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Numerical Methods for Differential and Integral Equations

Submission Deadline: 30 November 2019 (closed) View: 171

Guest Editors

Professor Şuayip Yüzbaşı, Department of Mathematics, Faculty of Science, Akdeniz University, Turkey
Professor Kamel Al-Khaled, Department of Mathematics, Jordan University of Science and Technology, Jordan
Professor Nurcan Baykuş Savaşaneril, Department of Mathematics, Faculty of Science, Dokuz Eylul University, Turkey
Professor Devendra Kumar, Department of Mathematics, University of Rajasthan, India

Summary

This special issue will focus on numerical solutions of diferential, integral and integro-differential equations, partial differential equations, fractional differential equations, fractional partial differential equations, stochastic partial differential equations and functional differential equations. The solutions of mentioned equations have a major role in many applied areas of science and engineering, For examle, physics, chemistry, astronomy, biology, mechanics, electronic, economics, potential theory, electrostatics. Since the mentioned equations are usually difficult to solve analytically, numerical methods are required. Therefore, this Special Issue will contribute to new numerical methods for solving the above mentioned equations and thus, many problems in science and engineering will be solved by means of new numerical methods in this special issue. We would like to invite researchers working on this topic to submit their articles to this Special Issue.

Keywords

• Numerical methods for ordinary differential equations
• Numerical methods for delay differential equations
• Numerical methods for partial differential equations
• Numerical methods for fractional differential equations
• Numerical methods for integral equations
• Numerical methods for integro-differential equations
• Numerical methods for model problems of differential equations
• Numerical methods for fractional PDE
• Numerical methods for Stochastic PDE
• Numerical methods for functional differential equations

Published Papers

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  • Open Access

    EDITORIAL

    Introduction to the Special Issue on Numerical Methods for Differential and Integral Equations

    Şuayip Yüzbaşı, Kamel Al-Khaled, Nurcan Baykuş Savaşaneril, Devendra Kumar
    CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.3, pp. 913-915, 2020, DOI:10.32604/cmes.2020.011225
    （This article belongs to the Special Issue: Numerical Methods for Differential and Integral Equations)
    Abstract This article has no abstract. More >

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    3238

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    2394

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    ARTICLE

    Crank-Nicolson ADI Galerkin Finite Element Methods for Two Classes of Riesz Space Fractional Partial Differential Equations

    An Chen
    CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.3, pp. 917-939, 2020, DOI:10.32604/cmes.2020.09224
    （This article belongs to the Special Issue: Numerical Methods for Differential and Integral Equations)
    Abstract In this paper, two classes of Riesz space fractional partial differential equations including space-fractional and space-time-fractional ones are considered. These two models can be regarded as the generalization of the classical wave equation in two space dimensions. Combining with the Crank-Nicolson method in temporal direction, efficient alternating direction implicit Galerkin finite element methods for solving these two fractional models are developed, respectively. The corresponding stability and convergence analysis of the numerical methods are discussed. Numerical results are provided to verify the theoretical analysis. More >

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    2429

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    ARTICLE

    A Galerkin-Type Fractional Approach for Solutions of Bagley-Torvik Equations

    Şuayip Yüzbaşı, Murat Karaçayır
    CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.3, pp. 941-956, 2020, DOI:10.32604/cmes.2020.08938
    （This article belongs to the Special Issue: Numerical Methods for Differential and Integral Equations)
    Abstract In this study, we present a numerical scheme similar to the Galerkin method in order to obtain numerical solutions of the Bagley Torvik equation of fractional order 3/2. In this approach, the approximate solution is assumed to have the form of a polynomial in the variable t = xα , where α is a positive real parameter of our choice. The problem is firstly expressed in vectoral form via substituting the matrix counterparts of the terms present in the equation. After taking inner product of this vector with nonnegative integer powers of t up to a More >

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    3073

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    2096

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    ARTICLE

    RBF Based Localized Method for Solving Nonlinear Partial Integro-Differential Equations

    Marjan Uddin, Najeeb Ullah, Syed Inayat Ali Shah
    CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.3, pp. 957-972, 2020, DOI:10.32604/cmes.2020.08911
    （This article belongs to the Special Issue: Numerical Methods for Differential and Integral Equations)
    Abstract In this work, a numerical scheme is constructed for solving nonlinear parabolictype partial-integro differential equations. The proposed numerical scheme is based on radial basis functions which are local in nature like finite difference numerical schemes. The radial basis functions are used to approximate the derivatives involved and the integral is approximated by equal width integration rule. The resultant differentiation matrices are sparse in nature. After spatial approximation using RBF the partial integro-differential equations reduce to the system of ODEs. Then ODEs system can be solved by various types of ODE solvers. The proposed numerical scheme More >

