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equations</title> <meta name="description" content="Search results for: integral equations"> <meta name="keywords" content="integral 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 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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="integral 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> 2573</div> </div> </div> </div> <h1 class="mt-3 mb-3 text-center" style="font-size:1.6rem;">Search results for: integral equations</h1> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">2573</span> Integral Image-Based Differential Filters</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Kohei%20Inoue">Kohei Inoue</a>, <a href="https://publications.waset.org/abstracts/search?q=Kenji%20Hara"> Kenji Hara</a>, <a href="https://publications.waset.org/abstracts/search?q=Kiichi%20Urahama"> Kiichi Urahama</a> </p> <p class="card-text"><strong>Abstract:</strong></p> We describe a relationship between integral images and differential images. First, we derive a simple difference filter from conventional integral image. In the derivation, we show that an integral image and the corresponding differential image are related to each other by simultaneous linear equations, where the numbers of unknowns and equations are the same, and therefore, we can execute the integration and differentiation by solving the simultaneous equations. We applied the relationship to an image fusion problem, and experimentally verified the effectiveness of the proposed method. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=integral%20images" title="integral images">integral images</a>, <a href="https://publications.waset.org/abstracts/search?q=differential%20images" title=" differential images"> differential images</a>, <a href="https://publications.waset.org/abstracts/search?q=differential%20filters" title=" differential filters"> differential filters</a>, <a href="https://publications.waset.org/abstracts/search?q=image%20fusion" title=" image fusion"> image fusion</a> </p> <a href="https://publications.waset.org/abstracts/8531/integral-image-based-differential-filters" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/8531.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">506</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">2572</span> The Finite Element Method for Nonlinear Fredholm Integral Equation of the Second Kind</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Melusi%20Khumalo">Melusi Khumalo</a>, <a href="https://publications.waset.org/abstracts/search?q=Anastacia%20Dlamini"> Anastacia Dlamini</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper, we consider a numerical solution for nonlinear Fredholm integral equations of the second kind. We work with uniform mesh and use the Lagrange polynomials together with the Galerkin finite element method, where the weight function is chosen in such a way that it takes the form of the approximate solution but with arbitrary coefficients. We implement the finite element method to the nonlinear Fredholm integral equations of the second kind. We consider the error analysis of the method. Furthermore, we look at a specific example to illustrate the implementation of the finite element method. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=finite%20element%20method" title="finite element method">finite element method</a>, <a href="https://publications.waset.org/abstracts/search?q=Galerkin%20approach" title=" Galerkin approach"> Galerkin approach</a>, <a href="https://publications.waset.org/abstracts/search?q=Fredholm%20integral%20equations" title=" Fredholm integral equations"> Fredholm integral equations</a>, <a href="https://publications.waset.org/abstracts/search?q=nonlinear%20integral%20equations" title=" nonlinear integral equations"> nonlinear integral equations</a> </p> <a href="https://publications.waset.org/abstracts/140832/the-finite-element-method-for-nonlinear-fredholm-integral-equation-of-the-second-kind" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/140832.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">376</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">2571</span> Numerical Solution of Integral Equations by Using Discrete GHM Multiwavelet</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Archit%20Yajnik">Archit Yajnik</a>, <a href="https://publications.waset.org/abstracts/search?q=Rustam%20Ali"> Rustam Ali</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper, numerical method based on discrete GHM multiwavelets is presented for solving the Fredholm integral equations of second kind. There is hardly any article available in the literature in which the integral equations are numerically solved using discrete GHM multiwavelet. A number of examples are demonstrated to justify the applicability of the method. In GHM multiwavelets, the values of scaling and wavelet functions are calculated only at t = 0, 0.5 and 1. The numerical solution obtained by the present approach is compared with the traditional Quadrature method. It is observed that the present approach is more accurate and computationally efficient as compared to quadrature method. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=GHM%20multiwavelet" title="GHM multiwavelet">GHM multiwavelet</a>, <a href="https://publications.waset.org/abstracts/search?q=fredholm%20integral%20equations" title=" fredholm integral equations"> fredholm integral equations</a>, <a href="https://publications.waset.org/abstracts/search?q=quadrature%20method" title=" quadrature method"> quadrature method</a>, <a href="https://publications.waset.org/abstracts/search?q=function%20approximation" title=" function approximation"> function approximation</a> </p> <a href="https://publications.waset.org/abstracts/36311/numerical-solution-of-integral-equations-by-using-discrete-ghm-multiwavelet" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/36311.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">462</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">2570</span> Numerical Solutions of Fredholm Integral Equations by B-Spline Wavelet Method</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Ritu%20Rani">Ritu Rani</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper, we apply minimalistically upheld linear semi-orthogonal B-spline wavelets, exceptionally developed for the limited interim to rough the obscure function present in the integral equations. Semi-orthogonal wavelets utilizing B-spline uniquely developed for the limited interim and these wavelets can be spoken to in a shut frame. This gives a minimized help. Semi-orthogonal wavelets frame the premise in the space L²(R). Utilizing this premise, an arbitrary function in L²(R) can be communicated as the wavelet arrangement. For the limited interim, the wavelet arrangement cannot be totally introduced by utilizing this premise. This is on the grounds that backings of some premise are truncated at the left or right end purposes of the interim. Subsequently, an uncommon premise must be brought into the wavelet development on the limited interim. These functions are alluded to as the limit scaling functions and limit wavelet functions. B-spline wavelet method has been connected to fathom linear and nonlinear integral equations and their systems. The above method diminishes the integral equations to systems of algebraic equations and afterward these systems can be illuminated by any standard numerical methods. Here, we have connected Newton's method with suitable starting speculation for solving these systems. