-method-for-solving-system-of-nonlinear-equations-in-two-variables" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/87602.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">257</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">5989</span> A Spectral Decomposition Method for Ordinary Differential Equation Systems with Constant or Linear Right Hand Sides</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=R.%20B.%20Ogunrinde">R. B. Ogunrinde</a>, <a href="https://publications.waset.org/abstracts/search?q=C.%20C.%20Jibunoh"> C. C. Jibunoh</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper, a spectral decomposition method is developed for the direct integration of stiff and nonstiff homogeneous linear (ODE) systems with linear, constant, or zero right hand sides (RHSs). The method does not require iteration but obtains solutions at any random points of t, by direct evaluation, in the interval of integration. All the numerical solutions obtained for the class of systems coincide with the exact theoretical solutions. In particular, solutions of homogeneous linear systems, i.e. with zero RHS, conform to the exact analytical solutions of the systems in terms of t. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=spectral%20decomposition" title="spectral decomposition">spectral decomposition</a>, <a href="https://publications.waset.org/abstracts/search?q=linear%20RHS" title=" linear RHS"> linear RHS</a>, <a href="https://publications.waset.org/abstracts/search?q=homogeneous%20linear%20systems" title=" homogeneous linear systems"> homogeneous linear systems</a>, <a href="https://publications.waset.org/abstracts/search?q=eigenvalues%20of%20the%20Jacobian" title=" eigenvalues of the Jacobian"> eigenvalues of the Jacobian</a> </p> <a href="https://publications.waset.org/abstracts/54215/a-spectral-decomposition-method-for-ordinary-differential-equation-systems-with-constant-or-linear-right-hand-sides" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/54215.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">330</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">5988</span> Global Stability Of Nonlinear Itô Equations And N. V. Azbelev's W-method</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Arcady%20Ponosov.">Arcady Ponosov.</a>, <a href="https://publications.waset.org/abstracts/search?q=Ramazan%20Kadiev"> Ramazan Kadiev</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The work studies the global moment stability of solutions of systems of nonlinear differential Itô equations with delays. A modified regularization method (W-method) for the analysis of various types of stability of such systems, based on the choice of the auxiliaryequations and applications of the theory of positive invertible matrices, is proposed and justified. Development of this method for deterministic functional differential equations is due to N.V. Azbelev and his students. Sufficient conditions for the moment stability of solutions in terms of the coefficients for sufficiently general as well as specific classes of Itô equations are given. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=asymptotic%20stability" title="asymptotic stability">asymptotic stability</a>, <a href="https://publications.waset.org/abstracts/search?q=delay%20equations" title=" delay equations"> delay equations</a>, <a href="https://publications.waset.org/abstracts/search?q=operator%20methods" title=" operator methods"> operator methods</a>, <a href="https://publications.waset.org/abstracts/search?q=stochastic%20noise" title=" stochastic noise"> stochastic noise</a> </p> <a href="https://publications.waset.org/abstracts/143260/global-stability-of-nonlinear-ito-equations-and-n-v-azbelevs-w-method" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/143260.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">224</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">5987</span> On a Continuous Formulation of Block Method for Solving First Order Ordinary Differential Equations (ODEs)</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=A.%20M.%20Sagir">A. M. Sagir</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The aim of this paper is to investigate the performance of the developed linear multistep block method for solving first order initial value problem of Ordinary Differential Equations (ODEs). The method calculates the numerical solution at three points simultaneously and produces three new equally spaced solution values within a block. The continuous formulations enable us to differentiate and evaluate at some selected points to obtain three discrete schemes, which were used in block form for parallel or sequential solutions of the problems. A stability analysis and efficiency of the block method are tested on ordinary differential equations involving practical applications, and the results obtained compared favorably with the exact solution. Furthermore, comparison of error analysis has been developed with the help of computer software. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=block%20method" title="block method">block method</a>, <a href="https://publications.waset.org/abstracts/search?q=first%20order%20ordinary%20differential%20equations" title=" first order ordinary differential equations"> first order ordinary differential equations</a>, <a href="https://publications.waset.org/abstracts/search?q=linear%20multistep" title=" linear multistep"> linear multistep</a>, <a href="https://publications.waset.org/abstracts/search?q=self-starting" title=" self-starting"> self-starting</a> </p> <a href="https://publications.waset.org/abstracts/3622/on-a-continuous-formulation-of-block-method-for-solving-first-order-ordinary-differential-equations-odes" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/3622.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">306</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">5986</span> Development of Extended Trapezoidal Method for Numerical Solution of Volterra Integro-Differential Equations</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Fuziyah%20Ishak">Fuziyah Ishak</a>, <a href="https://publications.waset.org/abstracts/search?q=Siti%20Norazura%20Ahmad"> Siti Norazura Ahmad</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Volterra integro-differential equations appear in many models for real life phenomena. Since analytical solutions for this type of differential equations are hard and at times impossible to attain, engineers and scientists resort to numerical solutions that can be made as accurately as possible. Conventionally, numerical methods for ordinary differential equations are adapted to solve Volterra integro-differential equations. In this paper, numerical solution for solving Volterra integro-differential equation using extended trapezoidal method is described. Formulae for the integral and differential parts of the equation are presented. Numerical results show that the extended method is suitable for solving first order Volterra integro-differential equations. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=accuracy" title="accuracy">accuracy</a>, <a href="https://publications.waset.org/abstracts/search?q=extended%20trapezoidal%20method" title=" extended trapezoidal method"> extended trapezoidal method</a>, <a href="https://publications.waset.org/abstracts/search?q=numerical%20solution" title=" numerical solution"> numerical solution</a>, <a href="https://publications.waset.org/abstracts/search?q=Volterra%20integro-differential%20equations" title=" Volterra integro-differential equations"> Volterra integro-differential equations</a> </p> <a href="https://publications.waset.org/abstracts/52856/development-of-extended-trapezoidal-method-for-numerical-solution-of-volterra-integro-differential-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/52856.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">426</span> </span> </div> </div> <ul class="pagination"> <li class="page-item disabled"><span class="page-link">&lsaquo;</span></li> <li class="page-item active"><span class="page-link">1</span></li> <li class="page-item"><a class="page-link" href="https://publications.waset.org/abstracts/search?q=exact%20solutions%20of%20the%20Einstein%20equations&page=2">2</a></li> <li class="page-item"><a class="page-link" href="https://publications.waset.org/abstracts/search?q=exact%20solutions%20of%20the%20Einstein%20equations&page=3">3</a></li> <li class="page-item"><a class="page-link" href="https://publications.waset.org/abstracts/search?q=exact%20solutions%20of%20the%20Einstein%20equations&page=4">4</a></li> <li class="page-item"><a class="page-link" href="https://publications.waset.org/abstracts/search?q=exact%20solutions%20of%20the%20Einstein%20equations&page=5">5</a></li> <li class="page-item"><a class="page-link" href="https://publications.waset.org/abstracts/search?q=exact%20solutions%20of%20the%20Einstein%20equations&page=6">6</a></li> <li class="page-item"><a class="page-link" href="https://publications.waset.org/abstracts/search?q=exact%20solutions%20of%20the%20Einstein%20equations&page=7">7</a></li> <li class="page-item"><a class="page-link" href="https://publications.waset.org/abstracts/search?q=exact%20solutions%20of%20the%20Einstein%20equations&page=8">8</a></li> <li class="page-item"><a class="page-link" href="https://publications.waset.org/abstracts/search?q=exact%20solutions%20of%20the%20Einstein%20equations&page=9">9</a></li> <li class="page-item"><a class="page-link" href="https://publications.waset.org/abstracts/search?q=exact%20solutions%20of%20the%20Einstein%20equations&page=10">10</a></li> <li class="page-item disabled"><span class="page-link">...