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methods</title> <meta name="description" content="Search results for: numerical methods"> <meta name="keywords" content="numerical methods"> <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="numerical methods"> <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> 18175</div> </div> </div> </div> <h1 class="mt-3 mb-3 text-center" style="font-size:1.6rem;">Search results for: numerical methods</h1> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">18175</span> Improvement of the Numerical Integration's Quality in Meshless Methods</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Ahlem%20Mougaida">Ahlem Mougaida</a>, <a href="https://publications.waset.org/abstracts/search?q=Hedi%20Bel%20Hadj%20Salah"> Hedi Bel Hadj Salah</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Several methods are suggested to improve the numerical integration in Galerkin weak form for Meshless methods. In fact, integrating without taking into account the characteristics of the shape functions reproduced by Meshless methods (rational functions, with compact support etc.), causes a large integration error that influences the PDE’s approximate solution. Comparisons between different methods of numerical integration for rational functions are discussed and compared. The algorithms are implemented in Matlab. Finally, numerical results were presented to prove the efficiency of our algorithms in improving results. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=adaptive%20methods" title="adaptive methods">adaptive methods</a>, <a href="https://publications.waset.org/abstracts/search?q=meshless" title=" meshless"> meshless</a>, <a href="https://publications.waset.org/abstracts/search?q=numerical%20integration" title=" numerical integration"> numerical integration</a>, <a href="https://publications.waset.org/abstracts/search?q=rational%20quadrature" title=" rational quadrature"> rational quadrature</a> </p> <a href="https://publications.waset.org/abstracts/46442/improvement-of-the-numerical-integrations-quality-in-meshless-methods" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/46442.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">364</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">18174</span> Numerical Methods versus Bjerksund and Stensland Approximations for American Options Pricing</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Marasovic%20Branka">Marasovic Branka</a>, <a href="https://publications.waset.org/abstracts/search?q=Aljinovic%20Zdravka"> Aljinovic Zdravka</a>, <a href="https://publications.waset.org/abstracts/search?q=Poklepovic%20Tea"> Poklepovic Tea</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Numerical methods like binomial and trinomial trees and finite difference methods can be used to price a wide range of options contracts for which there are no known analytical solutions. American options are the most famous of that kind of options. Besides numerical methods, American options can be valued with the approximation formulas, like Bjerksund-Stensland formulas from 1993 and 2002. When the value of American option is approximated by Bjerksund-Stensland formulas, the computer time spent to carry out that calculation is very short. The computer time spent using numerical methods can vary from less than one second to several minutes or even hours. However to be able to conduct a comparative analysis of numerical methods and Bjerksund-Stensland formulas, we will limit computer calculation time of numerical method to less than one second. Therefore, we ask the question: Which method will be most accurate at nearly the same computer calculation time? <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=Bjerksund%20and%20Stensland%20approximations" title="Bjerksund and Stensland approximations">Bjerksund and Stensland approximations</a>, <a href="https://publications.waset.org/abstracts/search?q=computational%20analysis" title=" computational analysis"> computational analysis</a>, <a href="https://publications.waset.org/abstracts/search?q=finance" title=" finance"> finance</a>, <a href="https://publications.waset.org/abstracts/search?q=options%20pricing" title=" options pricing"> options pricing</a>, <a href="https://publications.waset.org/abstracts/search?q=numerical%20methods" title=" numerical methods"> numerical methods</a> </p> <a href="https://publications.waset.org/abstracts/2832/numerical-methods-versus-bjerksund-and-stensland-approximations-for-american-options-pricing" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/2832.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">18173</span> Solving Mean Field Problems: A Survey of Numerical Methods and Applications</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Amal%20Machtalay">Amal Machtalay</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this survey, we aim to review the rapidly growing literature on numerical methods to solve different forms of mean field problems, namely mean field games (MFG), mean field controls (MFC), potential MFGs, and master equations, as well as their corresponding recent applications. Here, we distinguish two families of numerical methods: iterative methods based on mesh generation and those called mesh-free, normally related to neural networking and learning frameworks. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=mean-field%20games" title="mean-field games">mean-field games</a>, <a href="https://publications.waset.org/abstracts/search?q=numerical%20schemes" title=" numerical schemes"> numerical schemes</a>, <a href="https://publications.waset.org/abstracts/search?q=partial%20differential%20equations" title=" partial differential equations"> partial differential equations</a>, <a href="https://publications.waset.org/abstracts/search?q=complex%20systems" title=" complex systems"> complex systems</a>, <a href="https://publications.waset.org/abstracts/search?q=machine%20learning" title=" machine learning"> machine learning</a> </p> <a href="https://publications.waset.org/abstracts/153744/solving-mean-field-problems-a-survey-of-numerical-methods-and-applications" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/153744.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">113</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">18172</span> Groundwater Seepage Estimation into Amirkabir Tunnel Using Analytical Methods and DEM and SGR Method</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Hadi%20Farhadian">Hadi Farhadian</a>, <a href="https://publications.waset.org/abstracts/search?q=Homayoon%20Katibeh"> Homayoon Katibeh</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this paper, groundwater seepage into Amirkabir tunnel has been estimated using analytical and numerical methods for 14 different sections of the tunnel. Site Groundwater Rating (SGR) method also has been performed for qualitative and quantitative classification of the tunnel sections. The obtained results of above-mentioned methods were compared together. The study shows reasonable accordance with results of the all methods unless for two sections of tunnel. In these two sections there are some significant discrepancies between numerical and analytical results mainly originated from model geometry and high overburden. SGR and the analytical and numerical calculations, confirm the high concentration of seepage inflow in fault zones. Maximum seepage flow into tunnel has been estimated 0.425 lit/sec/m using analytical method and 0.628 lit/sec/m using numerical method occurred in crashed zone. Based on SGR method, six sections of 14 sections in Amirkabir tunnel axis are found to be in "No Risk" class that is supported by the analytical and numerical seepage value of less than 0.04 lit/sec/m. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=water%20Seepage" title="water Seepage">water Seepage</a>, <a href="https://publications.waset.org/abstracts/search?q=Amirkabir%20Tunnel" title=" Amirkabir Tunnel"> Amirkabir Tunnel</a>, <a href="https://publications.waset.org/abstracts/search?q=analytical%20method" title=" analytical method"> analytical method</a>, <a href="https://publications.waset.org/abstracts/search?q=DEM" title=" DEM"> DEM</a>, <a href="https://publications.waset.org/abstracts/search?q=SGR" title=" SGR"> SGR</a> </p> <a href="https://publications.waset.org/abstracts/26677/groundwater-seepage-estimation-into-amirkabir-tunnel-using-analytical-methods-and-dem-and-sgr-method" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/26677.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">476</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">18171</span> Numerical and Analytical Approach for Film Condensation on Different Forms of Surfaces</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=A.%20Kazemi%20Jouybari">A. Kazemi Jouybari</a>, <a href="https://publications.waset.org/abstracts/search?q=A.%20Mirabdolah%20Lavasani"> A. Mirabdolah Lavasani</a> </p> <p class="card-text"><strong>Abstract:</strong></p> This paper seeks to the solution of condensation around of a flat plate, circular and elliptical tube in way of numerical and analytical methods. Also, it calculates the entropy production rates. The first, problem was solved by using mesh dynamic and rational assumptions, next it was compared with the numerical solution that the result had acceptable errors. An additional supporting relation was applied based on a characteristic of condensation phenomenon for condensing elements. As it has been shown here, due to higher rates of heat transfer for elliptical tubes, they have more entropy production rates, in comparison to circular ones. Findings showed that two methods were efficient. Furthermore, analytical methods can be used to optimize the problem and reduce the entropy production rate. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=condensation" title="condensation">condensation</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=analytical%20solution" title=" analytical solution"> analytical solution</a>, <a href="https://publications.waset.org/abstracts/search?q=entropy%20rate" title=" entropy rate"> entropy rate</a> </p> <a href="https://publications.waset.org/abstracts/94520/numerical-and-analytical-approach-for-film-condensation-on-different-forms-of-surfaces" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/94520.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">216</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">18170</span> A Comparative Study between FEM and Meshless Methods</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Jay%20N.%20Vyas">Jay N. Vyas</a>, <a href="https://publications.waset.org/abstracts/search?q=Sachin%20Daxini"> Sachin Daxini</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Numerical simulation techniques are widely used now in product development and testing instead of expensive, time-consuming and sometimes dangerous laboratory experiments. Numerous numerical methods are available for performing simulation of physical problems of different engineering fields. Grid based methods, like Finite Element Method, are extensively used in performing various kinds of static, dynamic, structural and non-structural analysis during product development phase. Drawbacks of grid based methods in terms of discontinuous secondary field variable, dealing fracture mechanics and large deformation problems led to development of a relatively a new class of numerical simulation techniques in last few years, which are popular as Meshless methods or Meshfree Methods. Meshless Methods are expected to be more adaptive and flexible than Finite Element Method because domain descretization in Meshless Method requires only nodes. Present paper introduces Meshless Methods and differentiates it with Finite Element Method in terms of following aspects: Shape functions used, role of weight function, techniques to impose essential boundary conditions, integration techniques for discrete system equations, convergence rate, accuracy of solution and computational effort. Capabilities, benefits and limitations of Meshless Methods are discussed and concluded at the end of paper. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=numerical%20simulation" title="numerical simulation">numerical simulation</a>, <a href="https://publications.waset.org/abstracts/search?q=Grid-based%20methods" title=" Grid-based methods"> Grid-based methods</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=Meshless%20Methods" title=" Meshless Methods"> Meshless Methods</a> </p> <a href="https://publications.waset.org/abstracts/43544/a-comparative-study-between-fem-and-meshless-methods" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/43544.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">18169</span> Assessment of Slope Stability by Continuum and Discontinuum Methods</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Taleb%20Hosni%20Abderrahmane">Taleb Hosni Abderrahmane</a>, <a href="https://publications.waset.org/abstracts/search?q=Berga%20Abdelmadjid"> Berga Abdelmadjid</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The development of numerical analysis and its application to geomechanics problems have provided geotechnical engineers with extremely powerful tools. One of the most important problems in geotechnical engineering is the slope stability assessment. It is a very difficult task due to several aspects such the nature of the problem, experimental consideration, monitoring, controlling, and assessment. The main objective of this paper is to perform a comparative numerical study between the following methods: The Limit Equilibrium (LEM), Finite Element (FEM), Limit Analysis (LAM) and Distinct Element (DEM). The comparison is conducted in terms of the safety factors and the critical slip surfaces. Through the results, we see the feasibility to analyse slope stability by many methods. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=comparison" title="comparison">comparison</a>, <a href="https://publications.waset.org/abstracts/search?q=factor%20of%20safety" title=" factor of safety"> factor of safety</a>, <a href="https://publications.waset.org/abstracts/search?q=geomechanics" title=" geomechanics"> geomechanics</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=slope%20analysis" title=" slope analysis"> slope analysis</a>, <a href="https://publications.waset.org/abstracts/search?q=slip%20surfaces" title=" slip surfaces"> slip surfaces</a> </p> <a href="https://publications.waset.org/abstracts/35737/assessment-of-slope-stability-by-continuum-and-discontinuum-methods" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/35737.