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    3146

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    2218

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    ARTICLE

    Bell Polynomial Approach for the Solutions of Fredholm Integro-Differential Equations with Variable Coefficients

    Gökçe Yıldız, Gültekin Tınaztepe, Mehmet Sezer
    CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.3, pp. 973-993, 2020, DOI:10.32604/cmes.2020.09329
    （This article belongs to the Special Issue: Numerical Methods for Differential and Integral Equations)
    Abstract In this article, we approximate the solution of high order linear Fredholm integro-differential equations with a variable coefficient under the initial-boundary conditions by Bell polynomials. Using collocation points and treating the solution as a linear combination of Bell polynomials, the problem is reduced to linear system of equations whose unknown variables are Bell coefficients. The solution to this algebraic system determines the approximate solution. Error estimation of approximate solution is done. Some examples are provided to illustrate the performance of the method. The numerical results are compared with the collocation method based on Legendre polynomials More >

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    2002

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    ARTICLE

    An ADI Finite Volume Element Method for a Viscous Wave Equation with Variable Coefficients

    Mengya Su, Zhihao Ren, Zhiyue Zhang
    CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.2, pp. 739-776, 2020, DOI:10.32604/cmes.2020.08563
    （This article belongs to the Special Issue: Numerical Methods for Differential and Integral Equations)
    Abstract Based on rectangular partition and bilinear interpolation, we construct an alternating-direction implicit (ADI) finite volume element method, which combined the merits of finite volume element method and alternating direction implicit method to solve a viscous wave equation with variable coefficients. This paper presents a general procedure to construct the alternating-direction implicit finite volume element method and gives computational schemes. Optimal error estimate in L2 norm is obtained for the schemes. Compared with the finite volume element method of the same convergence order, our method is more effective in terms of running time with the increasing of More >

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    2251

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    2

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    ARTICLE

    A Discrete Model of TB Dynamics in Khyber Pakhtunkhwa-Pakistan

    Muhammad Altaf Khan, Khanadan Khan, Mohammad A. Safi, Mahmoud H. DarAssi
    CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.2, pp. 777-795, 2020, DOI:10.32604/cmes.2020.08208
    （This article belongs to the Special Issue: Numerical Methods for Differential and Integral Equations)
    Abstract The present paper investigates the theoretical analysis of the tuberculosis (TB) model in the discrete-time case. The model is parameterized by the TB infection cases in the Pakistani province of Khyber Pakhtunkhwa between 2002 and 2017. The model is parameterized and the basic reproduction number is obtained and it is found R0 ¼ 1:5853. The stability analysis for the model is presented and it is shown that the discrete-time tuberculosis model is stable at the disease-free equilibrium whenever R0 < 1 and further we establish the results for the endemic equilibria and prove that the model More >

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    3464

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    2100

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    1

  • Open Access

    ARTICLE

    RBF-FD Method for Some Dispersive Wave Equations and Their Eventual Periodicity

    Marjan Uddin, Hameed Ullah Jan, Muhammad Usman
    CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.2, pp. 797-819, 2020, DOI:10.32604/cmes.2020.08717
    （This article belongs to the Special Issue: Numerical Methods for Differential and Integral Equations)
    Abstract In this paper, we approximate the solution and also discuss the periodic behavior termed as eventual periodicity of solutions of (IBVPs) for some dispersive wave equations on a bounded domain corresponding to periodic forcing. The constructed numerical scheme is based on radial kernels and local in nature like finite difference method. The temporal variable is executed through RK4 scheme. Due to the local nature and sparse differentiation matrices our numerical scheme efficiently recovers the solution. The results achieved are validated and examined with other methods accessible in the literature. More >

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    2048

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  • Open Access

    ARTICLE

    Modelling of Energy Storage Photonic Medium by WavelengthBased Multivariable Second-Order Differential Equation

    T. Binesh
    CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.1, pp. 333-351, 2020, DOI:10.32604/cmes.2020.08097
    （This article belongs to the Special Issue: Numerical Methods for Differential and Integral Equations)
    Abstract Wavelength-dependent mathematical modelling of the differential energy change of a photon has been performed inside a proposed hypothetical optical medium. The existence of this medium demands certain mathematical constraints, which have been derived in detail. Using reverse modelling, a medium satisfying the derived conditions is proven to store energy as the photon propagates from the entry to exit point. A single photon with a given intensity is considered in the analysis and hypothesized to possess a definite non-zero probability of maintaining its energy and velocity functions analytic inside the proposed optical medium, despite scattering, absorption, More >