</span></li> <li class="page-item"><a class="page-link" href="https://publications.waset.org/abstracts/search?q=exact%20solutions%20of%20the%20Einstein%20equations&page=200">200</a></li> <li class="page-item"><a class="page-link" href="https://publications.waset.org/abstracts/search?q=exact%20solutions%20of%20the%20Einstein%20equations&page=201">201</a></li> <li class="page-item"><a class="page-link" href="https://publications.waset.org/abstracts/search?q=exact%20solutions%20of%20the%20Einstein%20equations&page=2" rel="next">&rsaquo;</a></li> </ul> </div> </main> <footer> <div id="infolinks" class="pt-3 pb-2"> <div class="container"> <div style="background-color:#f5f5f5;" class="p-3"> <div class="row"> <div class="col-md-2"> <ul class="list-unstyled"> About <li><a href="https://waset.org/page/support">About Us</a></li> <li><a href="https://waset.org/page/support#legal-information">Legal</a></li> <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/WASET-16th-foundational-anniversary.pdf">WASET celebrates its 16th foundational anniversary</a></li> </ul> </div> <div class="col-md-2"> <ul class="list-unstyled"> Account <li><a href="https://waset.org/profile">My Account</a></li> </ul> </div> <div class="col-md-2"> <ul class="list-unstyled"> Explore <li><a href="https://waset.org/disciplines">Disciplines</a></li> <li><a href="https://waset.org/conferences">Conferences</a></li> <li><a href="https://waset.org/conference-programs">Conference Program</a></li> <li><a href="https://waset.org/committees">Committees</a></li> <li><a href="https://publications.waset.org">Publications</a></li> </ul> </div> <div class="col-md-2"> <ul class="list-unstyled"> Research <li><a href="https://publications.waset.org/abstracts">Abstracts</a></li> <li><a href="https://publications.waset.org">Periodicals</a></li> <li><a href="https://publications.waset.org/archive">Archive</a></li> </ul> </div> <div class="col-md-2"> <ul class="list-unstyled"> Open Science <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/Open-Science-Philosophy.pdf">Open Science Philosophy</a></li> <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/Open-Science-Award.pdf">Open Science Award</a></li> <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/Open-Society-Open-Science-and-Open-Innovation.pdf">Open Innovation</a></li> <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/Postdoctoral-Fellowship-Award.pdf">Postdoctoral Fellowship Award</a></li> <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/Scholarly-Research-Review.pdf">Scholarly Research Review</a></li> </ul> </div> <div class="col-md-2"> <ul class="list-unstyled"> Support <li><a href="https://waset.org/page/support">Support</a></li> <li><a href="https://waset.org/profile/messages/create">Contact Us</a></li> <li><a href="https://waset.org/profile/messages/create">Report Abuse</a></li> </ul> </div> </div> </div> </div> </div> <div class="container text-center"> <hr style="margin-top:0;margin-bottom:.3rem;"> <a href="https://creativecommons.org/licenses/by/4.0/" target="_blank" class="text-muted small">Creative Commons Attribution 4.0 International License</a> <div id="copy" class="mt-2">© 2024 World Academy of Science, Engineering and Technology</div> </div> </footer> <a href="javascript:" id="return-to-top"><i class="fas fa-arrow-up"></i></a> <div class="modal" id="modal-template"> <div class="modal-dialog"> <div class="modal-content"> <div class="row m-0 mt-1"> <div class="col-md-12"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> </div> </div> <div class="modal-body"></div> </div> </div> </div> <script src="https://cdn.waset.org/static/plugins/jquery-3.3.1.min.js"></script> <script src="https://cdn.waset.org/static/plugins/bootstrap-4.2.1/js/bootstrap.bundle.min.js"></script> <script src="https://cdn.waset.org/static/js/site.js?v=150220211556"></script> <script> jQuery(document).ready(function() { /*jQuery.get("https://publications.waset.org/xhr/user-menu", function (response) { jQuery('#mainNavMenu').append(response); });*/ jQuery.get({ url: "https://publications.waset.org/xhr/user-menu", cache: false }).then(function(response){ jQuery('#mainNavMenu').append(response); }); }); </script> </body> </html>