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equations</title> <meta name="description" content="Search results for: Boussinesq equations"> <meta name="keywords" content="Boussinesq 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="Boussinesq 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> 1843</div> </div> </div> </div> <h1 class="mt-3 mb-3 text-center" style="font-size:1.6rem;">Search results for: Boussinesq equations</h1> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">1843</span> Numerical Modeling of Storm Swells in Harbor by Boussinesq Equations Model</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Mustapha%20Kamel%20Mihoubi">Mustapha Kamel Mihoubi</a>, <a href="https://publications.waset.org/abstracts/search?q=Hocine%20Dahmani"> Hocine Dahmani </a> </p> <p class="card-text"><strong>Abstract:</strong></p> The purpose of work is to study the phenomenon of agitation of storm waves at basin caused by different directions of waves relative to the current provision thrown numerical model based on the equation in shallow water using Boussinesq model MIKE 21 BW. According to the diminishing effect of penetration of a wave optimal solution will be available to be reproduced in reduced model. Another alternative arrangement throws will be proposed to reduce the agitation and the effects of the swell reflection caused by the penetration of waves in the harbor. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=agitation" title="agitation">agitation</a>, <a href="https://publications.waset.org/abstracts/search?q=Boussinesq%20equations" title=" Boussinesq equations"> Boussinesq equations</a>, <a href="https://publications.waset.org/abstracts/search?q=combination" title=" combination"> combination</a>, <a href="https://publications.waset.org/abstracts/search?q=harbor" title=" harbor"> harbor</a> </p> <a href="https://publications.waset.org/abstracts/16182/numerical-modeling-of-storm-swells-in-harbor-by-boussinesq-equations-model" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/16182.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">389</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">1842</span> A Comparative Evaluation of Finite Difference Methods for the Extended Boussinesq Equations and Application to Tsunamis Modelling</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Aurore%20Cauquis">Aurore Cauquis</a>, <a href="https://publications.waset.org/abstracts/search?q=Philippe%20Heinrich"> Philippe Heinrich</a>, <a href="https://publications.waset.org/abstracts/search?q=Mario%20Ricchiuto"> Mario Ricchiuto</a>, <a href="https://publications.waset.org/abstracts/search?q=Audrey%20Gailler"> Audrey Gailler</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this talk, we look for an accurate time scheme to model the propagation of waves. Several numerical schemes have been developed to solve the extended weakly nonlinear weakly dispersive Boussinesq Equations. The temporal schemes used are two Lax-Wendroff schemes, second or third order accurate, two Runge-Kutta schemes of second and third order and a simplified third order accurate Lax-Wendroff scheme. Spatial derivatives are evaluated with fourth order accuracy. The numerical model is applied to two monodimensional benchmarks on a flat bottom. It is also applied to the simulation of the Algerian tsunami generated by a Mw=6 seism on the 18th March 2021. The tsunami propagation was highly dispersive and propagated across the Mediterranean Sea. We study here the effects of the order of temporal discretization on the accuracy of the results and on the time of computation. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=numerical%20analysis" title="numerical analysis">numerical analysis</a>, <a href="https://publications.waset.org/abstracts/search?q=tsunami%20propagation" title=" tsunami propagation"> tsunami propagation</a>, <a href="https://publications.waset.org/abstracts/search?q=water%20wave" title=" water wave"> water wave</a>, <a href="https://publications.waset.org/abstracts/search?q=boussinesq%20equations" title=" boussinesq equations"> boussinesq equations</a> </p> <a href="https://publications.waset.org/abstracts/141542/a-comparative-evaluation-of-finite-difference-methods-for-the-extended-boussinesq-equations-and-application-to-tsunamis-modelling" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/141542.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">242</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">1841</span> Sensitivity Analysis and Solitary Wave Solutions to the (2+1)-Dimensional Boussinesq Equation in Dispersive Media</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Naila%20Nasreen">Naila Nasreen</a>, <a href="https://publications.waset.org/abstracts/search?q=Dianchen%20Lu"> Dianchen Lu</a> </p> <p class="card-text"><strong>Abstract:</strong></p> This paper explores the dynamical behavior of the (2+1)-dimensional Boussinesq equation, which is a nonlinear water wave equation and is used to model wave packets in dispersive media with weak nonlinearity. This equation depicts how long wave made in shallow water propagates due to the influence of gravity. The (2+1)- dimensional Boussinesq equation combines the two-way propagation of the classical Boussinesq equation with the dependence on a second spatial variable, as that occurs in the two-dimensional Kadomstev- Petviashvili equation. This equation provides a description of head- on collision of oblique waves and it possesses some interesting properties. The governing model is discussed by the assistance of Ricatti equation mapping method, a relatively integration tool. The solutions have been extracted in different forms the solitary wave solutions as well as hyperbolic and periodic solutions. Moreover, the sensitivity analysis is demonstrated for the designed dynamical structural system’s wave profiles, where the soliton wave velocity and wave number parameters regulate the water wave singularity. In addition to being helpful for elucidating nonlinear partial differential equations, the method in use gives previously extracted solutions and extracts fresh exact solutions. Assuming the right values for the parameters, various graph in different shapes are sketched to provide information about the visual format of the earned results. This paper’s findings support the efficacy of the approach taken in enhancing nonlinear dynamical behavior. We believe this research will be of interest to a wide variety of engineers that work with engineering models. Findings show the effectiveness simplicity, and generalizability of the chosen computational approach, even when applied to complicated systems in a variety of fields, especially in ocean engineering. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=%282%2B1%29-dimensional%20Boussinesq%20equation" title="(2+1)-dimensional Boussinesq equation">(2+1)-dimensional Boussinesq equation</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=Ricatti%20equation%20mapping%20approach" title=" Ricatti equation mapping approach"> Ricatti equation mapping approach</a>, <a href="https://publications.waset.org/abstracts/search?q=nonlinear%20phenomena" title=" nonlinear phenomena"> nonlinear phenomena</a> </p> <a href="https://publications.waset.org/abstracts/165781/sensitivity-analysis-and-solitary-wave-solutions-to-the-21-dimensional-boussinesq-equation-in-dispersive-media" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/165781.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">100</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">1840</span> Boussinesq Model for Dam-Break Flow Analysis</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Najibullah%20M">Najibullah M</a>, <a href="https://publications.waset.org/abstracts/search?q=Soumendra%20Nath%20Kuiry"> Soumendra Nath Kuiry</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Dams and reservoirs are perceived for their estimable alms to irrigation, water supply, flood control, electricity generation, etc. which civilize the prosperity and wealth of society across the world. Meantime the dam breach could cause devastating flood that can threat to the human lives and properties. Failures of large dams remain fortunately very seldom events. Nevertheless, a number of occurrences have been recorded in the world, corresponding in an average to one to two failures worldwide every year. Some of those accidents have caused catastrophic consequences. So it is decisive to predict the dam break flow for emergency planning and preparedness, as it poses high risk to life and property. To mitigate the adverse impact of dam break, modeling is necessary to gain a good understanding of the temporal and spatial evolution of the dam-break floods. This study will mainly deal with one-dimensional (1D) dam break modeling. Less commonly used in the hydraulic research community, another possible option for modeling the rapidly varied dam-break flows is the extended Boussinesq equations (BEs), which can describe the dynamics of short waves with a reasonable accuracy. Unlike the Shallow Water Equations (SWEs), the BEs taken into account the wave dispersion and non-hydrostatic pressure distribution. To capture the dam-break oscillations accurately it is very much needed of at least fourth-order accurate numerical scheme to discretize the third-order dispersion terms present in the extended BEs. The scope of this work is therefore to develop an 1D fourth-order accurate in both space and time Boussinesq model for dam-break flow analysis by using finite-volume / finite difference scheme. The spatial discretization of the flux and dispersion terms achieved through a combination of finite-volume and finite difference approximations. The flux term, was solved using a finite-volume discretization whereas the bed source and dispersion term, were discretized using centered finite-difference scheme. Time integration achieved in two stages, namely the third-order Adams Basforth predictor stage and the fourth-order Adams Moulton corrector stage. Implementation of the 1D Boussinesq model done using PYTHON 2.7.5. Evaluation of the performance of the developed model predicted as compared with the volume of fluid (VOF) based commercial model ANSYS-CFX. The developed model is used to analyze the risk of cascading dam failures similar to the Panshet dam failure in 1961 that took place in Pune, India. Nevertheless, this model can be used to predict wave overtopping accurately compared to shallow water models for designing coastal protection structures. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=Boussinesq%20equation" title="Boussinesq equation">Boussinesq equation</a>, <a href="https://publications.waset.org/abstracts/search?q=Coastal%20protection" title=" Coastal protection"> Coastal protection</a>, <a href="https://publications.waset.org/abstracts/search?q=Dam-break%20flow" title=" Dam-break flow"> Dam-break flow</a>, <a href="https://publications.waset.org/abstracts/search?q=One-dimensional%20model" title=" One-dimensional model"> One-dimensional model</a> </p> <a href="https://publications.waset.org/abstracts/59450/boussinesq-model-for-dam-break-flow-analysis" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/59450.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">232</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">1839</span> Modeling of Landslide-Generated Tsunamis in Georgia Strait, Southern British Columbia</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Fatemeh%20Nemati">Fatemeh Nemati</a>, <a href="https://publications.waset.org/abstracts/search?q=Lucinda%20%20Leonard"> Lucinda Leonard</a>, <a href="https://publications.waset.org/abstracts/search?q=Gwyn%20Lintern"> Gwyn Lintern</a>, <a href="https://publications.waset.org/abstracts/search?q=Richard%20Thomson"> Richard Thomson</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this study, we will use modern numerical modeling approaches to estimate tsunami risks to the southern coast of British Columbia from landslides. Wave generation is to be simulated using the NHWAVE model, which solves the Navier-Stokes equations due to the more complex behavior of flow near the landslide source; far-field wave propagation will be simulated using the simpler model FUNWAVE_TVD with high-order Boussinesq-type wave equations, with a focus on the accurate simulation of wave propagation and regional- or coastal-scale inundation predictions. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=FUNWAVE-TVD" title="FUNWAVE-TVD">FUNWAVE-TVD</a>, <a href="https://publications.waset.org/abstracts/search?q=landslide-generated%20tsunami" title=" landslide-generated tsunami"> landslide-generated tsunami</a>, <a href="https://publications.waset.org/abstracts/search?q=NHWAVE" title=" NHWAVE"> NHWAVE</a>, <a href="https://publications.waset.org/abstracts/search?q=tsunami%20risk" title=" tsunami risk "> tsunami risk </a> </p> <a href="https://publications.waset.org/abstracts/129053/modeling-of-landslide-generated-tsunamis-in-georgia-strait-southern-british-columbia" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/129053.