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    2177

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    ARTICLE

    On Caputo-Type Cable Equation: Analysis and Computation

    Zhen Wang
    CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.1, pp. 353-376, 2020, DOI:10.32604/cmes.2020.08776
    （This article belongs to the Special Issue: Numerical Methods for Differential and Integral Equations)
    Abstract In this paper, a special case of nonlinear time fractional cable equation is studied. For the equation defined on a bounded domain, the existence, uniqueness, and regularity of the solution are firstly studied. Furthermore, it is numerically studied via the weighted and shifted Grünwald difference (WSGD) methods/the local discontinuous Galerkin (LDG) finite element methods. The derived numerical scheme has been proved to be stable and convergent with order O(∆t2 + hk+1), where ∆t, h, k are the time stepsize, the spatial stepsize, and the degree of piecewise polynomials, respectively. Finally, a numerical experiment is presented to verify the More >

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    3250

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    2113

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    ARTICLE

    Fractional Analysis of Viscous Fluid Flow with Heat and Mass Transfer Over a Flexible Rotating Disk

    Muhammad Shuaib, Muhammad Bilal, Muhammad Altaf Khan, Sharaf J. Malebary
    CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.1, pp. 377-400, 2020, DOI:10.32604/cmes.2020.08076
    （This article belongs to the Special Issue: Numerical Methods for Differential and Integral Equations)
    Abstract An unsteady viscous fluid flow with Dufour and Soret effect, which results in heat and mass transfer due to upward and downward motion of flexible rotating disk, has been studied. The upward or downward motion of non rotating disk results in two dimensional flow, while the vertical action and rotation of the disk results in three dimensional flow. By using an appropriate transformation the governing equations are transformed into the system of ordinary differential equations. The system of ordinary differential equations is further converted into first order differential equation by selecting suitable variables. Then, we More >

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    3279

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    2647

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  • Open Access

    ARTICLE

    A Numerical Algorithm Based on Quadratic Finite Element for Two-Dimensional Nonlinear Time Fractional Thermal Diffusion Model

    Yanlong Zhang, Baoli Yin, Yue Cao, Yang Liu, Hong Li
    CMES-Computer Modeling in Engineering & Sciences, Vol.122, No.3, pp. 1081-1098, 2020, DOI:10.32604/cmes.2020.07822
    （This article belongs to the Special Issue: Numerical Methods for Differential and Integral Equations)
    Abstract In this article, a high-order scheme, which is formulated by combining the quadratic finite element method in space with a second-order time discrete scheme, is developed for looking for the numerical solution of a two-dimensional nonlinear time fractional thermal diffusion model. The time Caputo fractional derivative is approximated by using the L2 -1 σ formula, the first-order derivative and nonlinear term are discretized by some second-order approximation formulas, and the quadratic finite element is used to approximate the spatial direction. The error accuracy O(h 3 + ∆t 2 ) is obtained, which is verified by the numerical More >

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    2229

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    ARTICLE

    Integral Transform Method for a Porous Slider with Magnetic Field and Velocity Slip

    Naeem Faraz, Yasir Khan, Amna Anjum, Anwar Hussain
    CMES-Computer Modeling in Engineering & Sciences, Vol.122, No.3, pp. 1099-1118, 2020, DOI:10.32604/cmes.2020.08389
    （This article belongs to the Special Issue: Numerical Methods for Differential and Integral Equations)
    Abstract Current research is about the injection of a viscous fluid in the presence of a transverse uniform magnetic field to reduce the sliding drag. There is a slip-on both the slider and the ground in the two cases, for example, a long porous slider and a circular porous slider. By utilizing similarity transformation Navier-Stokes equations are converted into coupled equations which are tackled by Integral Transform Method. Solutions are obtained for different values of Reynolds numbers, velocity slip, and magnetic field. We found that surface slip and Reynolds number has a substantial influence on the More >

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    2086

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    ARTICLE

    Analytical and Numerical Investigation for the DMBBM Equation

    Abdulghani Alharbi, Mahmoud A. E. Abdelrahman, M. B. Almatrafi
    CMES-Computer Modeling in Engineering & Sciences, Vol.122, No.2, pp. 743-756, 2020, DOI:10.32604/cmes.2020.07996
    （This article belongs to the Special Issue: Numerical Methods for Differential and Integral Equations)
    Abstract The nonlinear dispersive modified Benjamin-Bona-Mahony (DMBBM) equation is solved numerically using adaptive moving mesh PDEs (MMPDEs) method. Indeed, the exact solution of the DMBBM equation is obtained by using the extended Jacobian elliptic function expansion method. The current methods give a wider applicability for handling nonlinear wave equations in engineering and mathematical physics. The adaptive moving mesh method is compared with exact solution by numerical examples, where the explicit solutions are known. The numerical results verify the accuracy of the proposed method. More >

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