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=semi-orthogonal" title="semi-orthogonal">semi-orthogonal</a>, <a href="https://publications.waset.org/abstracts/search?q=wavelet%20arrangement" title=" wavelet arrangement"> wavelet arrangement</a>, <a href="https://publications.waset.org/abstracts/search?q=integral%20equations" title=" integral equations"> integral equations</a>, <a href="https://publications.waset.org/abstracts/search?q=wavelet%20development" title=" wavelet development "> wavelet development </a> </p> <a href="https://publications.waset.org/abstracts/125473/numerical-solutions-of-fredholm-integral-equations-by-b-spline-wavelet-method" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/125473.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">174</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">2569</span> Existence of positive periodic solutions for certain delay differential equations </h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Farid%20Nouioua">Farid Nouioua</a>, <a href="https://publications.waset.org/abstracts/search?q=Abdelouaheb%20Ardjouni"> Abdelouaheb Ardjouni</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this article, we study the existence of positive periodic solutions of certain delay differential equations. In the process we convert the differential equation into an equivalent integral equation after which appropriate mappings are constructed. We then employ Krasnoselskii's fixed point theorem to obtain sufficient conditions for the existence of a positive periodic solution of the differential equation. The obtained results improve and extend the results in the literature. Finally, an example is given to illustrate our results. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=delay%20differential%20equations" title="delay differential equations">delay differential equations</a>, <a href="https://publications.waset.org/abstracts/search?q=positive%20periodic%20solutions" title=" positive periodic solutions"> positive periodic solutions</a>, <a href="https://publications.waset.org/abstracts/search?q=integral%20equations" title=" integral equations"> integral equations</a>, <a href="https://publications.waset.org/abstracts/search?q=Krasnoselskii%20fixed%20point%20theorem" title=" Krasnoselskii fixed point theorem"> Krasnoselskii fixed point theorem</a> </p> <a href="https://publications.waset.org/abstracts/40904/existence-of-positive-periodic-solutions-for-certain-delay-differential-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/40904.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">438</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">2568</span> Superconvergence of the Iterated Discrete Legendre Galerkin Method for Fredholm-Hammerstein Equations</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Payel%20Das">Payel Das</a>, <a href="https://publications.waset.org/abstracts/search?q=Gnaneshwar%20Nelakanti"> Gnaneshwar Nelakanti</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper we analyse the iterated discrete Legendre Galerkin method for Fredholm-Hammerstein integral equations with smooth kernel. Using sufficiently accurate numerical quadrature rule, we obtain superconvergence rates for the iterated discrete Legendre Galerkin solutions in both infinity and $L^2$-norm. Numerical examples are given to illustrate the theoretical results. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=hammerstein%20integral%20equations" title="hammerstein integral equations">hammerstein integral equations</a>, <a href="https://publications.waset.org/abstracts/search?q=spectral%20method" title=" spectral method"> spectral method</a>, <a href="https://publications.waset.org/abstracts/search?q=discrete%20galerkin" title=" discrete galerkin"> discrete galerkin</a>, <a href="https://publications.waset.org/abstracts/search?q=numerical%0D%0Aquadrature" title=" numerical quadrature"> numerical quadrature</a>, <a href="https://publications.waset.org/abstracts/search?q=superconvergence" title=" superconvergence"> superconvergence</a> </p> <a href="https://publications.waset.org/abstracts/22260/superconvergence-of-the-iterated-discrete-legendre-galerkin-method-for-fredholm-hammerstein-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/22260.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">471</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">2567</span> Symbolic Partial Differential Equations Analysis Using Mathematica</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Davit%20Shahnazaryan">Davit Shahnazaryan</a>, <a href="https://publications.waset.org/abstracts/search?q=Diogo%20Gomes"> Diogo Gomes</a>, <a href="https://publications.waset.org/abstracts/search?q=Mher%20%20Safaryan"> Mher Safaryan</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Many symbolic computations and manipulations required in the analysis of partial differential equations (PDE) or systems of PDEs are tedious and error-prone. These computations arise when determining conservation laws, entropies or integral identities, which are essential tools for the study of PDEs. Here, we discuss a new Mathematica package for the symbolic analysis of PDEs that automate multiple tasks, saving time and effort. Methodologies: During the research, we have used concepts of linear algebra and partial differential equations. We have been working on creating algorithms based on theoretical mathematics to find results mentioned below. Major Findings: Our package provides the following functionalities; finding symmetry group of different PDE systems, generation of polynomials invariant with respect to different symmetry groups; simplification of integral quantities by integration by parts and null Lagrangian cleaning, computing general forms of expressions by integration by parts; finding equivalent forms of an integral expression that are simpler or more symmetric form; determining necessary and sufficient conditions on the coefficients for the positivity of a given symbolic expression. Conclusion: Using this package, we can simplify integral identities, find conserved and dissipated quantities of time-dependent PDE or system of PDEs. Some examples in the theory of mean-field games and semiconductor equations are discussed. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=partial%20differential%20equations" title="partial differential equations">partial differential equations</a>, <a href="https://publications.waset.org/abstracts/search?q=symbolic%20computation" title=" symbolic computation"> symbolic computation</a>, <a href="https://publications.waset.org/abstracts/search?q=conserved%20and%20dissipated%20quantities" title=" conserved and dissipated quantities"> conserved and dissipated quantities</a>, <a href="https://publications.waset.org/abstracts/search?q=mathematica" title=" mathematica"> mathematica</a> </p> <a href="https://publications.waset.org/abstracts/95037/symbolic-partial-differential-equations-analysis-using-mathematica" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/95037.