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">533</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">18168</span> Numerical Evolution Methods of Rational Form for Diffusion Equations</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Said%20Algarni">Said Algarni</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The purpose of this study was to investigate selected numerical methods that demonstrate good performance in solving PDEs. We adapted alternative method that involve rational polynomials. Padé time stepping (PTS) method, which is highly stable for the purposes of the present application and is associated with lower computational costs, was applied. Furthermore, PTS was modified for our study which focused on diffusion equations. Numerical runs were conducted to obtain the optimal local error control threshold. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=Pad%C3%A9%20time%20stepping" title="Padé time stepping">Padé time stepping</a>, <a href="https://publications.waset.org/abstracts/search?q=finite%20difference" title=" finite difference"> finite difference</a>, <a href="https://publications.waset.org/abstracts/search?q=reaction%20diffusion%20equation" title=" reaction diffusion equation"> reaction diffusion equation</a>, <a href="https://publications.waset.org/abstracts/search?q=PDEs" title=" PDEs"> PDEs</a> </p> <a href="https://publications.waset.org/abstracts/7176/numerical-evolution-methods-of-rational-form-for-diffusion-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/7176.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">18167</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">18166</span> Induction Heating Process Design Using Comsol® Multiphysics Software Version 4.2a</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=K.%20Djellabi">K. Djellabi</a>, <a href="https://publications.waset.org/abstracts/search?q=M.%20E.%20H.%20Latreche"> M. E. H. Latreche</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Induction heating computer simulation is a powerful tool for process design and optimization, induction coil design, equipment selection, as well as education and business presentations. The authors share their vast experience in the practical use of computer simulation for different induction heating and heat treating processes. In this paper deals with mathematical modeling and numerical simulation of induction heating furnaces with axisymmetric geometries. For the numerical solution, we propose finite element methods combined with boundary (FEM) for the electromagnetic model using COMSOL® Multiphysics Software. Some numerical results for an industrial furnace are shown with high frequency. <p class="card-text"><strong>Keywords:</strong> <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=induction%20furnaces" title=" induction furnaces"> induction furnaces</a>, <a href="https://publications.waset.org/abstracts/search?q=induction%20heating" title=" induction heating"> induction heating</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=Comsol%20multiphysics%20software" title=" Comsol multiphysics software"> Comsol multiphysics software</a> </p> <a href="https://publications.waset.org/abstracts/3469/induction-heating-process-design-using-comsol-multiphysics-software-version-42a" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/3469.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">449</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">18165</span> Quartic Spline Method for Numerical Solution of Self-Adjoint Singularly Perturbed Boundary Value Problems</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Reza%20Mohammadi">Reza Mohammadi</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Using quartic spline, we develop a method for numerical solution of singularly perturbed two-point boundary-value problems. The purposed method is fourth-order accurate and applicable to problems both in singular and non-singular cases. The convergence analysis of the method is given. The resulting linear system of equations has been solved by using a tri-diagonal solver. We applied the presented method to test problems which have been solved by other existing methods in references, for comparison of presented method with the existing methods. Numerical results are given to illustrate the efficiency of our methods. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=second-order%20ordinary%20differential%20equation" title="second-order ordinary differential equation">second-order ordinary differential equation</a>, <a href="https://publications.waset.org/abstracts/search?q=singularly-perturbed" title=" singularly-perturbed"> singularly-perturbed</a>, <a href="https://publications.waset.org/abstracts/search?q=quartic%20spline" title=" quartic spline"> quartic spline</a>, <a href="https://publications.waset.org/abstracts/search?q=convergence%20analysis" title=" convergence analysis"> convergence analysis</a> </p> <a href="https://publications.waset.org/abstracts/46459/quartic-spline-method-for-numerical-solution-of-self-adjoint-singularly-perturbed-boundary-value-problems" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/46459.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">360</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">18164</span> A Continuous Boundary Value Method of Order 8 for Solving the General Second Order Multipoint Boundary Value Problems</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=T.%20A.%20Biala">T. A. Biala</a> </p> <p class="card-text"><strong>Abstract:</strong></p> This paper deals with the numerical integration of the general second order multipoint boundary value problems. This has been achieved by the development of a continuous linear multistep method (LMM). The continuous LMM is used to construct a main discrete method to be used with some initial and final methods (also obtained from the continuous LMM) so that they form a discrete analogue of the continuous second order boundary value problems. These methods are used as boundary value methods and adapted to cope with the integration of the general second order multipoint boundary value problems. The convergence, the use and the region of absolute stability of the methods are discussed. Several numerical examples are implemented to elucidate our solution process. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=linear%20multistep%20methods" title="linear multistep methods">linear multistep methods</a>, <a href="https://publications.waset.org/abstracts/search?q=boundary%20value%20methods" title=" boundary value methods"> boundary value methods</a>, <a href="https://publications.waset.org/abstracts/search?q=second%20order%20multipoint%20boundary%20value%20problems" title=" second order multipoint boundary value problems"> second order multipoint boundary value problems</a>, <a href="https://publications.waset.org/abstracts/search?q=convergence" title=" convergence"> convergence</a> </p> <a href="https://publications.waset.org/abstracts/50749/a-continuous-boundary-value-method-of-order-8-for-solving-the-general-second-order-multipoint-boundary-value-problems" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/50749.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">18163</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">424</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">18162</span> Optimization of the Numerical Fracture Mechanics</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=H.%20Hentati">H. Hentati</a>, <a href="https://publications.waset.org/abstracts/search?q=R.%20Abdelmoula"> R. Abdelmoula</a>, <a href="https://publications.waset.org/abstracts/search?q=Li%20Jia"> Li Jia</a>, <a href="https://publications.waset.org/abstracts/search?q=A.