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">1838</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">1837</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">1836</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> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">1835</span> MHD Mixed Convection in a Vertical Porous Channel</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Brahim%20Fersadou">Brahim Fersadou</a>, <a href="https://publications.waset.org/abstracts/search?q=Henda%20Kahalerras"> Henda Kahalerras</a> </p> <p class="card-text"><strong>Abstract:</strong></p> This work deals with the problem of MHD mixed convection in a completely porous and differentially heated vertical channel. The model of Darcy-Brinkman-Forchheimer with the Boussinesq approximation is adopted and the governing equations are solved by the finite volume method. The effects of magnetic field and buoyancy force intensities are given by the Hartmann and Richardson numbers respectively, as well as the Joule heating represented by Eckert number on the velocity and temperature fields, are examined. The main results show an augmentation of heat transfer rate with the decrease of Darcy number and the increase of Ri and Ha when Joule heating is neglected. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=heat%20sources" title="heat sources">heat sources</a>, <a href="https://publications.waset.org/abstracts/search?q=magnetic%20field" title=" magnetic field"> magnetic field</a>, <a href="https://publications.waset.org/abstracts/search?q=mixed%20convection" title=" mixed convection"> mixed convection</a>, <a href="https://publications.waset.org/abstracts/search?q=porous%20channel" title=" porous channel "> porous channel </a> </p> <a href="https://publications.waset.org/abstracts/41679/mhd-mixed-convection-in-a-vertical-porous-channel" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/41679.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">377</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">1834</span> Numerical Approach to a Mathematical Modeling of Bioconvection Due to Gyrotactic Micro-Organisms over a Nonlinear Inclined Stretching Sheet</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Madhu%20Aneja">Madhu Aneja</a>, <a href="https://publications.waset.org/abstracts/search?q=Sapna%20Sharma"> Sapna Sharma</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The water-based bioconvection of a nanofluid containing motile gyrotactic micro-organisms over nonlinear inclined stretching sheet has been investigated. The governing nonlinear boundary layer equations of the model are reduced to a system of ordinary differential equations via Oberbeck-Boussinesq approximation and similarity transformations. Further, the modified set of equations with associated boundary conditions are solved using Finite Element Method. The impact of various pertinent parameters on the velocity, temperature, nanoparticles concentration, density of motile micro-organisms profiles are obtained and analyzed in details. The results show that with the increase in angle of inclination δ, velocity decreases while temperature, nanoparticles concentration, a density of motile micro-organisms increases. Additionally, the skin friction coefficient, Nusselt number, Sherwood number, density number are computed for various thermophysical parameters. It is noticed that increasing Brownian motion and thermophoresis parameter leads to an increase in temperature of fluid which results in a reduction in Nusselt number. On the contrary, Sherwood number rises with an increase in Brownian motion and thermophoresis parameter. The findings have been validated by comparing the results of special cases with existing studies. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=bioconvection" title="bioconvection">bioconvection</a>, <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=gyrotactic%20micro-organisms" title=" gyrotactic micro-organisms"> gyrotactic micro-organisms</a>, <a href="https://publications.waset.org/abstracts/search?q=inclined%20stretching%20sheet" title=" inclined stretching sheet"> inclined stretching sheet</a>, <a href="https://publications.waset.org/abstracts/search?q=nanofluid" title=" nanofluid"> nanofluid</a> </p> <a href="https://publications.waset.org/abstracts/73917/numerical-approach-to-a-mathematical-modeling-of-bioconvection-due-to-gyrotactic-micro-organisms-over-a-nonlinear-inclined-stretching-sheet" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/73917.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">189</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">1833</span> Human Intraocular Thermal Field in Action with Different Boundary Conditions Considering Aqueous Humor and Vitreous Humor Fluid Flow</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Dara%20Singh">Dara Singh</a>, <a href="https://publications.waset.org/abstracts/search?q=Keikhosrow%20Firouzbakhsh"> Keikhosrow Firouzbakhsh</a>, <a href="https://publications.waset.org/abstracts/search?q=Mohammad%20Taghi%20Ahmadian"> Mohammad Taghi Ahmadian</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this study, a validated 3D finite volume model of human eye is developed to study the fluid flow and heat transfer in the human eye at steady state conditions. For this purpose, discretized bio-heat transfer equation coupled with Boussinesq equation is analyzed with different anatomical, environmental, and physiological conditions. It is demonstrated that the fluid circulation is formed as a result of thermal gradients in various regions of eye. It is also shown that posterior region of the human eye is less affected by the ambient conditions compared to the anterior segment which is sensitive to the ambient conditions and also to the way the gravitational field is defined compared to the geometry of the eye making the circulations and the thermal field complicated in transient states. The effect of variation in material and boundary conditions guides us to the conclusion that thermal field of a healthy and non-healthy eye can be distinguished via computer simulations. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=bio-heat" title="bio-heat">bio-heat</a>, <a href="https://publications.waset.org/abstracts/search?q=boussinesq" title=" boussinesq"> boussinesq</a>, <a href="https://publications.waset.org/abstracts/search?q=conduction" title=" conduction"> conduction</a>, <a href="https://publications.waset.org/abstracts/search?q=convection" title=" convection"> convection</a>, <a href="https://publications.waset.org/abstracts/search?q=eye" title=" eye"> eye</a> </p> <a href="https://publications.waset.org/abstracts/60991/human-intraocular-thermal-field-in-action-with-different-boundary-conditions-considering-aqueous-humor-and-vitreous-humor-fluid-flow" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/60991.