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">163</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">2566</span> Multidimensional Integral and Discrete Opial–Type Inequalities</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Maja%20Andri%C4%87">Maja Andrić</a>, <a href="https://publications.waset.org/abstracts/search?q=Josip%20Pe%C4%8Dari%C4%87"> Josip Pečarić</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Over the last five decades, an enormous amount of work has been done on Opial’s integral inequality, dealing with new proofs, various generalizations, extensions and discrete analogs. The Opial inequality is recognized as a fundamental result in the analysis of qualitative properties of solution of differential equations. We use submultiplicative convex functions, appropriate representations of functions and inequalities involving means to obtain generalizations and extensions of certain known multidimensional integral and discrete Opial-type inequalities. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=Opial%27s%20inequality" title="Opial's inequality">Opial's inequality</a>, <a href="https://publications.waset.org/abstracts/search?q=Jensen%27s%20inequality" title=" Jensen's inequality"> Jensen's inequality</a>, <a href="https://publications.waset.org/abstracts/search?q=integral%20inequality" title=" integral inequality"> integral inequality</a>, <a href="https://publications.waset.org/abstracts/search?q=discrete%20inequality" title=" discrete inequality"> discrete inequality</a> </p> <a href="https://publications.waset.org/abstracts/41583/multidimensional-integral-and-discrete-opial-type-inequalities" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/41583.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">439</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">2565</span> Globally Attractive Mild Solutions for Non-Local in Time Subdiffusion Equations of Neutral Type</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Jorge%20Gonzalez%20Camus">Jorge Gonzalez Camus</a>, <a href="https://publications.waset.org/abstracts/search?q=Carlos%20Lizama"> Carlos Lizama</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this work is proved the existence of at least one globally attractive mild solution to the Cauchy problem, for fractional evolution equation of neutral type, involving the fractional derivate in Caputo sense. An almost sectorial operator on a Banach space X and a kernel belonging to a large class appears in the equation, which covers many relevant cases from physics applications, in particular, the important case of time - fractional evolution equations of neutral type. The main tool used in this work was the Hausdorff measure of noncompactness and fixed point theorems, specifically Darbo-type. Initially, the equation is a Cauchy problem, involving a fractional derivate in Caputo sense. Then, is formulated the equivalent integral version, and defining a convenient functional, using the analytic integral resolvent operator, and verifying the hypothesis of the fixed point theorem of Darbo type, give us the existence of mild solution for the initial problem. Furthermore, each mild solution is globally attractive, a property that is desired in asymptotic behavior for that solution. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=attractive%20mild%20solutions" title="attractive mild solutions">attractive mild solutions</a>, <a href="https://publications.waset.org/abstracts/search?q=integral%20Volterra%20equations" title=" integral Volterra equations"> integral Volterra equations</a>, <a href="https://publications.waset.org/abstracts/search?q=neutral%20type%20equations" title=" neutral type equations"> neutral type equations</a>, <a href="https://publications.waset.org/abstracts/search?q=non-local%20in%20time%20equations" title=" non-local in time equations"> non-local in time equations</a> </p> <a href="https://publications.waset.org/abstracts/99925/globally-attractive-mild-solutions-for-non-local-in-time-subdiffusion-equations-of-neutral-type" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/99925.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">160</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">2564</span> Closed Form Solution for 4-D Potential Integrals for Arbitrary Coplanar Polygonal Surfaces</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Damir%20Latypov">Damir Latypov</a> </p> <p class="card-text"><strong>Abstract:</strong></p> A closed-form solution for 4-D double surface integrals arising in boundary integrals equations of a potential theory is obtained for arbitrary coplanar polygonal surfaces. The solution method is based on the construction of exact differential forms followed by the application of Stokes' theorem for each surface integral. As a result, the 4-D double surface integral is reduced to a 2-D double line integral. By an appropriate change of variables, the integrand is transformed into a separable function of integration variables. The closed-form solutions to the corresponding 1-D integrals are readily available in the integration tables. Previously closed-form solutions were known only for the case of coincident triangle surfaces and coplanar rectangles. Solutions for these cases were obtained by surface-specific ad-hoc methods, while the present method is general. The method also works for non-polygonal surfaces. As an example, we compute in closed form the 4-D integral for the case of coincident surfaces in the shape of a circular disk. For an arbitrarily shaped surface, the proposed method provides an efficient quadrature rule. Extensions of the method for non-coplanar surfaces and other than 1/R integral kernels are also discussed. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=boundary%20integral%20equations" title="boundary integral equations">boundary integral equations</a>, <a href="https://publications.waset.org/abstracts/search?q=differential%20forms" title=" differential forms"> differential forms</a>, <a href="https://publications.waset.org/abstracts/search?q=integration" title=" integration"> integration</a>, <a href="https://publications.waset.org/abstracts/search?q=stokes%27%20theorem" title=" stokes' theorem"> stokes' theorem</a> </p> <a href="https://publications.waset.org/abstracts/130006/closed-form-solution-for-4-d-potential-integrals-for-arbitrary-coplanar-polygonal-surfaces" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/130006.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">311</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">2563</span> A Coupled System of Caputo-Type Katugampola Fractional Differential Equations with Integral Boundary Conditions</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Yacine%20Arioua">Yacine Arioua</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper, we investigate the existence and uniqueness of solutions for a coupled system of nonlinear Caputo-type Katugampola fractional differential equations with integral boundary conditions. Based upon a contraction mapping principle, Schauders fixed point theorems, some new existence and uniqueness results of solutions for the given problems are obtained. For application, some examples are given to illustrate the usefulness of our main results. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=fractional%20differential%20equations" title="fractional differential equations">fractional differential equations</a>, <a href="https://publications.waset.org/abstracts/search?q=coupled%20system" title=" coupled system"> coupled system</a>, <a href="https://publications.waset.org/abstracts/search?q=Caputo-Katugampola%20derivative" title=" Caputo-Katugampola derivative"> Caputo-Katugampola derivative</a>, <a href="https://publications.waset.org/abstracts/search?q=fixed%20point%20theorems" title=" fixed point theorems"> fixed point theorems</a>, <a href="https://publications.waset.org/abstracts/search?q=existence" title=" existence"> existence</a>, <a href="https://publications.waset.org/abstracts/search?q=uniqueness" title=" uniqueness"> uniqueness</a> </p> <a href="https://publications.waset.org/abstracts/124953/a-coupled-system-of-caputo-type-katugampola-fractional-differential-equations-with-integral-boundary-conditions" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/124953.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">264</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">2562</span> Displacement Solution for a Static Vertical Rigid Movement of an Interior Circular Disc in a Transversely Isotropic Tri-Material Full-Space</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=D.%20Mehdizadeh">D. Mehdizadeh</a>, <a href="https://publications.waset.org/abstracts/search?q=M.%20Rahimian"> M. Rahimian</a>, <a href="https://publications.waset.org/abstracts/search?q=M.%20Eskandari-Ghadi"> M. Eskandari-Ghadi</a> </p> <p class="card-text"><strong>Abstract:</strong></p> This article is concerned with the determination of the static interaction of a vertically loaded rigid circular disc embedded at the interface of a horizontal layer sandwiched in between two different transversely isotropic half-spaces called as tri-material full-space. The axes of symmetry of different regions are assumed to be normal to the horizontal interfaces and parallel to the movement direction. With the use of a potential function method, and by implementing Hankel integral transforms in the radial direction, the government partial differential equation for the solely scalar potential function is transformed to an ordinary 4th order differential equation, and the mixed boundary conditions are transformed into a pair of integral equations called dual integral equations, which can be reduced to a Fredholm integral equation of the second kind, which is solved analytically. Then, the displacements and stresses are given in the form of improper line integrals, which is due to inverse Hankel integral transforms. It is shown that the present solutions are in exact agreement with the existing solutions for a homogeneous full-space with transversely isotropic material. To confirm the accuracy of the numerical evaluation of the integrals involved, the numerical results are compared with the solutions exists for the homogeneous full-space. Then, some different cases with different degrees of material anisotropy are compared to portray the effect of degree of anisotropy. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=transversely%20isotropic" title="transversely isotropic">transversely isotropic</a>, <a href="https://publications.waset.org/abstracts/search?q=rigid%20disc" title=" rigid disc"> rigid disc</a>, <a href="https://publications.waset.org/abstracts/search?q=elasticity" title=" elasticity"> elasticity</a>, <a href="https://publications.waset.org/abstracts/search?q=dual%20integral%20equations" title=" dual integral equations"> dual integral equations</a>, <a href="https://publications.waset.org/abstracts/search?q=tri-material%20full-space" title=" tri-material full-space"> tri-material full-space</a> </p> <a href="https://publications.waset.org/abstracts/5469/displacement-solution-for-a-static-vertical-rigid-movement-of-an-interior-circular-disc-in-a-transversely-isotropic-tri-material-full-space" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/5469.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">440</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">2561</span> Generalization of Tau Approximant and Error Estimate of Integral Form of Tau Methods for Some Class 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.%20I.%20Ma%E2%80%99ali">A. I. Ma’ali</a>, <a href="https://publications.waset.org/abstracts/search?q=R.%20B.%20Adeniyi"> R. B. Adeniyi</a>, <a href="https://publications.waset.org/abstracts/search?q=A.%20Y.%20Badeggi"> A. Y. Badeggi</a>, <a href="https://publications.waset.org/abstracts/search?q=U.%20Mohammed"> U. Mohammed </a> </p> <p class="card-text"><strong>Abstract:</strong></p> An error estimation of the integrated formulation of the Lanczos tau method for some class of ordinary differential equations was reported. This paper is concern with the generalization of tau approximants and their corresponding error estimates for some class of ordinary differential equations (ODEs) characterized by m + s =3 (i.e for m =1, s=2; m=2, s=1; and m=3, s=0) where m and s are the order of differential equations and number of overdetermination, respectively. The general result obtained were validated with some numerical examples. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=approximant" title="approximant">approximant</a>, <a href="https://publications.waset.org/abstracts/search?q=error%20estimate" title=" error estimate"> error estimate</a>, <a href="https://publications.waset.org/abstracts/search?q=tau%20method" title=" tau method"> tau method</a>, <a href="https://publications.waset.org/abstracts/search?q=overdetermination" title=" overdetermination"> overdetermination</a> </p> <a href="https://publications.waset.org/abstracts/16442/generalization-of-tau-approximant-and-error-estimate-of-integral-form-of-tau-methods-for-some-class-of-ordinary-differential-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/16442.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">606</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">2560</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> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">2559</span> Existence of Minimal and Maximal Mild Solutions for Non-Local in Time Subdiffusion Equations of Neutral Type</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Jorge%20Gonzalez-Camus">Jorge Gonzalez-Camus</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this work is proved the existence of at least one minimal and maximal mild solutions to the Cauchy problem, for fractional evolution equation of neutral type, involving a general kernel. An operator A generating a resolvent family and integral resolvent family on a Banach space X and a kernel belonging to a large class appears in the equation, which covers many relevant cases from physics applications, in particular, the important case of time - fractional evolution equations of neutral type. The main tool used in this work was the Kuratowski measure of noncompactness and fixed point theorems, specifically Darbo-type, and an iterative method of lower and upper solutions, based in an order in X induced by a normal cone P. Initially, the equation is a Cauchy problem, involving a fractional derivate in Caputo sense. Then, is formulated the equivalent integral version, and defining a convenient functional, using the theory of resolvent families, and verifying the hypothesis of the fixed point theorem of Darbo type, give us the existence of mild solution for the initial problem. Furthermore, the existence of minimal and maximal mild solutions was proved through in an iterative method of lower and upper solutions, using the Azcoli-Arzela Theorem, and the Gronwall’s inequality. Finally, we recovered the case derivate in Caputo sense. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=fractional%20evolution%20equations" title="fractional evolution equations">fractional evolution equations</a>, <a href="https://publications.waset.org/abstracts/search?q=Volterra%20integral%20equations" title=" Volterra integral equations"> Volterra integral equations</a>, <a href="https://publications.waset.org/abstracts/search?q=minimal%20and%20maximal%20mild%20solutions" title=" minimal and maximal mild solutions"> minimal and maximal mild solutions</a>, <a href="https://publications.waset.org/abstracts/search?q=neutral%20type%20equations" title=" neutral type equations"> neutral type equations</a>, <a href="https://publications.waset.org/abstracts/search?q=non-local%20in%20time%20equations" title=" non-local in time equations"> non-local in time equations</a> </p> <a href="https://publications.waset.org/abstracts/105179/existence-of-minimal-and-maximal-mild-solutions-for-non-local-in-time-subdiffusion-equations-of-neutral-type" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/105179.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">176</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">2558</span> Closed Form Exact Solution for Second Order Linear Differential Equations</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Saeed%20Otarod">Saeed Otarod</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In a different simple and straight forward analysis a closed-form integral solution is found for nonhomogeneous second order linear ordinary differential equations, in terms of a particular solution of their corresponding homogeneous part. To find the particular solution of the homogeneous part, the equation is transformed into a simple Riccati equation from which the general solution of non-homogeneouecond order differential equation, in the form of a closed integral equation is inferred. The method works well in manyimportant cases, such as Schrödinger equation for hydrogen-like atoms. A non-homogenous second order linear differential equation has been solved as an extra example <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=explicit" title="explicit">explicit</a>, <a href="https://publications.waset.org/abstracts/search?q=linear" title=" linear"> linear</a>, <a href="https://publications.waset.org/abstracts/search?q=differential" title=" differential"> differential</a>, <a href="https://publications.waset.org/abstracts/search?q=closed%20form" title=" closed form"> closed form</a> </p> <a href="https://publications.waset.org/abstracts/185365/closed-form-exact-solution-for-second-order-linear-differential-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/185365.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">2557</span> On Fourier Type Integral Transform for a Class of Generalized Quotients</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=A.%20S.%20Issa">A. S. Issa</a>, <a href="https://publications.waset.org/abstracts/search?q=S.%20K.%20Q.%20AL-Omari"> S. K. Q. AL-Omari</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper, we investigate certain spaces of generalized functions for the Fourier and Fourier type integral transforms. We discuss convolution theorems and establish certain spaces of distributions for the considered integrals. The new Fourier type integral is well-defined, linear, one-to-one and continuous with respect to certain types of convergences. Many properties and an inverse problem are also discussed in some details. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=Boehmian" title="Boehmian">Boehmian</a>, <a href="https://publications.waset.org/abstracts/search?q=Fourier%20integral" title=" Fourier integral"> Fourier integral</a>, <a href="https://publications.waset.org/abstracts/search?q=Fourier%20type%20integral" title=" Fourier type integral"> Fourier type integral</a>, <a href="https://publications.waset.org/abstracts/search?q=generalized%20quotient" title=" generalized quotient"> generalized quotient</a> </p> <a href="https://publications.waset.org/abstracts/45947/on-fourier-type-integral-transform-for-a-class-of-generalized-quotients" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/45947.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">365</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">2556</span> Solution of Some Boundary Value Problems of the Generalized Theory of Thermo-Piezoelectricity</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Manana%20Chumburidze">Manana Chumburidze</a> </p> <p class="card-text"><strong>Abstract:</strong></p> We have considered a non-classical model of dynamical problems for a conjugated system of differential equations arising in thermo-piezoelectricity, which was formulated by Toupin – Mindlin. The basic concepts and the general theory of solvability for isotropic homogeneous elastic media is considered. They are worked by using the methods the Laplace integral transform, potential method and singular integral equations. Approximate solutions of mixed boundary value problems for finite domain, bounded by the some closed surface are constructed. They are solved in explicitly by using the generalized Fourier's series method. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=thermo-piezoelectricity" title="thermo-piezoelectricity">thermo-piezoelectricity</a>, <a href="https://publications.waset.org/abstracts/search?q=boundary%20value%20problems" title=" boundary value problems"> boundary value problems</a>, <a href="https://publications.waset.org/abstracts/search?q=Fourier%27s%20series" title=" Fourier's series"> Fourier's series</a>, <a href="https://publications.waset.org/abstracts/search?q=isotropic%20homogeneous%20elastic%20media" title=" isotropic homogeneous elastic media "> isotropic homogeneous elastic media </a> </p> <a href="https://publications.waset.org/abstracts/33983/solution-of-some-boundary-value-problems-of-the-generalized-theory-of-thermo-piezoelectricity" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/33983.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">465</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">2555</span> A Hybrid Classical-Quantum Algorithm for Boundary Integral Equations of Scattering Theory</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Damir%20Latypov">Damir Latypov</a> </p> <p class="card-text"><strong>Abstract:</strong></p> A hybrid classical-quantum algorithm to solve boundary integral equations (BIE) arising in problems of electromagnetic and acoustic scattering is proposed. The quantum speed-up is due to a Quantum Linear System Algorithm (QLSA). The original QLSA of Harrow et al. provides an exponential speed-up over the best-known classical algorithms but only in the case of sparse systems. Due to the non-local nature of integral operators, matrices arising from discretization of BIEs, are, however, dense. A QLSA for dense matrices was introduced in 2017. Its runtime as function of the system's size N is bounded by O(√Npolylog(N)). The run time of the best-known classical algorithm for an arbitrary dense matrix scales as O(N².