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solution of the Navier-Stokes equations</title> <meta name="description" content="Search results for: numerical solution of the Navier-Stokes equations"> <meta name="keywords" content="numerical solution of the Navier-Stokes 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 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</div> </div> <h1 class="mt-3 mb-3 text-center" style="font-size:1.6rem;">Search results for: numerical solution of the Navier-Stokes equations</h1> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">9668</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">425</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">9667</span> Inverse Polynomial Numerical Scheme for the Solution of Initial Value Problems in Ordinary Differential Equations </h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Ogunrinde%20Roseline%20Bosede">Ogunrinde Roseline Bosede</a> </p> <p class="card-text"><strong>Abstract:</strong></p> This paper presents the development, analysis and implementation of an inverse polynomial numerical method which is well suitable for solving initial value problems in first order ordinary differential equations with applications to sample problems. We also present some basic concepts and fundamental theories which are vital to the analysis of the scheme. We analyzed the consistency, convergence, and stability properties of the scheme. Numerical experiments were carried out and the results compared with the theoretical or exact solution and the algorithm was later coded using MATLAB programming language. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=differential%20equations" title="differential equations">differential equations</a>, <a href="https://publications.waset.org/abstracts/search?q=numerical" title=" numerical"> numerical</a>, <a href="https://publications.waset.org/abstracts/search?q=polynomial" title=" polynomial"> polynomial</a>, <a href="https://publications.waset.org/abstracts/search?q=initial%20value%20problem" title=" initial value problem"> initial value problem</a>, <a href="https://publications.waset.org/abstracts/search?q=differential%20equation" title=" differential equation"> differential equation</a> </p> <a href="https://publications.waset.org/abstracts/23505/inverse-polynomial-numerical-scheme-for-the-solution-of-initial-value-problems-in-ordinary-differential-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/23505.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">447</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">9666</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">9665</span> Solution of Hybrid Fuzzy Differential Equations</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Mahmood%20Otadi">Mahmood Otadi</a>, <a href="https://publications.waset.org/abstracts/search?q=Maryam%20Mosleh"> Maryam Mosleh</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The hybrid differential equations have a wide range of applications in science and engineering. In this paper, the homotopy analysis method (HAM) is applied to obtain the series solution of the hybrid differential equations. Using the homotopy analysis method, it is possible to find the exact solution or an approximate solution of the problem. Comparisons are made between improved predictor-corrector method, homotopy analysis method and the exact solution. Finally, we illustrate our approach by some numerical example. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=fuzzy%20number" title="fuzzy number">fuzzy number</a>, <a href="https://publications.waset.org/abstracts/search?q=fuzzy%20ODE" title=" fuzzy ODE"> fuzzy ODE</a>, <a href="https://publications.waset.org/abstracts/search?q=HAM" title=" HAM"> HAM</a>, <a href="https://publications.waset.org/abstracts/search?q=approximate%20method" title=" approximate method"> approximate method</a> </p> <a href="https://publications.waset.org/abstracts/31754/solution-of-hybrid-fuzzy-differential-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/31754.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">511</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">9664</span> Impact of the Time Interval in the Numerical Solution of Incompressible Flows</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=M.%20Salmanzadeh">M. Salmanzadeh</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In paper, we will deal with incompressible Couette flow, which represents an exact analytical solution of the Navier-Stokes equations. Couette flow is perhaps the simplest of all viscous flows, while at the same time retaining much of the same physical characteristics of a more complicated boundary-layer flow. The numerical technique that we will employ for the solution of the Couette flow is the Crank-Nicolson implicit method. Parabolic partial differential equations lend themselves to a marching solution; in addition, the use of an implicit technique allows a much larger marching step size than would be the case for an explicit solution. Hence, in the present paper we will have the opportunity to explore some aspects of CFD different from those discussed in the other papers. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=incompressible%20couette%20flow" title="incompressible couette flow">incompressible couette flow</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=partial%20differential%20equation" title=" partial differential equation"> partial differential equation</a>, <a href="https://publications.waset.org/abstracts/search?q=Crank-Nicolson%20implicit" title=" Crank-Nicolson implicit"> Crank-Nicolson implicit</a> </p> <a href="https://publications.waset.org/abstracts/23787/impact-of-the-time-interval-in-the-numerical-solution-of-incompressible-flows" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/23787.