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">345</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">1832</span> A Novel Method for Solving Nonlinear Whitham–Broer–Kaup Equation System</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Ayda%20Nikkar">Ayda Nikkar</a>, <a href="https://publications.waset.org/abstracts/search?q=Roghayye%20Ahmadiasl"> Roghayye Ahmadiasl</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this letter, a new analytical method called homotopy perturbation method, which does not need small parameter in the equation is implemented for solving the nonlinear Whitham–Broer–Kaup (WBK) partial differential equation. In this method, a homotopy is introduced to be constructed for the equation. The initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary and initial conditions. Comparison of the results with those of exact solution has led us to significant consequences. The results reveal that the HPM is very effective, convenient and quite accurate to systems of nonlinear equations. It is predicted that the HPM can be found widely applicable in engineering. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=homotopy%20perturbation%20method" title="homotopy perturbation method">homotopy perturbation method</a>, <a href="https://publications.waset.org/abstracts/search?q=Whitham%E2%80%93Broer%E2%80%93Kaup%20%28WBK%29%20equation" title=" Whitham–Broer–Kaup (WBK) equation"> Whitham–Broer–Kaup (WBK) equation</a>, <a href="https://publications.waset.org/abstracts/search?q=Modified%20Boussinesq" title=" Modified Boussinesq"> Modified Boussinesq</a>, <a href="https://publications.waset.org/abstracts/search?q=Approximate%20Long%20Wave" title=" Approximate Long Wave"> Approximate Long Wave</a> </p> <a href="https://publications.waset.org/abstracts/35317/a-novel-method-for-solving-nonlinear-whitham-broer-kaup-equation-system" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/35317.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">1831</span> Unsteady Natural Convection in a Square Cavity Partially Filled with Porous Media Using a Thermal Non-Equilibrium Model</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Ammar%20Alsabery">Ammar Alsabery</a>, <a href="https://publications.waset.org/abstracts/search?q=Habibis%20Saleh"> Habibis Saleh</a>, <a href="https://publications.waset.org/abstracts/search?q=Norazam%20Arbin"> Norazam Arbin</a>, <a href="https://publications.waset.org/abstracts/search?q=Ishak%20Hashim"> Ishak Hashim</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Unsteady natural convection and heat transfer in a square cavity partially filled with porous media using a thermal non-equilibrium model is studied in this paper. The left vertical wall is maintained at a constant hot temperature and the right vertical wall is maintained at a constant cold temperature, while the horizontal walls are adiabatic. The governing equations are obtained by applying the Darcy model and Boussinesq approximation. COMSOL's finite element method is used to solve the non-dimensional governing equations together with specified boundary conditions. The governing parameters of this study are the Rayleigh number, the modified thermal conductivity ratio, the inter-phase heat transfer coefficien and the time independent. The results presented for values of the governing parameters in terms of streamlines in both fluid/porous layer, isotherms of fluid and solid porous layer, isotherms of fluid layer, and average Nusselt number. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=unsteady%20natural%20convection" title="unsteady natural convection">unsteady natural convection</a>, <a href="https://publications.waset.org/abstracts/search?q=thermal%20non-equilibrium%20model" title=" thermal non-equilibrium model"> thermal non-equilibrium model</a>, <a href="https://publications.waset.org/abstracts/search?q=Darcy%20model" title=" Darcy model"> Darcy model</a> </p> <a href="https://publications.waset.org/abstracts/12724/unsteady-natural-convection-in-a-square-cavity-partially-filled-with-porous-media-using-a-thermal-non-equilibrium-model" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/12724.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">1830</span> Further Results on Modified Variational Iteration Method for the Analytical Solution of Nonlinear Advection Equations</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=A.%20W.%20Gbolagade">A. W. Gbolagade</a>, <a href="https://publications.waset.org/abstracts/search?q=M.%20O.%20Olayiwola"> M. O. Olayiwola</a>, <a href="https://publications.waset.org/abstracts/search?q=K.%20O.%20Kareem"> K. O. Kareem</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper, further to our result on recent paper on the solution of nonlinear advection equations, we present further results on the nonlinear nonhomogeneous advection equations using a modified variational iteration method. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=lagrange%20multiplier" title="lagrange multiplier">lagrange multiplier</a>, <a href="https://publications.waset.org/abstracts/search?q=non-homogeneous%20equations" title=" non-homogeneous equations"> non-homogeneous equations</a>, <a href="https://publications.waset.org/abstracts/search?q=advection%20equations" title=" advection equations"> advection equations</a>, <a href="https://publications.waset.org/abstracts/search?q=mathematics" title=" mathematics"> mathematics</a> </p> <a href="https://publications.waset.org/abstracts/3945/further-results-on-modified-variational-iteration-method-for-the-analytical-solution-of-nonlinear-advection-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/3945.