³⁷³). Instead of exponential as in case of sparse matrices, here we have only a polynomial speed-up. Nevertheless, sufficiently high power of this polynomial, ~4.7, should make QLSA an appealing alternative. Unfortunately for the QLSA, the asymptotic separability of the Green's function leads to high compressibility of the BIEs matrices. Classical fast algorithms such as Multilevel Fast Multipole Method (MLFMM) take advantage of this fact and reduce the runtime to O(Nlog(N)), i.e., the QLSA is only quadratically faster than the MLFMM. To be truly impactful for computational electromagnetics and acoustics engineers, QLSA must provide more substantial advantage than that. We propose a computational scheme which combines elements of the classical fast algorithms with the QLSA to achieve the required performance. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=quantum%20linear%20system%20algorithm" title="quantum linear system algorithm">quantum linear system algorithm</a>, <a href="https://publications.waset.org/abstracts/search?q=boundary%20integral%20equations" title=" boundary integral equations"> boundary integral equations</a>, <a href="https://publications.waset.org/abstracts/search?q=dense%20matrices" title=" dense matrices"> dense matrices</a>, <a href="https://publications.waset.org/abstracts/search?q=electromagnetic%20scattering%20theory" title=" electromagnetic scattering theory"> electromagnetic scattering theory</a> </p> <a href="https://publications.waset.org/abstracts/130056/a-hybrid-classical-quantum-algorithm-for-boundary-integral-equations-of-scattering-theory" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/130056.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">154</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">2554</span> A Proof of the N. Davydov Theorem for Douglis Algebra Valued Functions</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Jean-Marie%20Vilaire">Jean-Marie Vilaire</a>, <a href="https://publications.waset.org/abstracts/search?q=Ricardo%20Abreu-Blaya"> Ricardo Abreu-Blaya</a>, <a href="https://publications.waset.org/abstracts/search?q=Juan%20Bory-Reyes"> Juan Bory-Reyes</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The classical Beltrami system of elliptic equations generalizes the Cauchy Riemann equation in the complex plane and offers the possibility to consider homogeneous system with no terms of zero order. The theory of Douglis-valued functions, called Hyper-analytic functions, is special case of the above situation. In this note, we prove an analogue of the N. Davydov theorem in the framework of the theory of hyperanalytic functions. The used methodology contemplates characteristic methods of the hypercomplex analysis as well as the singular integral operators and elliptic systems of the partial differential equations theories. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=Beltrami%20equation" title="Beltrami equation">Beltrami equation</a>, <a href="https://publications.waset.org/abstracts/search?q=Douglis%20algebra-valued%20function" title=" Douglis algebra-valued function"> Douglis algebra-valued function</a>, <a href="https://publications.waset.org/abstracts/search?q=Hypercomplex%20Cauchy%20type%20integral" title=" Hypercomplex Cauchy type integral"> Hypercomplex Cauchy type integral</a>, <a href="https://publications.waset.org/abstracts/search?q=Sokhotski-Plemelj%20formulae" title=" Sokhotski-Plemelj formulae"> Sokhotski-Plemelj formulae</a> </p> <a href="https://publications.waset.org/abstracts/92078/a-proof-of-the-n-davydov-theorem-for-douglis-algebra-valued-functions" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/92078.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">250</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">2553</span> Integral Form Solutions of the Linearized Navier-Stokes Equations without Deviatoric Stress Tensor Term in the Forward Modeling for FWI</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Anyeres%20N.%20Atehortua%20Jimenez">Anyeres N. Atehortua Jimenez</a>, <a href="https://publications.waset.org/abstracts/search?q=J.%20David%20Lambra%C3%B1o"> J. David Lambraño</a>, <a href="https://publications.waset.org/abstracts/search?q=Juan%20Carlos%20Mu%C3%B1oz"> Juan Carlos Muñoz</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Navier-Stokes equations (NSE), which describe the dynamics of a fluid, have an important application on modeling waves used for data inversion techniques as full waveform inversion (FWI). In this work a linearized version of NSE and its variables, neglecting deviatoric terms of stress tensor, is presented. In order to get a theoretical modeling of pressure p(x,t) and wave velocity profile c(x,t), a wave equation of visco-acoustic medium (VAE) is written. A change of variables p(x,t)=q(x,t)h(ρ), is made on the equation for the VAE leading to a well known Klein-Gordon equation (KGE) describing waves propagating in variable density medium (ρ) with dispersive term α^2(x). KGE is reduced to a Poisson equation and solved by proposing a specific function for α^2(x) accounting for the energy dissipation and dispersion. Finally, an integral form solution is derived for p(x,t), c(x,t) and kinematics variables like particle velocity v(x,t), displacement u(x,t) and bulk modulus function k_b(x,t). Further, it is compared this visco-acoustic formulation with another form broadly used in the geophysics; it is argued that this formalism is more general and, given its integral form, it may offer several advantages from the modern parallel computing point of view. Applications to minimize the errors in modeling for FWI applied to oils resources in geophysics are discussed. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=Navier-Stokes%20equations" title="Navier-Stokes equations">Navier-Stokes equations</a>, <a href="https://publications.waset.org/abstracts/search?q=modeling" title=" modeling"> modeling</a>, <a href="https://publications.waset.org/abstracts/search?q=visco-acoustic" title=" visco-acoustic"> visco-acoustic</a>, <a href="https://publications.waset.org/abstracts/search?q=inversion%20FWI" title=" inversion FWI "> inversion FWI </a> </p> <a href="https://publications.waset.org/abstracts/33620/integral-form-solutions-of-the-linearized-navier-stokes-equations-without-deviatoric-stress-tensor-term-in-the-forward-modeling-for-fwi" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/33620.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">520</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">2552</span> Three-Dimensional Numerical Analysis of the Harmfulness of Defects in Oil Pipes</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=B.%20Medjadji">B. Medjadji</a>, <a href="https://publications.waset.org/abstracts/search?q=L.%20Aminallah"> L. Aminallah</a>, <a href="https://publications.waset.org/abstracts/search?q=B.%20Serier"> B. Serier</a>, <a href="https://publications.waset.org/abstracts/search?q=M.%20Benlebna"> M. Benlebna</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this study, the finite element method in 3-D is used to calculate the integral J in the semi-elliptical crack in a pipe subjected to internal pressure. The stress-strain curve of the pipe has been determined experimentally. The J-integral was calculated in two fronts crack (Ф = 0 and Ф = π/2). The effect of the configuration of the crack on the J integral is analysed. The results show that an external longitudinal crack in a pipe is the most dangerous. It also shows that the increase in the applied pressure causes a remarkable increase of the integral J. The effect of the depth of the crack becomes important when the ratio between the depth of the crack and the thickness of the pipe (a / t) tends to 1. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=J%20integral" title="J integral">J integral</a>, <a href="https://publications.waset.org/abstracts/search?q=pipeline" title=" pipeline"> pipeline</a>, <a href="https://publications.waset.org/abstracts/search?q=crack" title=" crack"> crack</a>, <a href="https://publications.waset.org/abstracts/search?q=MEF" title=" MEF"> MEF</a> </p> <a href="https://publications.waset.org/abstracts/4115/three-dimensional-numerical-analysis-of-the-harmfulness-of-defects-in-oil-pipes" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/4115.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">409</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">2551</span> The Solution of the Direct Problem of Electrical Prospecting with Direct Current Under Conditions of Ground Surface Relief</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Balgaisha%20Mukanova">Balgaisha Mukanova</a>, <a href="https://publications.waset.org/abstracts/search?q=Tolkyn%20Mirgalikyzy"> Tolkyn Mirgalikyzy</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Theory of interpretation of electromagnetic fields studied in the electrical prospecting with direct current is mainly developed for the case of a horizontal surface observation. However in practice we often have to work in difficult terrain surface. Conducting interpretation without the influence of topography can cause non-existent anomalies on sections. This raises the problem of studying the impact of different shapes of ground surface relief on the results of electrical prospecting's research. This research examines the numerical solutions of the direct problem of electrical prospecting for two-dimensional and three-dimensional media, taking into account the terrain. The problem is solved using the method of integral equations. The density of secondary currents on the relief surface is obtained. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=ground%20surface%20relief" title="ground surface relief">ground surface relief</a>, <a href="https://publications.waset.org/abstracts/search?q=method%20of%20integral%20equations" title=" method of integral equations"> method of integral equations</a>, <a href="https://publications.waset.org/abstracts/search?q=numerical%20method" title=" numerical method"> numerical method</a>, <a href="https://publications.waset.org/abstracts/search?q=electromagnetic" title=" electromagnetic "> electromagnetic </a> </p> <a href="https://publications.waset.org/abstracts/27446/the-solution-of-the-direct-problem-of-electrical-prospecting-with-direct-current-under-conditions-of-ground-surface-relief" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/27446.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">363</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">2550</span> A Comparative Study on a Tilt-Integral-Derivative Controller with Proportional-Integral-Derivative Controller for a Pacemaker</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Aysan%20Esgandanian">Aysan Esgandanian</a>, <a href="https://publications.waset.org/abstracts/search?q=Sabalan%20Daneshvar"> Sabalan Daneshvar</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The study is done to determine the comparison between proportional-integral-derivative controller (PID controller) and tilt-integral-derivative (TID controller) for cardiac pacemaker systems, which can automatically control the heart rate to accurately track a desired preset profile. The controller offers good adaption of heart to the physiological needs of the patient. The parameters of the both controllers are tuned by particle swarm optimization (PSO) algorithm which uses the integral of time square error as a fitness function to be minimized. Simulation results are performed on the developed cardiovascular system of humans and results demonstrate that the TID controller produces superior control performance than PID controllers. In this paper, all simulations were performed in Matlab. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=integral%20of%20time%20square%20error" title="integral of time square error">integral of time square error</a>, <a href="https://publications.waset.org/abstracts/search?q=pacemaker%20systems" title=" pacemaker systems"> pacemaker systems</a>, <a href="https://publications.waset.org/abstracts/search?q=proportional-integral-derivative%20controller" title=" proportional-integral-derivative controller"> proportional-integral-derivative controller</a>, <a href="https://publications.waset.org/abstracts/search?q=PSO%20algorithm" title=" PSO algorithm"> PSO algorithm</a>, <a href="https://publications.waset.org/abstracts/search?q=tilt-integral-derivative%20controller" title=" tilt-integral-derivative controller"> tilt-integral-derivative controller</a> </p> <a href="https://publications.waset.org/abstracts/43351/a-comparative-study-on-a-tilt-integral-derivative-controller-with-proportional-integral-derivative-controller-for-a-pacemaker" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/43351.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">462</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">2549</span> Spectral Domain Fast Multipole Method for Solving Integral Equations of One and Two Dimensional Wave Scattering </h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Mohammad%20Ahmad">Mohammad Ahmad</a>, <a href="https://publications.waset.org/abstracts/search?q=Dayalan%20Kasilingam"> Dayalan Kasilingam</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper, a spectral domain implementation of the fast multipole method is presented. It is shown that the aggregation, translation, and disaggregation stages of the fast multipole method (FMM) can be performed using the spectral domain (SD) analysis. The spectral domain fast multipole method (SD-FMM) has the advantage of eliminating the near field/far field classification used in conventional FMM formulation. The study focuses on the application of SD-FMM to one-dimensional (1D) and two-dimensional (2D) electric field integral equation (EFIE). The case of perfectly conducting strip, circular and square cylinders are numerically analyzed and compared with the results from the standard method of moments (MoM). <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=electric%20field%20integral%20equation" title="electric field integral equation">electric field integral equation</a>, <a href="https://publications.waset.org/abstracts/search?q=fast%20multipole%20method" title=" fast multipole method"> fast multipole method</a>, <a href="https://publications.waset.org/abstracts/search?q=method%20of%20moments" title=" method of moments"> method of moments</a>, <a href="https://publications.waset.org/abstracts/search?q=wave%20scattering" title=" wave scattering"> wave scattering</a>, <a href="https://publications.waset.org/abstracts/search?q=spectral%20domain" title=" spectral domain"> spectral domain</a> </p> <a href="https://publications.waset.org/abstracts/65787/spectral-domain-fast-multipole-method-for-solving-integral-equations-of-one-and-two-dimensional-wave-scattering" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/65787.