%20Maalej"> A. Maalej</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this work, we present numerical simulations of the quasi-static crack propagation based on the variation approach. We perform numerical simulations of a piece of brittle material without initial crack. An alternate minimization algorithm is used. Based on these numerical results, we determine the influence of numerical parameters on the location of crack. We show the importance of trying to optimize the time of numerical computation and we present the first attempt to develop a simple numerical method to optimize this time. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=fracture%20mechanics" title="fracture mechanics">fracture mechanics</a>, <a href="https://publications.waset.org/abstracts/search?q=optimization" title=" optimization"> optimization</a>, <a href="https://publications.waset.org/abstracts/search?q=variation%20approach" title=" variation approach"> variation approach</a>, <a href="https://publications.waset.org/abstracts/search?q=mechanic" title=" mechanic"> mechanic</a> </p> <a href="https://publications.waset.org/abstracts/5187/optimization-of-the-numerical-fracture-mechanics" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/5187.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">606</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">18161</span> Three Dimensional Numerical Analysis for Longitudinal Seismic Response of Tunnels under Asynchronous Earthquake</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Peng%20Li">Peng Li</a>, <a href="https://publications.waset.org/abstracts/search?q=Er-xiang%20Song"> Er-xiang Song</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Numerical analysis of longitudinal tunnel seismic response due to spatial variation of earthquake ground motion is an important issue that cannot be ignored in the design and safety evaluation of tunnel structures. In this paper, numerical methods for analysis of tunnel longitudinal response under asynchronous seismic wave is extensively studied, including the improvement of the 1D time-domain finite element method, three dimensional numerical simulation technique for the site asynchronous earthquake response as well as the 3-D soil-tunnel structure interaction analysis. The study outcome will be beneficial to aid further research on the nonlinear meticulous numerical analysis and seismic response mechanism of tunnel structures under asynchronous earthquake motion. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=asynchronous%20input" title="asynchronous input">asynchronous input</a>, <a href="https://publications.waset.org/abstracts/search?q=longitudinal%20seismic%20response" title=" longitudinal seismic response"> longitudinal seismic response</a>, <a href="https://publications.waset.org/abstracts/search?q=tunnel%20structure" title=" tunnel structure"> tunnel structure</a>, <a href="https://publications.waset.org/abstracts/search?q=numerical%20simulation" title=" numerical simulation"> numerical simulation</a>, <a href="https://publications.waset.org/abstracts/search?q=traveling%20wave%20effect" title=" traveling wave effect"> traveling wave effect</a> </p> <a href="https://publications.waset.org/abstracts/9730/three-dimensional-numerical-analysis-for-longitudinal-seismic-response-of-tunnels-under-asynchronous-earthquake" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/9730.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">437</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">18160</span> On the Solution of Fractional-Order Dynamical Systems Endowed with Block Hybrid Methods</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Kizito%20Ugochukwu%20Nwajeri">Kizito Ugochukwu Nwajeri</a> </p> <p class="card-text"><strong>Abstract:</strong></p> This paper presents a distinct approach to solving fractional dynamical systems using hybrid block methods (HBMs). Fractional calculus extends the concept of derivatives and integrals to non-integer orders and finds increasing application in fields such as physics, engineering, and finance. However, traditional numerical techniques often struggle to accurately capture the complex behaviors exhibited by these systems. To address this challenge, we develop HBMs that integrate single-step and multi-step methods, enabling the simultaneous computation of multiple solution points while maintaining high accuracy. Our approach employs polynomial interpolation and collocation techniques to derive a system of equations that effectively models the dynamics of fractional systems. We also directly incorporate boundary and initial conditions into the formulation, enhancing the stability and convergence properties of the numerical solution. An adaptive step-size mechanism is introduced to optimize performance based on the local behavior of the solution. Extensive numerical simulations are conducted to evaluate the proposed methods, demonstrating significant improvements in accuracy and efficiency compared to traditional numerical approaches. The results indicate that our hybrid block methods are robust and versatile, making them suitable for a wide range of applications involving fractional dynamical systems. This work contributes to the existing literature by providing an effective numerical framework for analyzing complex behaviors in fractional systems, thereby opening new avenues for research and practical implementation across various disciplines. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=fractional%20calculus" title="fractional calculus">fractional calculus</a>, <a href="https://publications.waset.org/abstracts/search?q=numerical%20simulation" title=" numerical simulation"> numerical simulation</a>, <a href="https://publications.waset.org/abstracts/search?q=stability%20and%20convergence" title=" stability and convergence"> stability and convergence</a>, <a href="https://publications.waset.org/abstracts/search?q=Adaptive%20step-size%20mechanism" title=" Adaptive step-size mechanism"> Adaptive step-size mechanism</a>, <a href="https://publications.waset.org/abstracts/search?q=collocation%20methods" title=" collocation methods"> collocation methods</a> </p> <a href="https://publications.waset.org/abstracts/188911/on-the-solution-of-fractional-order-dynamical-systems-endowed-with-block-hybrid-methods" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/188911.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">43</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">18159</span> Pricing European Options under Jump Diffusion Models with Fast L-stable Padé Scheme</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Salah%20Alrabeei">Salah Alrabeei</a>, <a href="https://publications.waset.org/abstracts/search?q=Mohammad%20Yousuf"> Mohammad Yousuf</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The goal of option pricing theory is to help the investors to manage their money, enhance returns and control their financial future by theoretically valuing their options. Modeling option pricing by Black-School models with jumps guarantees to consider the market movement. However, only numerical methods can solve this model. Furthermore, not all the numerical methods are efficient to solve these models because they have nonsmoothing payoffs or discontinuous derivatives at the exercise price. In this paper, the exponential time differencing (ETD) method is applied for solving partial integrodifferential equations arising in pricing European options under Merton’s and Kou’s jump-diffusion models. Fast Fourier Transform (FFT) algorithm is used as a matrix-vector multiplication solver, which reduces the complexity from O(M2) into O(M logM). A partial fraction form of Pad`e schemes is used to overcome the complexity of inverting polynomial of matrices. These two tools guarantee to get efficient and accurate numerical solutions. We construct a parallel and easy to implement a version of the numerical scheme. Numerical experiments are given to show how fast and accurate is our scheme. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=Integral%20differential%20equations" title="Integral differential equations">Integral differential equations</a>, <a href="https://publications.waset.org/abstracts/search?q=" title=""></a>, <a href="https://publications.waset.org/abstracts/search?q=L-stable%20methods" title=" L-stable methods"> L-stable methods</a>, <a href="https://publications.waset.org/abstracts/search?q=pricing%20European%20options" title=" pricing European options"> pricing European options</a>, <a href="https://publications.waset.org/abstracts/search?q=Jump%E2%80%93diffusion%20model" title=" Jump–diffusion model"> Jump–diffusion model</a> </p> <a href="https://publications.waset.org/abstracts/126160/pricing-european-options-under-jump-diffusion-models-with-fast-l-stable-pade-scheme" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/126160.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">151</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">18158</span> Robust Numerical Method for Singularly Perturbed Semilinear Boundary Value Problem with Nonlocal Boundary Condition</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Habtamu%20Garoma%20Debela">Habtamu Garoma Debela</a>, <a href="https://publications.waset.org/abstracts/search?q=Gemechis%20File%20Duressa"> Gemechis File Duressa</a> </p> <p class="card-text"><strong>Abstract:</strong></p> In this work, our primary interest is to provide ε-uniformly convergent numerical techniques for solving singularly perturbed semilinear boundary value problems with non-local boundary condition. These singular perturbation problems are described by differential equations in which the highest-order derivative is multiplied by an arbitrarily small parameter ε (say) known as singular perturbation parameter. This leads to the existence of boundary layers, which are basically narrow regions in the neighborhood of the boundary of the domain, where the gradient of the solution becomes steep as the perturbation parameter tends to zero. Due to the appearance of the layer phenomena, it is a challenging task to provide ε-uniform numerical methods. The term 'ε-uniform' refers to identify those numerical methods in which the approximate solution converges to the corresponding exact solution (measured to the supremum norm) independently with respect to the perturbation parameter ε. Thus, the purpose of this work is to develop, analyze, and improve the ε-uniform numerical methods for solving singularly perturbed problems. These methods are based on nonstandard fitted finite difference method. The basic idea behind the fitted operator, finite difference method, is to replace the denominator functions of the classical derivatives with positive functions derived in such a way that they capture some notable properties of the governing differential equation. A uniformly convergent numerical method is constructed via nonstandard fitted operator numerical method and numerical integration methods to solve the problem. The non-local boundary condition is treated using numerical integration techniques. Additionally, Richardson extrapolation technique, which improves the first-order accuracy of the standard scheme to second-order convergence, is applied for singularly perturbed convection-diffusion problems using the proposed numerical method. Maximum absolute errors and rates of convergence for different values of perturbation parameter and mesh sizes are tabulated for the numerical example considered. The method is shown to be ε-uniformly convergent. Finally, extensive numerical experiments are conducted which support all of our theoretical findings. A concise conclusion is provided at the end of this work. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=nonlocal%20boundary%20condition" title="nonlocal boundary condition">nonlocal boundary condition</a>, <a href="https://publications.waset.org/abstracts/search?q=nonstandard%20fitted%20operator" title=" nonstandard fitted operator"> nonstandard fitted operator</a>, <a href="https://publications.waset.org/abstracts/search?q=semilinear%20problem" title=" semilinear problem"> semilinear problem</a>, <a href="https://publications.waset.org/abstracts/search?q=singular%20perturbation" title=" singular perturbation"> singular perturbation</a>, <a href="https://publications.waset.org/abstracts/search?q=uniformly%20convergent" title=" uniformly convergent"> uniformly convergent</a> </p> <a href="https://publications.waset.org/abstracts/130967/robust-numerical-method-for-singularly-perturbed-semilinear-boundary-value-problem-with-nonlocal-boundary-condition" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/130967.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">143</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">18157</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">18156</span> FE Analysis of Blade-Disc Dovetail Joints Using Mortar Base Frictional Contact Formulation</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Abbas%20Moradi">Abbas Moradi</a>, <a href="https://publications.waset.org/abstracts/search?q=Mohsen%20Safajoy"> Mohsen Safajoy</a>, <a href="https://publications.waset.org/abstracts/search?q=Reza%20Yazdanparast"> Reza Yazdanparast</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Analysis of blade-disc dovetail joints is one of the biggest challenges facing designers of aero-engines. To avoid comparatively expensive experimental full-scale tests, numerical methods can be used to simulate loaded disc-blades assembly. Mortar method provides a powerful and flexible tool for solving frictional contact problems. In this study, 2D frictional contact in dovetail has been analysed based on the mortar algorithm. In order to model the friction, the classical law of coulomb and moving friction cone algorithm is applied. The solution is then obtained by solving the resulting set of non-linear equations using an efficient numerical algorithm based on Newton–Raphson Method. The numerical results show that this approach has better convergence rate and accuracy than other proposed numerical methods. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=computational%20contact%20mechanics" title="computational contact mechanics">computational contact mechanics</a>, <a href="https://publications.waset.org/abstracts/search?q=dovetail%20joints" title=" dovetail joints"> dovetail joints</a>, <a href="https://publications.waset.org/abstracts/search?q=nonlinear%20FEM" title=" nonlinear FEM"> nonlinear FEM</a>, <a href="https://publications.waset.org/abstracts/search?q=mortar%20approach" title=" mortar approach"> mortar approach</a> </p> <a href="https://publications.waset.org/abstracts/4936/fe-analysis-of-blade-disc-dovetail-joints-using-mortar-base-frictional-contact-formulation" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/4936.