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">301</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">1829</span> A Unified Fitting Method for the Set of Unified Constitutive Equations for Modelling Microstructure Evolution in Hot Deformation</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Chi%20Zhang">Chi Zhang</a>, <a href="https://publications.waset.org/abstracts/search?q=Jun%20Jiang"> Jun Jiang</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Constitutive equations are very important in finite element (FE) modeling, and the accuracy of the material constants in the equations have significant effects on the accuracy of the FE models. A wide range of constitutive equations are available; however, fitting the material constants in the constitutive equations could be complex and time-consuming due to the strong non-linearity and relationship between the constants. This work will focus on the development of a set of unified MATLAB programs for fitting the material constants in the constitutive equations efficiently. Users will only need to supply experimental data in the required format and run the program without modifying functions or precisely guessing the initial values, or finding the parameters in previous works and will be able to fit the material constants efficiently. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=constitutive%20equations" title="constitutive equations">constitutive equations</a>, <a href="https://publications.waset.org/abstracts/search?q=FE%20modelling" title=" FE modelling"> FE modelling</a>, <a href="https://publications.waset.org/abstracts/search?q=MATLAB%20program" title=" MATLAB program"> MATLAB program</a>, <a href="https://publications.waset.org/abstracts/search?q=non-linear%20curve%20fitting" title=" non-linear curve fitting"> non-linear curve fitting</a> </p> <a href="https://publications.waset.org/abstracts/162562/a-unified-fitting-method-for-the-set-of-unified-constitutive-equations-for-modelling-microstructure-evolution-in-hot-deformation" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/162562.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">99</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">1828</span> New Insight into Fluid Mechanics of Lorenz Equations</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Yu-Kai%20Ting">Yu-Kai Ting</a>, <a href="https://publications.waset.org/abstracts/search?q=Jia-Ying%20Tu"> Jia-Ying Tu</a>, <a href="https://publications.waset.org/abstracts/search?q=Chung-Chun%20Hsiao"> Chung-Chun Hsiao</a> </p> <p class="card-text"><strong>Abstract:</strong></p> New physical insights into the nonlinear Lorenz equations related to flow resistance is discussed in this work. The chaotic dynamics related to Lorenz equations has been studied in many papers, which is due to the sensitivity of Lorenz equations to initial conditions and parameter uncertainties. However, the physical implication arising from Lorenz equations about convectional motion attracts little attention in the relevant literature. Therefore, as a first step to understand the related fluid mechanics of convectional motion, this paper derives the Lorenz equations again with different forced conditions in the model. Simulation work of the modified Lorenz equations without the viscosity or buoyancy force is discussed. The time-domain simulation results may imply that the states of the Lorenz equations are related to certain flow speed and flow resistance. The flow speed of the underlying fluid system increases as the flow resistance reduces. This observation would be helpful to analyze the coupling effects of different fluid parameters in a convectional model in future work. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=Galerkin%20method" title="Galerkin method">Galerkin method</a>, <a href="https://publications.waset.org/abstracts/search?q=Lorenz%20equations" title=" Lorenz equations"> Lorenz equations</a>, <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=convectional%20motion" title=" convectional motion"> convectional motion</a> </p> <a href="https://publications.waset.org/abstracts/21427/new-insight-into-fluid-mechanics-of-lorenz-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/21427.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">393</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">1827</span> Comparison of Finite Difference Schemes for Numerical Study of Ripa Model</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Sidrah%20Ahmed">Sidrah Ahmed</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The river and lakes flows are modeled mathematically by shallow water equations that are depth-averaged Reynolds Averaged Navier-Stokes equations under Boussinesq approximation. The temperature stratification dynamics influence the water quality and mixing characteristics. It is mainly due to the atmospheric conditions including air temperature, wind velocity, and radiative forcing. The experimental observations are commonly taken along vertical scales and are not sufficient to estimate small turbulence effects of temperature variations induced characteristics of shallow flows. Wind shear stress over the water surface influence flow patterns, heat fluxes and thermodynamics of water bodies as well. Hence it is crucial to couple temperature gradients with shallow water model to estimate the atmospheric effects on flow patterns. The Ripa system has been introduced to study ocean currents as a variant of shallow water equations with addition of temperature variations within the flow. Ripa model is a hyperbolic system of partial differential equations because all the eigenvalues of the system’s Jacobian matrix are real and distinct. The time steps of a numerical scheme are estimated with the eigenvalues of the system. The solution to Riemann problem of the Ripa model is composed of shocks, contact and rarefaction waves. Solving Ripa model with Riemann initial data with the central schemes is difficult due to the eigen structure of the system.This works presents the comparison of four different finite difference schemes for the numerical solution of Riemann problem for Ripa model. These schemes include Lax-Friedrichs, Lax-Wendroff, MacCormack scheme and a higher order finite difference scheme with WENO method. The numerical flux functions in both dimensions are approximated according to these methods. The temporal accuracy is achieved by employing TVD Runge Kutta method. The numerical tests are presented to examine the accuracy and robustness of the applied methods. It is revealed that Lax-Freidrichs scheme produces results with oscillations while Lax-Wendroff and higher order difference scheme produce quite better results. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=finite%20difference%20schemes" title="finite difference schemes">finite difference schemes</a>, <a href="https://publications.waset.org/abstracts/search?q=Riemann%20problem" title=" Riemann problem"> Riemann problem</a>, <a href="https://publications.waset.org/abstracts/search?q=shallow%20water%20equations" title=" shallow water equations"> shallow water equations</a>, <a href="https://publications.waset.org/abstracts/search?q=temperature%20gradients" title=" temperature gradients"> temperature gradients</a> </p> <a href="https://publications.waset.org/abstracts/77732/comparison-of-finite-difference-schemes-for-numerical-study-of-ripa-model" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/77732.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">203</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">1826</span> On the Relation between λ-Symmetries and μ-Symmetries of Partial Differential Equations</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Teoman%20Ozer">Teoman Ozer</a>, <a href="https://publications.waset.org/abstracts/search?q=Ozlem%20Orhan"> Ozlem Orhan</a> </p> <p class="card-text"><strong>Abstract:</strong></p> This study deals with symmetry group properties and conservation laws of partial differential equations. We give a geometrical interpretation of notion of μ-prolongations of vector fields and of the related concept of μ-symmetry for partial differential equations. We show that these are in providing symmetry reduction of partial differential equations and systems and invariant solutions. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=%CE%BB-symmetry" title="λ-symmetry">λ-symmetry</a>, <a href="https://publications.waset.org/abstracts/search?q=%CE%BC-symmetry" title=" μ-symmetry"> μ-symmetry</a>, <a href="https://publications.waset.org/abstracts/search?q=classification" title=" classification"> classification</a>, <a href="https://publications.waset.org/abstracts/search?q=invariant%20solution" title=" invariant solution"> invariant solution</a> </p> <a href="https://publications.waset.org/abstracts/59662/on-the-relation-between-l-symmetries-and-m-symmetries-of-partial-differential-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/59662.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">319</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">1825</span> Equations of Pulse Propagation in Three-Layer Structure of As2S3 Chalcogenide Plasmonic Nano-Waveguides</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Leila%20Motamed-Jahromi">Leila Motamed-Jahromi</a>, <a href="https://publications.waset.org/abstracts/search?q=Mohsen%20Hatami"> Mohsen Hatami</a>, <a href="https://publications.waset.org/abstracts/search?q=Alireza%20Keshavarz"> Alireza Keshavarz </a> </p> <p class="card-text"><strong>Abstract:</strong></p> This research aims at obtaining the equations of pulse propagation in nonlinear plasmonic waveguides created with As<sub>2</sub>S<sub>3</sub> chalcogenide materials. Via utilizing Helmholtz equation and first-order perturbation theory, two components of electric field are determined within frequency domain. Afterwards, the equations are formulated in time domain. The obtained equations include two coupled differential equations that considers nonlinear dispersion<span dir="RTL">.</span> <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=nonlinear%20optics" title="nonlinear optics">nonlinear optics</a>, <a href="https://publications.waset.org/abstracts/search?q=plasmonic%20waveguide" title=" plasmonic waveguide"> plasmonic waveguide</a>, <a href="https://publications.waset.org/abstracts/search?q=chalcogenide" title=" chalcogenide"> chalcogenide</a>, <a href="https://publications.waset.org/abstracts/search?q=propagation%20equation" title=" propagation equation"> propagation equation</a> </p> <a href="https://publications.waset.org/abstracts/52758/equations-of-pulse-propagation-in-three-layer-structure-of-as2s3-chalcogenide-plasmonic-nano-waveguides" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/52758.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">1824</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">1823</span> Reduced Differential Transform Methods for Solving the Fractional Diffusion Equations</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Yildiray%20Keskin">Yildiray Keskin</a>, <a href="https://publications.waset.org/abstracts/search?q=Omer%20Acan"> Omer Acan</a>, <a href="https://publications.waset.org/abstracts/search?q=Murat%20Akkus"> Murat Akkus</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper, the solution of fractional diffusion equations is presented by means of the reduced differential transform method. Fractional partial differential equations have special importance in engineering and sciences. Application of reduced differential transform method to this problem shows the rapid convergence of the sequence constructed by this method to the exact solution. The numerical results show that the approach is easy to implement and accurate when applied to fractional diffusion equations. The method introduces a promising tool for solving many fractional partial differential equations. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=fractional%20diffusion%20equations" title="fractional diffusion equations">fractional diffusion equations</a>, <a href="https://publications.waset.org/abstracts/search?q=Caputo%20fractional%20derivative" title=" Caputo fractional derivative"> Caputo fractional derivative</a>, <a href="https://publications.waset.org/abstracts/search?q=reduced%20differential%20transform%20method" title=" reduced differential transform method"> reduced differential transform method</a>, <a href="https://publications.waset.org/abstracts/search?q=partial" title=" partial"> partial</a> </p> <a href="https://publications.waset.org/abstracts/17526/reduced-differential-transform-methods-for-solving-the-fractional-diffusion-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/17526.