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">406</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">2548</span> Non-Local Behavior of a Mixed-Mode Crack in a Functionally Graded Piezoelectric Medium</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Nidhal%20Jamia">Nidhal Jamia</a>, <a href="https://publications.waset.org/abstracts/search?q=Sami%20El-Borgi"> Sami El-Borgi</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper, the problem of a mixed-Mode crack embedded in an infinite medium made of a functionally graded piezoelectric material (FGPM) with crack surfaces subjected to electro-mechanical loadings is investigated. Eringen’s non-local theory of elasticity is adopted to formulate the governing electro-elastic equations. The properties of the piezoelectric material are assumed to vary exponentially along a perpendicular plane to the crack. Using Fourier transform, three integral equations are obtained in which the unknown variables are the jumps of mechanical displacements and electric potentials across the crack surfaces. To solve the integral equations, the unknowns are directly expanded as a series of Jacobi polynomials, and the resulting equations solved using the Schmidt method. In contrast to the classical solutions based on the local theory, it is found that no mechanical stress and electric displacement singularities are present at the crack tips when nonlocal theory is employed to investigate the problem. A direct benefit is the ability to use the calculated maximum stress as a fracture criterion. The primary objective of this study is to investigate the effects of crack length, material gradient parameter describing FGPMs, and lattice parameter on the mechanical stress and electric displacement field near crack tips. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=functionally%20graded%20piezoelectric%20material%20%28FGPM%29" title="functionally graded piezoelectric material (FGPM)">functionally graded piezoelectric material (FGPM)</a>, <a href="https://publications.waset.org/abstracts/search?q=mixed-mode%20crack" title=" mixed-mode crack"> mixed-mode crack</a>, <a href="https://publications.waset.org/abstracts/search?q=non-local%20theory" title=" non-local theory"> non-local theory</a>, <a href="https://publications.waset.org/abstracts/search?q=Schmidt%20method" title=" Schmidt method"> Schmidt method</a> </p> <a href="https://publications.waset.org/abstracts/50684/non-local-behavior-of-a-mixed-mode-crack-in-a-functionally-graded-piezoelectric-medium" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/50684.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">308</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">2547</span> Classification of Equations of Motion</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Amritpal%20Singh%20Nafria">Amritpal Singh Nafria</a>, <a href="https://publications.waset.org/abstracts/search?q=Rohit%20Sharma"> Rohit Sharma</a>, <a href="https://publications.waset.org/abstracts/search?q=Md.%20Shami%20Ansari"> Md. Shami Ansari</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Up to now only five different equations of motion can be derived from velocity time graph without needing to know the normal and frictional forces acting at the point of contact. In this paper we obtained all possible requisite conditions to be considering an equation as an equation of motion. After that we classified equations of motion by considering two equations as fundamental kinematical equations of motion and other three as additional kinematical equations of motion. After deriving these five equations of motion, we examine the easiest way of solving a wide variety of useful numerical problems. At the end of the paper, we discussed the importance and educational benefits of classification of equations of motion. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=velocity-time%20graph" title="velocity-time graph">velocity-time graph</a>, <a href="https://publications.waset.org/abstracts/search?q=fundamental%20equations" title=" fundamental equations"> fundamental equations</a>, <a href="https://publications.waset.org/abstracts/search?q=additional%20equations" title=" additional equations"> additional equations</a>, <a href="https://publications.waset.org/abstracts/search?q=requisite%20conditions" title=" requisite conditions"> requisite conditions</a>, <a href="https://publications.waset.org/abstracts/search?q=importance%20and%20educational%20benefits" title=" importance and educational benefits"> importance and educational benefits</a> </p> <a href="https://publications.waset.org/abstracts/15102/classification-of-equations-of-motion" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/15102.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">787</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">2546</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">2545</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">2544</span> Integrable Heisenberg Ferromagnet Equations with Self-Consistent Potentials</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Gulgassyl%20Nugmanova">Gulgassyl Nugmanova</a>, <a href="https://publications.waset.org/abstracts/search?q=Zhanat%20Zhunussova"> Zhanat Zhunussova</a>, <a href="https://publications.waset.org/abstracts/search?q=Kuralay%20Yesmakhanova"> Kuralay Yesmakhanova</a>, <a href="https://publications.waset.org/abstracts/search?q=Galya%20Mamyrbekova"> Galya Mamyrbekova</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 paper, we consider some integrable Heisenberg Ferromagnet Equations with self-consistent potentials. We study their Lax representations. In particular we derive their equivalent counterparts in the form of nonlinear Schr\"odinger type equations. We present the integrable reductions of the Heisenberg Ferromagnet Equations with self-consistent potentials. These integrable Heisenberg Ferromagnet Equations with self-consistent potentials describe nonlinear waves in ferromagnets with some additional physical fields. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=Heisenberg%20Ferromagnet%20equations" title="Heisenberg Ferromagnet equations">Heisenberg Ferromagnet equations</a>, <a href="https://publications.waset.org/abstracts/search?q=soliton%20equations" title=" soliton equations"> soliton equations</a>, <a href="https://publications.waset.org/abstracts/search?q=equivalence" title=" equivalence"> equivalence</a>, <a href="https://publications.waset.org/abstracts/search?q=Lax%20representation" title=" Lax representation"> Lax representation</a> </p> <a href="https://publications.waset.org/abstracts/27440/integrable-heisenberg-ferromagnet-equations-with-self-consistent-potentials" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/27440.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">457</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=integral%20equations&page=2">2</a></li> <li class="page-item"><a class="page-link" href="https://publications.waset.org/abstracts/search?q=integral%20equations&page=3">3</a></li> <li class="page-item"><a class="page-link" href="https://publications.waset.org/abstracts/search?q=integral%20equations&page=4">4</a></li> <li class="page-item"><a class="page-link" href="https://publications.waset.org/abstracts/search?q=integral%20equations&page=5">5</a></li> <li class="page-item"><a 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