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">352</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">18155</span> A New Class of Conjugate Gradient Methods Based on a Modified Search Direction for Unconstrained Optimization</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Belloufi%20Mohammed">Belloufi Mohammed</a>, <a href="https://publications.waset.org/abstracts/search?q=Sellami%20Badreddine"> Sellami Badreddine</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Conjugate gradient methods have played a special role for solving large scale optimization problems due to the simplicity of their iteration, convergence properties and their low memory requirements. In this work, we propose a new class of conjugate gradient methods which ensures sufficient descent. Moreover, we propose a new search direction with the Wolfe line search technique for solving unconstrained optimization problems, a global convergence result for general functions is established provided that the line search satisfies the Wolfe conditions. Our numerical experiments indicate that our proposed methods are preferable and in general superior to the classical conjugate gradient methods in terms of efficiency and robustness. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=unconstrained%20optimization" title="unconstrained optimization">unconstrained optimization</a>, <a href="https://publications.waset.org/abstracts/search?q=conjugate%20gradient%20method" title=" conjugate gradient method"> conjugate gradient method</a>, <a href="https://publications.waset.org/abstracts/search?q=sufficient%20descent%20property" title=" sufficient descent property"> sufficient descent property</a>, <a href="https://publications.waset.org/abstracts/search?q=numerical%20comparisons" title=" numerical comparisons"> numerical comparisons</a> </p> <a href="https://publications.waset.org/abstracts/41725/a-new-class-of-conjugate-gradient-methods-based-on-a-modified-search-direction-for-unconstrained-optimization" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/41725.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">18154</span> Development of Residual Power Series Methods for Efficient Solutions of Stiff Differential Equations</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Gebreegziabher%20Hailu">Gebreegziabher Hailu</a> </p> <p class="card-text"><strong>Abstract:</strong></p> This paper presents the development of residual power series methods aimed at efficiently solving stiff differential equations, which pose significant challenges in numerical analysis due to their rapid changes in solution behavior. The RPSM is a numerical approach that generates polynomial-based approximate solutions without the need for linearization, discretization, or perturbation techniques, making it straightforward to implement and less prone to computational errors. We introduce an approach that utilizes power series expansions combined with residual minimization techniques to enhance convergence and stability. By analyzing the theoretical foundations of stiffness, we delve into the formulation of the residual power series method, detailing how it effectively captures the dynamics of stiff systems while maintaining computational efficiency. Numerical experiments demonstrate the method's superiority in terms of accuracy and computational cost when compared to traditional methods like implicit Runge-Kutta or multistep techniques. We also explore adaptive strategies within our framework to automatically adjust parameters based on the stiffness characteristics of the problem at hand. Ultimately, our findings contribute to the broader toolkit for tackling stiff differential equations, offering a robust alternative that promises to streamline computational workflows in various applied mathematics and engineering contexts. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=residual%20power%20series%20methods" title="residual power series methods">residual power series methods</a>, <a href="https://publications.waset.org/abstracts/search?q=stiff%20differential%20equoations" title=" stiff differential equoations"> stiff differential equoations</a>, <a href="https://publications.waset.org/abstracts/search?q=numerical%20approach" title=" numerical approach"> numerical approach</a>, <a href="https://publications.waset.org/abstracts/search?q=Runge%20Kutta%20methods" title=" Runge Kutta methods"> Runge Kutta methods</a> </p> <a href="https://publications.waset.org/abstracts/192125/development-of-residual-power-series-methods-for-efficient-solutions-of-stiff-differential-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/192125.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">22</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">18153</span> The Impact of Cooperative Learning on Numerical Methods Course</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Sara%20Bilal">Sara Bilal</a>, <a href="https://publications.waset.org/abstracts/search?q=Abdi%20Omar%20Shuriye"> Abdi Omar Shuriye</a>, <a href="https://publications.waset.org/abstracts/search?q=Raihan%20Othman"> Raihan Othman</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Numerical Methods is a course that can be conducted using workshops and group discussion. This study has been implemented on undergraduate students of level two at the Faculty of Engineering, International Islamic University Malaysia. The Numerical Method course has been delivered to two Sections 1 and 2 with 44 and 22 students in each section, respectively. Systematic steps have been followed to apply the student centered learning approach in teaching Numerical Method course. Initially, the instructor has chosen the topic which was Euler’s Method to solve Ordinary Differential Equations (ODE) to be learned. The students were then divided into groups with five members in each group. Initial instructions have been given to the group members to prepare their subtopics before meeting members from other groups to discuss the subtopics in an expert group inside the classroom. For the time assigned for the classroom discussion, the setting of the classroom was rearranged to accommodate the student centered learning approach. Teacher strength was by monitoring the process of learning inside and outside the class. The students have been assessed during the migrating to the expert groups, recording of a video explanation outside the classroom and during the final examination. Euler’s Method to solve the ODE was set as part of Question 3(b) in the final exam. It is observed that none of the students from both sections obtained a zero grade in Q3(b), compared to Q3(a) and Q3(c). Also, for Section 1(44 students), 29 students obtained the full mark of 7/7, while only 10 obtained 7/7 for Q3(a) and no students obtained 6/6 for Q3(c). Finally, we can recommend that the Numerical Method course be moved toward more student-centered Learning classrooms where the students will be engaged in group discussion rather than having a teacher one man show. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=teacher%20centered%20learning" title="teacher centered learning">teacher centered learning</a>, <a href="https://publications.waset.org/abstracts/search?q=student%20centered%20learning" title=" student centered learning"> student centered learning</a>, <a href="https://publications.waset.org/abstracts/search?q=mathematic" title=" mathematic"> mathematic</a>, <a href="https://publications.waset.org/abstracts/search?q=numerical%20methods" title=" numerical methods"> numerical methods</a> </p> <a href="https://publications.waset.org/abstracts/60012/the-impact-of-cooperative-learning-on-numerical-methods-course" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/60012.