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  • Open Access


    A Comparative Numerical Study of Parabolic Partial Integro-Differential Equation Arising from Convection-Diffusion

    Kamil Khan1, Arshed Ali1,*, Fazal-i-Haq2, Iltaf Hussain3, Nudrat Amir4

    CMES-Computer Modeling in Engineering & Sciences, Vol.126, No.2, pp. 673-692, 2021, DOI:10.32604/cmes.2021.012730

    Abstract This article studies the development of two numerical techniques for solving convection-diffusion type partial integro-differential equation (PIDE) with a weakly singular kernel. Cubic trigonometric B-spline (CTBS) functions are used for interpolation in both methods. The first method is CTBS based collocation method which reduces the PIDE to an algebraic tridiagonal system of linear equations. The other method is CTBS based differential quadrature method which converts the PIDE to a system of ODEs by computing spatial derivatives as weighted sum of function values. An efficient tridiagonal solver is used for the solution of the linear system… More >

  • Open Access


    A Differential Quadrature Based Approach for Volterra Partial Integro-Differential Equation with a Weakly Singular Kernel

    Siraj-ul-Islam1, Arshed Ali2,*, Aqib Zafar1, Iltaf Hussain1

    CMES-Computer Modeling in Engineering & Sciences, Vol.124, No.3, pp. 915-935, 2020, DOI:10.32604/cmes.2020.011218

    Abstract Differential quadrature method is employed by numerous researchers due to its numerical accuracy and computational efficiency, and is mentioned as potential alternative of conventional numerical methods. In this paper, a differential quadrature based numerical scheme is developed for solving volterra partial integro-differential equation of second order having a weakly singular kernel. The scheme uses cubic trigonometric B-spline functions to determine the weighting coefficients in the differential quadrature approximation of the second order spatial derivative. The advantage of this approximation is that it reduces the problem to a first order time dependent integro-differential equation (IDE). The More >

  • Open Access


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

    Marjan Uddin1, *, Najeeb Ullah2, Syed Inayat Ali Shah2

    CMES-Computer Modeling in Engineering & Sciences, Vol.123, No.3, pp. 957-972, 2020, DOI:10.32604/cmes.2020.08911

    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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