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">536</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">9663</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">9662</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">9661</span> Numerical Treatment of Block Method for the Solution of Ordinary Differential Equations</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=A.%20M.%20Sagir">A. M. Sagir</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Discrete linear multistep block method of uniform order for the solution of first order Initial Value Problems (IVPs) in Ordinary Differential Equations (ODEs) is presented in this paper. The approach of interpolation and collocation approximation are adopted in the derivation of the method which is then applied to first order ordinary differential equations with associated initial conditions. The continuous hybrid formulations enable us to differentiate and evaluate at some grids and off – grid points to obtain four discrete schemes, which were used in block form for parallel or sequential solutions of the problems. Furthermore, a stability analysis and efficiency of the block method are tested on ordinary differential equations, and the results obtained compared favorably with the exact solution. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=block%20method" title="block method">block method</a>, <a href="https://publications.waset.org/abstracts/search?q=first%20order%20ordinary%20differential%20equations" title=" first order ordinary differential equations"> first order ordinary differential equations</a>, <a href="https://publications.waset.org/abstracts/search?q=hybrid" title=" hybrid"> hybrid</a>, <a href="https://publications.waset.org/abstracts/search?q=self-starting" title=" self-starting "> self-starting </a> </p> <a href="https://publications.waset.org/abstracts/3426/numerical-treatment-of-block-method-for-the-solution-of-ordinary-differential-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/3426.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">482</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">9660</span> On the Approximate Solution of Continuous Coefficients for Solving Third Order Ordinary Differential Equations</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=A.%20M.%20Sagir">A. M. Sagir</a> </p> <p class="card-text"><strong>Abstract:</strong></p> This paper derived four newly schemes which are combined in order to form an accurate and efficient block method for parallel or sequential solution of third order ordinary differential equations of the form y^'''= f(x,y,y^',y^'' ), y(α)=y_0,〖y〗^' (α)=β,y^('' ) (α)=μ with associated initial or boundary conditions. The implementation strategies of the derived method have shown that the block method is found to be consistent, zero stable and hence convergent. The derived schemes were tested on stiff and non-stiff ordinary differential equations, and the numerical results obtained compared favorably with the exact solution. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=block%20method" title="block method">block method</a>, <a href="https://publications.waset.org/abstracts/search?q=hybrid" title=" hybrid"> hybrid</a>, <a href="https://publications.waset.org/abstracts/search?q=linear%20multistep" title=" linear multistep"> linear multistep</a>, <a href="https://publications.waset.org/abstracts/search?q=self-starting" title=" self-starting"> self-starting</a>, <a href="https://publications.waset.org/abstracts/search?q=third%20order%20ordinary%20differential%20equations" title=" third order ordinary differential equations"> third order ordinary differential equations</a> </p> <a href="https://publications.waset.org/abstracts/3659/on-the-approximate-solution-of-continuous-coefficients-for-solving-third-order-ordinary-differential-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/3659.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">271</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">9659</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">9658</span> Approximations of Fractional Derivatives and Its Applications in Solving Non-Linear Fractional Variational Problems</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Harendra%20Singh">Harendra Singh</a>, <a href="https://publications.waset.org/abstracts/search?q=Rajesh%20Pandey"> Rajesh Pandey</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The paper presents a numerical method based on operational matrix of integration and Ryleigh method for the solution of a class of non-linear fractional variational problems (NLFVPs). Chebyshev first kind polynomials are used for the construction of operational matrix. Using operational matrix and Ryleigh method the NLFVP is converted into a system of non-linear algebraic equations, and solving these equations we obtained approximate solution for NLFVPs. Convergence analysis of the proposed method is provided. Numerical experiment is done to show the applicability of the proposed numerical method. The obtained numerical results are compared with exact solution and solution obtained from Chebyshev third kind. Further the results are shown graphically for different fractional order involved in the problems. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=non-linear%20fractional%20variational%20problems" title="non-linear fractional variational problems">non-linear fractional variational problems</a>, <a href="https://publications.waset.org/abstracts/search?q=Rayleigh-Ritz%20method" title=" Rayleigh-Ritz method"> Rayleigh-Ritz method</a>, <a href="https://publications.waset.org/abstracts/search?q=convergence%20analysis" title=" convergence analysis"> convergence analysis</a>, <a href="https://publications.waset.org/abstracts/search?q=error%20analysis" title=" error analysis"> error analysis</a> </p> <a href="https://publications.waset.org/abstracts/57497/approximations-of-fractional-derivatives-and-its-applications-in-solving-non-linear-fractional-variational-problems" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/57497.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">298</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">9657</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">134</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">9656</span> A Series Solution of Fuzzy Integro-Differential Equation</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Maryam%20Mosleh">Maryam Mosleh</a>, <a href="https://publications.waset.org/abstracts/search?q=Mahmood%20Otadi"> Mahmood Otadi</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The hybrid differential equations have a wide range of applications in science and engineering. In this paper, the homotopy analysis method (HAM) is applied to obtain the series solution of the hybrid differential equations. Using the homotopy analysis method, it is possible to find the exact solution or an approximate solution of the problem. Comparisons are made between improved predictor-corrector method, homotopy analysis method and the exact solution. Finally, we illustrate our approach by some numerical example. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=Fuzzy%20number" title="Fuzzy number">Fuzzy number</a>, <a href="https://publications.waset.org/abstracts/search?q=parametric%20form%20of%20a%20fuzzy%20number" title=" parametric form of a fuzzy number"> parametric form of a fuzzy number</a>, <a href="https://publications.waset.org/abstracts/search?q=fuzzy%20integrodifferential%20equation" title=" fuzzy integrodifferential equation"> fuzzy integrodifferential equation</a>, <a href="https://publications.waset.org/abstracts/search?q=homotopy%20analysis%20method" title=" homotopy analysis method"> homotopy analysis method</a> </p> <a href="https://publications.waset.org/abstracts/31775/a-series-solution-of-fuzzy-integro-differential-equation" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/31775.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">557</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">9655</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">9654</span> A Numerical Solution Based on Operational Matrix of Differentiation of Shifted Second Kind Chebyshev Wavelets for a Stefan Problem</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Rajeev">Rajeev</a>, <a href="https://publications.waset.org/abstracts/search?q=N.%20K.%20Raigar"> N. K. Raigar</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this study, one dimensional phase change problem (a Stefan problem) is considered and a numerical solution of this problem is discussed. First, we use similarity transformation to convert the governing equations into ordinary differential equations with its boundary conditions. The solutions of ordinary differential equation with the associated boundary conditions and interface condition (Stefan condition) are obtained by using a numerical approach based on operational matrix of differentiation of shifted second kind Chebyshev wavelets. The obtained results are compared with existing exact solution which is sufficiently accurate. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=operational%20matrix%20of%20differentiation" title="operational matrix of differentiation">operational matrix of differentiation</a>, <a href="https://publications.waset.org/abstracts/search?q=similarity%20transformation" title=" similarity transformation"> similarity transformation</a>, <a href="https://publications.waset.org/abstracts/search?q=shifted%20second%20kind%20chebyshev%20wavelets" title=" shifted second kind chebyshev wavelets"> shifted second kind chebyshev wavelets</a>, <a href="https://publications.waset.org/abstracts/search?q=stefan%20problem" title=" stefan problem"> stefan problem</a> </p> <a href="https://publications.waset.org/abstracts/30738/a-numerical-solution-based-on-operational-matrix-of-differentiation-of-shifted-second-kind-chebyshev-wavelets-for-a-stefan-problem" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/30738.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">403</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">9653</span> A Numerical Method for Diffusion and Cahn-Hilliard Equations on Evolving Spherical Surfaces</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Jyh-Yang%20Wu">Jyh-Yang Wu</a>, <a href="https://publications.waset.org/abstracts/search?q=Sheng-Gwo%20Chen"> Sheng-Gwo Chen</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper, we present a simple effective numerical geometric method to estimate the divergence of a vector field over a curved surface. The conservation law is an important principle in physics and mathematics. However, many well-known numerical methods for solving diffusion equations do not obey conservation laws. Our presented method in this paper combines the divergence theorem with a generalized finite difference method and obeys the conservation law on discrete closed surfaces. We use the similar method to solve the Cahn-Hilliard equations on evolving spherical surfaces and observe stability results in our numerical simulations. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=conservation%20laws" title="conservation laws">conservation laws</a>, <a href="https://publications.waset.org/abstracts/search?q=diffusion%20equations" title=" diffusion equations"> diffusion equations</a>, <a href="https://publications.waset.org/abstracts/search?q=Cahn-Hilliard%20equations" title=" Cahn-Hilliard equations"> Cahn-Hilliard equations</a>, <a href="https://publications.waset.org/abstracts/search?q=evolving%20surfaces" title=" evolving surfaces"> evolving surfaces</a> </p> <a href="https://publications.waset.org/abstracts/56432/a-numerical-method-for-diffusion-and-cahn-hilliard-equations-on-evolving-spherical-surfaces" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/56432.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">494</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">9652</span> On a Continuous Formulation of Block Method for Solving First Order Ordinary Differential Equations (ODEs)</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=A.%20M.%20Sagir">A. M. Sagir</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The aim of this paper is to investigate the performance of the developed linear multistep block method for solving first order initial value problem of Ordinary Differential Equations (ODEs). The method calculates the numerical solution at three points simultaneously and produces three new equally spaced solution values within a block. The continuous formulations enable us to differentiate and evaluate at some selected points to obtain three discrete schemes, which were used in block form for parallel or sequential solutions of the problems. A stability analysis and efficiency of the block method are tested on ordinary differential equations involving practical applications, and the results obtained compared favorably with the exact solution. Furthermore, comparison of error analysis has been developed with the help of computer software. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=block%20method" title="block method">block method</a>, <a href="https://publications.waset.org/abstracts/search?q=first%20order%20ordinary%20differential%20equations" title=" first order ordinary differential equations"> first order ordinary differential equations</a>, <a href="https://publications.waset.org/abstracts/search?q=linear%20multistep" title=" linear multistep"> linear multistep</a>, <a href="https://publications.waset.org/abstracts/search?q=self-starting" title=" self-starting"> self-starting</a> </p> <a href="https://publications.waset.org/abstracts/3622/on-a-continuous-formulation-of-block-method-for-solving-first-order-ordinary-differential-equations-odes" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/3622.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">306</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">9651</span> Convergence of Sinc Methods Applied to Kuramoto-Sivashinsky Equation</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> A comparative study of the Sinc-Galerkin and Sinc-Collocation methods for solving the Kuramoto-Sivashinsky equation is given. Both approaches depend on using Sinc basis functions. Firstly, a numerical scheme using Sinc-Galerkin method is developed to approximate the solution of Kuramoto-Sivashinsky equation. Sinc approximations to both derivatives and indefinite integrals reduces the solution to an explicit system of algebraic equations. The error in the solution is shown to converge to the exact solution at an exponential. The convergence proof of the solution for the discrete system is given using fixed-point iteration. Secondly, a combination of a Crank-Nicolson formula in the time direction, with the Sinc-collocation in the space direction is presented, where the derivatives in the space variable are replaced by the necessary matrices to produce a system of algebraic equations. The methods are tested on two examples. The demonstrated results show that both of the presented methods more or less have the same accuracy. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=Sinc-Collocation" title="Sinc-Collocation">Sinc-Collocation</a>, <a href="https://publications.waset.org/abstracts/search?q=nonlinear%20PDEs" title=" nonlinear PDEs"> nonlinear PDEs</a>, <a href="https://publications.waset.org/abstracts/search?q=numerical%20methods" title=" numerical methods"> numerical methods</a>, <a href="https://publications.waset.org/abstracts/search?q=fixed-point" title=" fixed-point"> fixed-point</a> </p> <a href="https://publications.waset.org/abstracts/9717/convergence-of-sinc-methods-applied-to-kuramoto-sivashinsky-equation" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/9717.