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">366</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">18152</span> Generic Hybrid Models for Two-Dimensional Ultrasonic Guided Wave Problems</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Manoj%20Reghu">Manoj Reghu</a>, <a href="https://publications.waset.org/abstracts/search?q=Prabhu%20Rajagopal"> Prabhu Rajagopal</a>, <a href="https://publications.waset.org/abstracts/search?q=C.%20V.%20Krishnamurthy"> C. V. Krishnamurthy</a>, <a href="https://publications.waset.org/abstracts/search?q=Krishnan%20Balasubramaniam"> Krishnan Balasubramaniam</a> </p> <p class="card-text"><strong>Abstract:</strong></p> A thorough understanding of guided ultrasonic wave behavior in structures is essential for the application of existing Non Destructive Evaluation (NDE) technologies, as well as for the development of new methods. However, the analysis of guided wave phenomena is challenging because of their complex dispersive and multimodal nature. Although numerical solution procedures have proven to be very useful in this regard, the increasing complexity of features and defects to be considered, as well as the desire to improve the accuracy of inspection often imposes a large computational cost. Hybrid models that combine numerical solutions for wave scattering with faster alternative methods for wave propagation have long been considered as a solution to this problem. However usually such models require modification of the base code of the solution procedure. Here we aim to develop Generic Hybrid models that can be directly applied to any two different solution procedures. With this goal in mind, a Numerical Hybrid model and an Analytical-Numerical Hybrid model has been developed. The concept and implementation of these Hybrid models are discussed in this paper. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=guided%20ultrasonic%20waves" title="guided ultrasonic waves">guided ultrasonic waves</a>, <a href="https://publications.waset.org/abstracts/search?q=Finite%20Element%20Method%20%28FEM%29" title=" Finite Element Method (FEM)"> Finite Element Method (FEM)</a>, <a href="https://publications.waset.org/abstracts/search?q=Hybrid%20model" title=" Hybrid model"> Hybrid model</a> </p> <a href="https://publications.waset.org/abstracts/16058/generic-hybrid-models-for-two-dimensional-ultrasonic-guided-wave-problems" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/16058.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">465</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">18151</span> Computational Approaches for Ballistic Impact Response of Stainless Steel 304</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=A.%20Mostafa">A. Mostafa</a> </p> <p class="card-text"><strong>Abstract:</strong></p> This paper presents a numerical study on determination of ballistic limit velocity (<em>V<sub>50</sub></em>) of stainless steel 304 (SS 304) used in manufacturing security screens. The simulated ballistic impact tests were conducted on clamped sheets with different thicknesses using ABAQUS/Explicit nonlinear finite element (FE) package. The ballistic limit velocity was determined using three approaches, namely: numerical tests based on material properties, FE calculated residual velocities and FE calculated residual energies. Johnson-Cook plasticity and failure criterion were utilized to simulate the dynamic behaviour of the SS 304 under various strain rates, while the well-known Lambert-Jonas equation was used for the data regression for the residual velocity and energy model. Good agreement between the investigated numerical methods was achieved. Additionally, the dependence of the ballistic limit velocity on the sheet thickness was observed. The proposed approaches present viable and cost-effective assessment methods of the ballistic performance of SS 304, which will support the development of robust security screen systems. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=ballistic%20velocity" title="ballistic velocity">ballistic velocity</a>, <a href="https://publications.waset.org/abstracts/search?q=stainless%20steel" title=" stainless steel"> stainless steel</a>, <a href="https://publications.waset.org/abstracts/search?q=numerical%20approaches" title=" numerical approaches"> numerical approaches</a>, <a href="https://publications.waset.org/abstracts/search?q=security%20screen" title=" security screen"> security screen</a> </p> <a href="https://publications.waset.org/abstracts/125088/computational-approaches-for-ballistic-impact-response-of-stainless-steel-304" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/125088.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">163</span> </span> </div> </div> <div class="card paper-listing mb-3 mt-3"> <h5 class="card-header" style="font-size:.9rem"><span class="badge badge-info">18150</span> Some Results on the Generalized Higher Rank Numerical Ranges</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Mohsen%20Zahraei">Mohsen Zahraei</a> </p> <p class="card-text"><strong>Abstract:</strong></p> ‎In this paper, ‎the notion of ‎rank-k numerical range of rectangular complex matrix polynomials‎ ‎are introduced. ‎Some algebraic and geometrical properties are investigated. ‎Moreover, ‎for ε>0 the notion of Birkhoff-James approximate orthogonality sets for ε-higher ‎rank numerical ranges of rectangular matrix polynomials is also introduced and studied. ‎The proposed definitions yield a natural generalization of the standard higher rank numerical ranges. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=%E2%80%8E%E2%80%8ERank-k%20numerical%20range%E2%80%8E" title="‎‎Rank-k numerical range‎">‎‎Rank-k numerical range‎</a>, <a href="https://publications.waset.org/abstracts/search?q=%E2%80%8Eisometry%E2%80%8E" title=" ‎isometry‎"> ‎isometry‎</a>, <a href="https://publications.waset.org/abstracts/search?q=%E2%80%8Enumerical%20range%E2%80%8E" title=" ‎numerical range‎"> ‎numerical range‎</a>, <a href="https://publications.waset.org/abstracts/search?q=%E2%80%8Erectangular%20matrix%20polynomials" title=" ‎rectangular matrix polynomials"> ‎rectangular matrix polynomials</a> </p> <a href="https://publications.waset.org/abstracts/28955/some-results-on-the-generalized-higher-rank-numerical-ranges" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/28955.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">459</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">18149</span> Numerical Modeling for Water Engineering and Obstacle Theory</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Mounir%20Adal">Mounir Adal</a>, <a href="https://publications.waset.org/abstracts/search?q=Baalal%20Azeddine"> Baalal Azeddine</a>, <a href="https://publications.waset.org/abstracts/search?q=Afifi%20Moulay%20Larbi"> Afifi Moulay Larbi</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Numerical analysis is a branch of mathematics devoted to the development of iterative matrix calculation techniques. We are searching for operations optimization as objective to calculate and solve systems of equations of order n with time and energy saving for computers that are conducted to calculate and analyze big data by solving matrix equations. Furthermore, this scientific discipline is producing results with a margin of error of approximation called rates. Thus, the results obtained from the numerical analysis techniques that are held on computer software such as MATLAB or Simulink offers a preliminary diagnosis of the situation of the environment or space targets. By this we can offer technical procedures needed for engineering or scientific studies exploitable by engineers for water. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=numerical%20analysis%20methods" title="numerical analysis methods">numerical analysis methods</a>, <a href="https://publications.waset.org/abstracts/search?q=obstacles%20solving" title=" obstacles solving"> obstacles solving</a>, <a href="https://publications.waset.org/abstracts/search?q=engineering" title=" engineering"> engineering</a>, <a href="https://publications.waset.org/abstracts/search?q=simulation" title=" simulation"> simulation</a>, <a href="https://publications.waset.org/abstracts/search?q=numerical%20modeling" title=" numerical modeling"> numerical modeling</a>, <a href="https://publications.waset.org/abstracts/search?q=iteration" title=" iteration"> iteration</a>, <a href="https://publications.waset.org/abstracts/search?q=computer" title=" computer"> computer</a>, <a href="https://publications.waset.org/abstracts/search?q=MATLAB" title=" MATLAB"> MATLAB</a>, <a href="https://publications.waset.org/abstracts/search?q=water" title=" water"> water</a>, <a href="https://publications.waset.org/abstracts/search?q=underground" title=" underground"> underground</a>, <a href="https://publications.waset.org/abstracts/search?q=velocity" title=" velocity"> velocity</a> </p> <a href="https://publications.waset.org/abstracts/32980/numerical-modeling-for-water-engineering-and-obstacle-theory" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/32980.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">18148</span> On the Solution of Boundary Value Problems Blended with Hybrid Block Methods</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Kizito%20Ugochukwu%20Nwajeri">Kizito Ugochukwu Nwajeri</a> </p> <p class="card-text"><strong>Abstract:</strong></p> This paper explores the application of hybrid block methods for solving boundary value problems (BVPs), which are prevalent in various fields such as science, engineering, and applied mathematics. Traditionally, numerical approaches such as finite difference and shooting methods, often encounter challenges related to stability and convergence, particularly in the context of complex and nonlinear BVPs. To address these challenges, we propose a hybrid block method that integrates features from both single-step and multi-step techniques. This method allows for the simultaneous computation of multiple solution points while maintaining high accuracy. Specifically, we employ a combination of polynomial interpolation and collocation strategies to derive a system of equations that captures the behavior of the solution across the entire domain. By directly incorporating boundary conditions into the formulation, we enhance the stability and convergence properties of the numerical solution. Furthermore, we introduce an adaptive step-size mechanism to optimize performance based on the local behavior of the solution. This adjustment allows the method to respond effectively to variations in solution behavior, improving both accuracy and computational efficiency. Numerical tests on a variety of boundary value problems demonstrate the effectiveness of the hybrid block methods. These tests showcase significant improvements in accuracy and computational efficiency compared to conventional methods, indicating that our approach is robust and versatile. The results suggest that this hybrid block method is suitable for a wide range of applications in real-world problems, offering a promising alternative to existing numerical techniques. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=hybrid%20block%20methods" title="hybrid block methods">hybrid block methods</a>, <a href="https://publications.waset.org/abstracts/search?q=boundary%20value%20problem" title=" boundary value problem"> boundary value problem</a>, <a href="https://publications.waset.org/abstracts/search?q=polynomial%20interpolation" title=" polynomial interpolation"> polynomial interpolation</a>, <a href="https://publications.waset.org/abstracts/search?q=adaptive%20step-size%20control" title=" adaptive step-size control"> adaptive step-size control</a>, <a href="https://publications.waset.org/abstracts/search?q=collocation%20methods" title=" collocation methods"> collocation methods</a> </p> <a href="https://publications.waset.org/abstracts/188910/on-the-solution-of-boundary-value-problems-blended-with-hybrid-block-methods" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/188910.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">31</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">18147</span> Optimal Relaxation Parameters for Obtaining Efficient Iterative Methods for the Solution of Electromagnetic Scattering Problems</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/abstracts/search?q=Nadaniela%20Egidi">Nadaniela Egidi</a>, <a href="https://publications.waset.org/abstracts/search?q=Pierluigi%20Maponi"> Pierluigi Maponi</a> </p> <p class="card-text"><strong>Abstract:</strong></p> The approximate solution of a time-harmonic electromagnetic scattering problem for inhomogeneous media is required in several application contexts, and its two-dimensional formulation is a Fredholm integral equation of the second kind. This integral equation provides a formulation for the direct scattering problem, but it has to be solved several times also in the numerical solution of the corresponding inverse scattering problem. The discretization of this Fredholm equation produces large and dense linear systems that are usually solved by iterative methods. In order to improve the efficiency of these iterative methods, we use the Symmetric SOR preconditioning, and we propose an algorithm for the evaluation of the associated relaxation parameter. We show the efficiency of the proposed algorithm by several numerical experiments, where we use two Krylov subspace methods, i.e., Bi-CGSTAB and GMRES. <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/abstracts/search?q=Fredholm%20integral%20equation" title="Fredholm integral equation">Fredholm integral equation</a>, <a href="https://publications.waset.org/abstracts/search?q=iterative%20method" title=" iterative method"> iterative method</a>, <a href="https://publications.waset.org/abstracts/search?q=preconditioning" title=" preconditioning"> preconditioning</a>, <a href="https://publications.waset.org/abstracts/search?q=scattering%20problem" title=" scattering problem"> scattering problem</a> </p> <a href="https://publications.waset.org/abstracts/142902/optimal-relaxation-parameters-for-obtaining-efficient-iterative-methods-for-the-solution-of-electromagnetic-scattering-problems" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/abstracts/142902.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">103</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">18146</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> <ul class="pagination"> <li class="page-item disabled"><span class="page-link">&lsaquo;</span></li> <li class="page-item active"><span class="page-link">1</span></li> <li class="page-item"><a class="page-link" href="https://publications.waset.org/abstracts/search?q=numerical%20methods&page=2">2</a></li> <li class="page-item"><a class="page-link" href="https://publications.waset.org/abstracts/search?q=numerical%20methods&page=3">3</a></li> <li class="page-item"><a class="page-link" href="https://publications.waset.org/abstracts/search?q=numerical%20methods&page=4">4</a></li> <li class="page-item"><a class="page-link" href="https://publications.waset.org/abstracts/search?q=numerical%20methods&page=5">5</a></li> <li class="page-item"><a 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