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">525</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">1822</span> Nanofluids and Hybrid Nanofluids: Comparative Study of Mixed Convection in a Round Bottom Flask</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Hicham%20Salhi">Hicham Salhi</a> </p> <p class="card-text"><strong>Abstract:</strong></p> This research project focuses on the numerical investigation of the mixed convection of Hybrid nanofluids in a round bottom flask commonly used in organic chemistry synthesis. The aim of this study is to improve the thermal properties of the reaction medium and enhance the rate of chemical reactions by using hybrid nanofluids. The flat bottom wall of the flask is maintained at a constant high temperature, while the top, left, and right walls are kept at a low temperature. The nanofluids used in this study contain suspended Cu and Al2O3 nanoparticles in pure water. The governing equations are solved numerically using the finite-volume approach and the Boussinesq approximation. The effects of the volume fraction of nanoparticles (φ) ranging from 0% to 5%, the Rayleigh number from 103 to 106, and the type of nanofluid (Cu and Al2O3) on the flow streamlines, isotherm distribution, and Nusselt number are examined in the simulation. The results indicate that the addition of Cu and Al2O3 nanoparticles increases the mean Nusselt number, which improves heat transfer and significantly alters the flow pattern. Moreover, the mean Nusselt number increases with increasing Rayleigh number and volume fraction, with Cu- Al2O3 hybrid nanofluid producing the best results. This research project focuses on the numerical investigation of the mixed convection of Hybrid nanofluids in a round bottom flask commonly used in organic chemistry synthesis. The aim of this study is to improve the thermal properties of the reaction medium and enhance the rate of chemical reactions by using hybrid nanofluids. The flat bottom wall of the flask is maintained at a constant high temperature, while the top, left, and right walls are kept at a low temperature. The nanofluids used in this study contain suspended Cu and Al2O3 nanoparticles in pure water. The governing equations are solved numerically using the finite-volume approach and the Boussinesq approximation. The effects of the volume fraction of nanoparticles (φ) ranging from 0% to 5%, the Rayleigh number from 103 to 106, and the type of nanofluid (Cu and Al2O3) on the flow streamlines, isotherm distribution, and Nusselt number are examined in the simulation. The results indicate that the addition of Cu and Al2O3 nanoparticles increases the mean Nusselt number, which improves heat transfer and significantly alters the flow pattern. Moreover, the mean Nusselt number increases with increasing Rayleigh number and volume fraction, with Cu- Al2O3 hybrid nanofluid producing the best results. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=bottom%20flask" title="bottom flask">bottom flask</a>, <a href="https://publications.waset.org/abstracts/search?q=mixed%20convection" title=" mixed convection"> mixed convection</a>, <a href="https://publications.waset.org/abstracts/search?q=hybrid%20nanofluids" title=" hybrid nanofluids"> hybrid nanofluids</a>, <a href="https://publications.waset.org/abstracts/search?q=numerical%20simulation" title=" numerical simulation"> numerical simulation</a> </p> <a href="https://publications.waset.org/abstracts/164986/nanofluids-and-hybrid-nanofluids-comparative-study-of-mixed-convection-in-a-round-bottom-flask" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/164986.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">88</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">1821</span> Serious Digital Video Game for Solving Algebraic Equations</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Liliana%20O.%20Mart%C3%ADnez">Liliana O. Martínez</a>, <a href="https://publications.waset.org/abstracts/search?q=Juan%20E%20Gonz%C3%A1lez"> Juan E González</a>, <a href="https://publications.waset.org/abstracts/search?q=Manuel%20Ram%C3%ADrez-Aranda"> Manuel Ramírez-Aranda</a>, <a href="https://publications.waset.org/abstracts/search?q=Ana%20Cervantes-Herrera"> Ana Cervantes-Herrera</a> </p> <p class="card-text"><strong>Abstract:</strong></p> A serious game category mobile application called Math Dominoes is presented. The main objective of this applications is to strengthen the teaching-learning process of solving algebraic equations and is based on the board game "Double 6" dominoes. Math Dominoes allows the practice of solving first, second-, and third-degree algebraic equations. This application is aimed to students who seek to strengthen their skills in solving algebraic equations in a dynamic, interactive, and fun way, to reduce the risk of failure in subsequent courses that require mastery of this algebraic tool. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=algebra" title="algebra">algebra</a>, <a href="https://publications.waset.org/abstracts/search?q=equations" title=" equations"> equations</a>, <a href="https://publications.waset.org/abstracts/search?q=dominoes" title=" dominoes"> dominoes</a>, <a href="https://publications.waset.org/abstracts/search?q=serious%20games" title=" serious games"> serious games</a> </p> <a href="https://publications.waset.org/abstracts/156059/serious-digital-video-game-for-solving-algebraic-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/156059.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">130</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">1820</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">1819</span> Double Diffusive Natural Convection in Horizontal Elliptical Annulus Containing a Fluid-Saturated Porous Medium: Effects of Lewis Number</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Hichem%20Boulechfar">Hichem Boulechfar</a>, <a href="https://publications.waset.org/abstracts/search?q=Mahfoud%20Djezzar"> Mahfoud Djezzar</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Two-dimensional double diffusive natural convection in an annular elliptical space filled with fluid-saturated porous medium, is analyzed by solving numerically the mass balance, momentum, energy and concentration equations, using Darcy's law and Boussinesq approximation. Both walls delimiting the annular space are maintained at two uniform different temperatures and concentrations. The external parameter considered is the Lewis number. For the present work, the heat and mass transfer for natural convection is studied for the case of aiding buoyancies, where the flow is generated in a cooperative mode by both temperature and solutal gradients. The local Nusselt and Sherwood numbers are presented in term of the external parameter. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=double%20diffusive" title="double diffusive">double diffusive</a>, <a href="https://publications.waset.org/abstracts/search?q=natural%20convection" title=" natural convection"> natural convection</a>, <a href="https://publications.waset.org/abstracts/search?q=porous%20media" title=" porous media"> porous media</a>, <a href="https://publications.waset.org/abstracts/search?q=elliptical%20annulus" title=" elliptical annulus"> elliptical annulus</a> </p> <a href="https://publications.waset.org/abstracts/38246/double-diffusive-natural-convection-in-horizontal-elliptical-annulus-containing-a-fluid-saturated-porous-medium-effects-of-lewis-number" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/38246.