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">9650</span> Comparing Numerical Accuracy of Solutions of Ordinary Differential Equations (ODE) Using Taylor's Series Method, Euler's Method and Runge-Kutta (RK) Method</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Palwinder%20Singh">Palwinder Singh</a>, <a href="https://publications.waset.org/abstracts/search?q=Munish%20Sandhir"> Munish Sandhir</a>, <a href="https://publications.waset.org/abstracts/search?q=Tejinder%20Singh"> Tejinder Singh</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The ordinary differential equations (ODE) represent a natural framework for mathematical modeling of many real-life situations in the field of engineering, control systems, physics, chemistry and astronomy etc. Such type of differential equations can be solved by analytical methods or by numerical methods. If the solution is calculated using analytical methods, it is done through calculus theories, and thus requires a longer time to solve. In this paper, we compare the numerical accuracy of the solutions given by the three main types of one-step initial value solvers: Taylor’s Series Method, Euler’s Method and Runge-Kutta Fourth Order Method (RK4). The comparison of accuracy is obtained through comparing the solutions of ordinary differential equation given by these three methods. Furthermore, to verify the accuracy; we compare these numerical solutions with the exact solutions. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=Ordinary%20differential%20equations%20%28ODE%29" title="Ordinary differential equations (ODE)">Ordinary differential equations (ODE)</a>, <a href="https://publications.waset.org/abstracts/search?q=Taylor%E2%80%99s%20Series%20Method" title=" Taylor’s Series Method"> Taylor’s Series Method</a>, <a href="https://publications.waset.org/abstracts/search?q=Euler%E2%80%99s%20Method" title=" Euler’s Method"> Euler’s Method</a>, <a href="https://publications.waset.org/abstracts/search?q=Runge-Kutta%20Fourth%20Order%20Method" title=" Runge-Kutta Fourth Order Method"> Runge-Kutta Fourth Order Method</a> </p> <a href="https://publications.waset.org/abstracts/56685/comparing-numerical-accuracy-of-solutions-of-ordinary-differential-equations-ode-using-taylors-series-method-eulers-method-and-runge-kutta-rk-method" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/56685.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">358</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">9649</span> Numerical Solution of Porous Media Equation Using Jacobi Operational Matrix</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Shubham%20Jaiswal">Shubham Jaiswal</a> </p> <p class="card-text"><strong>Abstract:</strong></p> During modeling of transport phenomena in porous media, many nonlinear partial differential equations (NPDEs) encountered which greatly described the convection, diffusion and reaction process. To solve such types of nonlinear problems, a reliable and efficient technique is needed. In this article, the numerical solution of NPDEs encountered in porous media is derived. Here Jacobi collocation method is used to solve the considered problems which convert the NPDEs in systems of nonlinear algebraic equations that can be solved using Newton-Raphson method. The numerical results of some illustrative examples are reported to show the efficiency and high accuracy of the proposed approach. The comparison of the numerical results with the existing analytical results already reported in the literature and the error analysis for each example exhibited through graphs and tables confirms the exponential convergence rate of the proposed method. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=nonlinear%20porous%20media%20equation" title="nonlinear porous media equation">nonlinear porous media equation</a>, <a href="https://publications.waset.org/abstracts/search?q=shifted%20Jacobi%20polynomials" title=" shifted Jacobi polynomials"> shifted Jacobi polynomials</a>, <a href="https://publications.waset.org/abstracts/search?q=operational%20matrix" title=" operational matrix"> operational matrix</a>, <a href="https://publications.waset.org/abstracts/search?q=spectral%20collocation%20method" title=" spectral collocation method"> spectral collocation method</a> </p> <a href="https://publications.waset.org/abstracts/80603/numerical-solution-of-porous-media-equation-using-jacobi-operational-matrix" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/80603.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">9648</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">9647</span> A Family of Second Derivative Methods for Numerical Integration of Stiff Initial Value Problems in Ordinary Differential Equations</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Luke%20Ukpebor">Luke Ukpebor</a>, <a href="https://publications.waset.org/abstracts/search?q=C.%20E.%20Abhulimen"> C. E. Abhulimen</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Stiff initial value problems in ordinary differential equations are problems for which a typical solution is rapidly decaying exponentially, and their numerical investigations are very tedious. Conventional numerical integration solvers cannot cope effectively with stiff problems as they lack adequate stability characteristics. In this article, we developed a new family of four-step second derivative exponentially fitted method of order six for the numerical integration of stiff initial value problem of general first order differential equations. In deriving our method, we employed the idea of breaking down the general multi-derivative multistep method into predator and corrector schemes which possess free parameters that allow for automatic fitting into exponential functions. The stability analysis of the method was discussed and the method was implemented with numerical examples. The result shows that the method is A-stable and competes favorably with existing methods in terms of efficiency and accuracy. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=A-stable" title="A-stable">A-stable</a>, <a href="https://publications.waset.org/abstracts/search?q=exponentially%20fitted" title=" exponentially fitted"> exponentially fitted</a>, <a href="https://publications.waset.org/abstracts/search?q=four%20step" title=" four step"> four step</a>, <a href="https://publications.waset.org/abstracts/search?q=predator-corrector" title=" predator-corrector"> predator-corrector</a>, <a href="https://publications.waset.org/abstracts/search?q=second%20derivative" title=" second derivative"> second derivative</a>, <a href="https://publications.waset.org/abstracts/search?q=stiff%20initial%20value%20problems" title=" stiff initial value problems"> stiff initial value problems</a> </p> <a href="https://publications.waset.org/abstracts/73388/a-family-of-second-derivative-methods-for-numerical-integration-of-stiff-initial-value-problems-in-ordinary-differential-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/73388.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">258</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">9646</span> Numerical Modeling of Wave Run-Up in Shallow Water Flows Using Moving Wet/Dry Interfaces</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Alia%20Alghosoun">Alia Alghosoun</a>, <a href="https://publications.waset.org/abstracts/search?q=Michael%20Herty"> Michael Herty</a>, <a href="https://publications.waset.org/abstracts/search?q=Mohammed%20Seaid"> Mohammed Seaid</a> </p> <p class="card-text"><strong>Abstract:</strong></p> We present a new class of numerical techniques to solve shallow water flows over dry areas including run-up. Many recent investigations on wave run-up in coastal areas are based on the well-known shallow water equations. Numerical simulations have also performed to understand the effects of several factors on tsunami wave impact and run-up in the presence of coastal areas. In all these simulations the shallow water equations are solved in entire domain including dry areas and special treatments are used for numerical solution of singularities at these dry regions. In the present study we propose a new method to deal with these difficulties by reformulating the shallow water equations into a new system to be solved only in the wetted domain. The system is obtained by a change in the coordinates leading to a set of equations in a moving domain for which the wet/dry interface is the reconstructed using the wave speed. To solve the new system we present a finite volume method of Lax-Friedrich type along with a modified method of characteristics. The method is well-balanced and accurately resolves dam-break problems over dry areas. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=dam-break%20problems" title="dam-break problems">dam-break problems</a>, <a href="https://publications.waset.org/abstracts/search?q=finite%20volume%20method" title=" finite volume method"> finite volume method</a>, <a href="https://publications.waset.org/abstracts/search?q=run-up%20waves" title=" run-up waves"> run-up waves</a>, <a href="https://publications.waset.org/abstracts/search?q=shallow%20water%20flows" title=" shallow water flows"> shallow water flows</a>, <a href="https://publications.waset.org/abstracts/search?q=wet%2Fdry%20interfaces" title=" wet/dry interfaces"> wet/dry interfaces</a> </p> <a href="https://publications.waset.org/abstracts/72559/numerical-modeling-of-wave-run-up-in-shallow-water-flows-using-moving-wetdry-interfaces" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/72559.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">145</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">9645</span> A Hybrid Block Multistep Method for Direct Numerical Integration of Fourth Order Initial Value Problems</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Adamu%20S.%20Salawu">Adamu S. Salawu</a>, <a href="https://publications.waset.org/abstracts/search?q=Ibrahim%20O.%20Isah"> Ibrahim O. Isah</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Direct solution to several forms of fourth-order ordinary differential equations is not easily obtained without first reducing them to a system of first-order equations. Thus, numerical methods are being developed with the underlying techniques in the literature, which seeks to approximate some classes of fourth-order initial value problems with admissible error bounds. Multistep methods present a great advantage of the ease of implementation but with a setback of several functions evaluation for every stage of implementation. However, hybrid methods conventionally show a slightly higher order of truncation for any k-step linear multistep method, with the possibility of obtaining solutions at off mesh points within the interval of solution. In the light of the foregoing, we propose the continuous form of a hybrid multistep method with Chebyshev polynomial as a basis function for the numerical integration of fourth-order initial value problems of ordinary differential equations. The basis function is interpolated and collocated at some points on the interval [0, 2] to yield a system of equations, which is solved to obtain the unknowns of the approximating polynomial. The continuous form obtained, its first and second derivatives are evaluated at carefully chosen points to obtain the proposed block method needed to directly approximate fourth-order initial value problems. The method is analyzed for convergence. Implementation of the method is done by conducting numerical experiments on some test problems. The outcome of the implementation of the method suggests that the method performs well on problems with oscillatory or trigonometric terms since the approximations at several points on the solution domain did not deviate too far from the theoretical solutions. The method also shows better performance compared with an existing hybrid method when implemented on a larger interval of solution. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=Chebyshev%20polynomial" title="Chebyshev polynomial">Chebyshev polynomial</a>, <a href="https://publications.waset.org/abstracts/search?q=collocation" title=" collocation"> collocation</a>, <a href="https://publications.waset.org/abstracts/search?q=hybrid%20multistep%20method" title=" hybrid multistep method"> hybrid multistep method</a>, <a href="https://publications.waset.org/abstracts/search?q=initial%20value%20problems" title=" initial value problems"> initial value problems</a>, <a href="https://publications.waset.org/abstracts/search?q=interpolation" title=" interpolation"> interpolation</a> </p> <a href="https://publications.waset.org/abstracts/113723/a-hybrid-block-multistep-method-for-direct-numerical-integration-of-fourth-order-initial-value-problems" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/113723.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">122</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">9644</span> Simulation of Turbulent Flow in Channel Using Generalized Hydrodynamic Equations</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Alex%20Fedoseyev">Alex Fedoseyev</a> </p> <p class="card-text"><strong>Abstract:</strong></p> This study explores Generalized Hydrodynamic Equations (GHE) for the simulation of turbulent flows. The GHE was derived from the Generalized Boltzmann Equation (GBE) by Alexeev (1994). GBE was obtained by first principles from the chain of Bogolubov kinetic equations and considered particles of finite dimensions, Alexeev (1994). The GHE has new terms, temporal and spatial fluctuations compared to the Navier-Stokes equations (NSE). These new terms have a timescale multiplier τ, and the GHE becomes the NSE when τ is zero. The nondimensional τ is a product of the Reynolds number and the squared length scale ratio, τ=Re*(l/L)², where l is the apparent Kolmogorov length scale, and L is a hydrodynamic length scale. The turbulence phenomenon is not well understood and is not described by NSE. An additional one or two equations are required for the turbulence model, which may have to be tuned for specific problems. We show that, in the case of the GHE, no additional turbulence model is needed, and the turbulent velocity profile is obtained from the GHE. The 2D turbulent channel and circular pipe flows were investigated using a numerical solution of the GHE for several cases. The solutions are compared with the experimental data in the circular pipes and 2D channels by Nicuradse (1932, Prandtl Lab), Hussain and Reynolds (1975), Wei and Willmarth (1989), Van Doorne (2007), theory by Wosnik, Castillo and George (2000), and the relevant experiments on Superpipe setup at Princeton, data by Zagarola (1996) and Zagarola and Smits (1998), the Reynolds number is from Re=7200 to Re=960000. The numerical solution data compared well with the experimental data, as well as with the approximate analytical solution for turbulent flow in channel Fedoseyev (2023). The obtained results confirm that the Alexeev generalized hydrodynamic theory (GHE) is in good agreement with the experiments for turbulent flows. The proposed approach is limited to 2D and 3D axisymmetric channel geometries. Further work will extend this approach by including channels with square and rectangular cross-sections. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=comparison%20with%20experimental%20data.%20generalized%20hydrodynamic%20equations" title="comparison with experimental data. generalized hydrodynamic equations">comparison with experimental data. generalized hydrodynamic equations</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=turbulent%20boundary%20layer" title=" turbulent boundary layer"> turbulent boundary layer</a>, <a href="https://publications.waset.org/abstracts/search?q=turbulent%20flow%20in%20channel" title=" turbulent flow in channel"> turbulent flow in channel</a> </p> <a href="https://publications.waset.org/abstracts/179060/simulation-of-turbulent-flow-in-channel-using-generalized-hydrodynamic-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/179060.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">65</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">9643</span> Study and Solving High Complex Non-Linear Differential Equations Applied in the Engineering Field by Analytical New Approach AGM </h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Mohammadreza%20Akbari">Mohammadreza Akbari</a>, <a href="https://publications.waset.org/abstracts/search?q=Sara%20Akbari"> Sara Akbari</a>, <a href="https://publications.waset.org/abstracts/search?q=Davood%20Domiri%20Ganji"> Davood Domiri Ganji</a>, <a href="https://publications.waset.org/abstracts/search?q=Pooya%20Solimani"> Pooya Solimani</a>, <a href="https://publications.waset.org/abstracts/search?q=Reza%20Khalili"> Reza Khalili</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper, three complicated nonlinear differential equations(PDE,ODE) in the field of engineering and non-vibration have been analyzed and solved completely by new method that we have named it Akbari-Ganji's Method (AGM) . As regards the previous published papers, investigating this kind of equations is a very hard task to do and the obtained solution is not accurate and reliable. This issue will be emerged after comparing the achieved solutions by Numerical Method. Based on the comparisons which have been made between the gained solutions by AGM and Numerical Method (Runge-Kutta 4th), it is possible to indicate that AGM can be successfully applied for various differential equations particularly for difficult ones. Furthermore, It is necessary to mention that a summary of the excellence of this method in comparison with the other approaches can be considered as follows: It is noteworthy that these results have been indicated that this approach is very effective and easy therefore it can be applied for other kinds of nonlinear equations, And also the reasons of selecting the mentioned method for solving differential equations in a wide variety of fields not only in vibrations but also in different fields of sciences such as fluid mechanics, solid mechanics, chemical engineering, etc. Therefore, a solution with high precision will be acquired. With regard to the afore-mentioned explanations, the process of solving nonlinear equation(s) will be very easy and convenient in comparison with the other methods. And also one of the important position that is explored in this paper is: Trigonometric and exponential terms in the differential equation (the method AGM) , is no need to use Taylor series Expansion to enhance the precision of the result. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=new%20method%20%28AGM%29" title="new method (AGM)">new method (AGM)</a>, <a href="https://publications.waset.org/abstracts/search?q=complex%20non-linear%20partial%20differential%20equations" title=" complex non-linear partial differential equations"> complex non-linear partial differential equations</a>, <a href="https://publications.waset.org/abstracts/search?q=damping%20ratio" title=" damping ratio"> damping ratio</a>, <a href="https://publications.waset.org/abstracts/search?q=energy%20lost%20per%20cycle" title=" energy lost per cycle"> energy lost per cycle</a> </p> <a href="https://publications.waset.org/abstracts/36424/study-and-solving-high-complex-non-linear-differential-equations-applied-in-the-engineering-field-by-analytical-new-approach-agm" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/36424.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">469</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">9642</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">9641</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">9640</span> Numerical Solution of Space Fractional Order Linear/Nonlinear Reaction-Advection Diffusion Equation Using Jacobi Polynomial</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Shubham%20Jaiswal">Shubham Jaiswal</a> </p> <p class="card-text"><strong>Abstract:</strong></p> During modelling of many physical problems and engineering processes, fractional calculus plays an important role. Those are greatly described by fractional differential equations (FDEs). So a reliable and efficient technique to solve such types of FDEs is needed. In this article, a numerical solution of a class of fractional differential equations namely space fractional order reaction-advection dispersion equations subject to initial and boundary conditions is derived. In the proposed approach shifted Jacobi polynomials are used to approximate the solutions together with shifted Jacobi operational matrix of fractional order and spectral collocation method. The main advantage of this approach is that it converts such problems in the systems of algebraic equations which are easier to be solved. The proposed approach is effective to solve the linear as well as non-linear FDEs. To show the reliability, validity and high accuracy of proposed approach, the numerical results of some illustrative examples are reported, which are compared with the existing analytical results already reported in the literature. The error analysis for each case exhibited through graphs and tables confirms the exponential convergence rate of the proposed method. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=space%20fractional%20order%20linear%2Fnonlinear%20reaction-advection%20diffusion%20equation" title="space fractional order linear/nonlinear reaction-advection diffusion equation">space fractional order linear/nonlinear reaction-advection diffusion equation</a>, <a href="https://publications.waset.org/abstracts/search?q=shifted%20Jacobi%20polynomials" title=" shifted Jacobi polynomials"> shifted Jacobi polynomials</a>, <a href="https://publications.waset.org/abstracts/search?q=operational%20matrix" title=" operational matrix"> operational matrix</a>, <a href="https://publications.waset.org/abstracts/search?q=collocation%20method" title=" collocation method"> collocation method</a>, <a href="https://publications.waset.org/abstracts/search?q=Caputo%20derivative" title=" Caputo derivative"> Caputo derivative</a> </p> <a href="https://publications.waset.org/abstracts/79521/numerical-solution-of-space-fractional-order-linearnonlinear-reaction-advection-diffusion-equation-using-jacobi-polynomial" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/79521.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">445</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">9639</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 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