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">210</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">1818</span> Solutions of Fractional Reaction-Diffusion Equations Used to Model the Growth and Spreading of Biological Species</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> Reaction-diffusion equations are commonly used in population biology to model the spread of biological species. In this paper, we propose a fractional reaction-diffusion equation, where the classical second derivative diffusion term is replaced by a fractional derivative of order less than two. Based on the symbolic computation system Mathematica, Adomian decomposition method, developed for fractional differential equations, is directly extended to derive explicit and numerical solutions of space fractional reaction-diffusion equations. The fractional derivative is described in the Caputo sense. Finally, the recent appearance of fractional reaction-diffusion equations as models in some ﬁelds such as cell biology, chemistry, physics, and ﬁnance, makes it necessary to apply the results reported here to some numerical examples. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=fractional%20partial%20differential%20equations" title="fractional partial differential equations">fractional partial differential equations</a>, <a href="https://publications.waset.org/abstracts/search?q=reaction-di%EF%AC%80usion%20equations" title=" reaction-diﬀusion equations"> reaction-diﬀusion equations</a>, <a href="https://publications.waset.org/abstracts/search?q=adomian%20decomposition" title=" adomian decomposition"> adomian decomposition</a>, <a href="https://publications.waset.org/abstracts/search?q=biological%20species" title=" biological species"> biological species</a> </p> <a href="https://publications.waset.org/abstracts/55994/solutions-of-fractional-reaction-diffusion-equations-used-to-model-the-growth-and-spreading-of-biological-species" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/55994.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">375</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">1817</span> Natural Convection in Wavy-Wall Cavities Filled with Power-Law Fluid</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Cha%E2%80%99o-Kuang%20Chen">Cha’o-Kuang Chen</a>, <a href="https://publications.waset.org/abstracts/search?q=Ching-Chang%20Cho"> Ching-Chang Cho</a> </p> <p class="card-text"><strong>Abstract:</strong></p> This paper investigates the natural convection heat transfer performance in a complex-wavy-wall cavity filled with power-law fluid. In performing the simulations, the continuity, Cauchy momentum and energy equations are solved subject to the Boussinesq approximation using a finite volume method. The simulations focus specifically on the effects of the flow behavior index in the power-law model and the Rayleigh number on the flow streamlines, isothermal contours and mean Nusselt number within the cavity. The results show that pseudoplastic fluids have a better heat transfer performance than Newtonian or dilatant fluids. Moreover, it is shown that for Rayleigh numbers greater than Ra=103, the mean Nusselt number has a significantly increase as the flow behavior index is decreased. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=non-Newtonian%20fluid" title="non-Newtonian fluid">non-Newtonian fluid</a>, <a href="https://publications.waset.org/abstracts/search?q=power-law%20fluid" title=" power-law fluid"> power-law fluid</a>, <a href="https://publications.waset.org/abstracts/search?q=natural%20convection" title=" natural convection"> natural convection</a>, <a href="https://publications.waset.org/abstracts/search?q=heat%20transfer%20enhancement" title=" heat transfer enhancement"> heat transfer enhancement</a>, <a href="https://publications.waset.org/abstracts/search?q=cavity" title=" cavity"> cavity</a>, <a href="https://publications.waset.org/abstracts/search?q=wavy%20wall" title=" wavy wall"> wavy wall</a> </p> <a href="https://publications.waset.org/abstracts/6789/natural-convection-in-wavy-wall-cavities-filled-with-power-law-fluid" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/6789.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">266</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">1816</span> Numerical Solution for Integro-Differential Equations by Using Quartic B-Spline Wavelet and Operational Matrices</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Khosrow%20Maleknejad">Khosrow Maleknejad</a>, <a href="https://publications.waset.org/abstracts/search?q=Yaser%20Rostami"> Yaser Rostami</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper, semi-orthogonal B-spline scaling functions and wavelets and their dual functions are presented to approximate the solutions of integro-differential equations.The B-spline scaling functions and wavelets, their properties and the operational matrices of derivative for this function are presented to reduce the solution of integro-differential equations to the solution of algebraic equations. Here we compute B-spline scaling functions of degree 4 and their dual, then we will show that by using them we have better approximation results for the solution of integro-differential equations in comparison with less degrees of scaling functions. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=%C4%B1ntegro-differential%20equations" title="ıntegro-differential equations">ıntegro-differential equations</a>, <a href="https://publications.waset.org/abstracts/search?q=quartic%20B-spline%20wavelet" title=" quartic B-spline wavelet"> quartic B-spline wavelet</a>, <a href="https://publications.waset.org/abstracts/search?q=operational%20matrices" title=" operational matrices"> operational matrices</a>, <a href="https://publications.waset.org/abstracts/search?q=dual%20functions" title=" dual functions"> dual functions</a> </p> <a href="https://publications.waset.org/abstracts/5002/numerical-solution-for-integro-differential-equations-by-using-quartic-b-spline-wavelet-and-operational-matrices" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/5002.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">456</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">1815</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">1814</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 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