Numerical solution of Volterra partial integro-differential equations based on sinc-collocation method

We provide the numerical solution of a Volterra integro-differential equation of parabolic type with memory term subject to initial boundary value conditions. Finite difference method in combination with product trapezoidal integration rule is used to discretize the equation in time and sinc-collocation method is employed in space. A weakly singular kernel has been viewed as an important case in this study. The convergence analysis has been discussed in detail, which shows that the approach exponentially converges to the solution. Furthermore, numerical examples and illustrations are presented to prove the validity of the suggested method.

[1]  Peter Linz,et al.  Analytical and numerical methods for Volterra equations , 1985, SIAM studies in applied and numerical mathematics.

[2]  Masaaki Sugihara,et al.  Near optimality of the sinc approximation , 2003, Math. Comput..

[3]  Kenneth L. Bowers,et al.  Sinc methods for quadrature and differential equations , 1987 .

[4]  I. Olkin,et al.  Inequalities: Theory of Majorization and Its Applications , 1980 .

[5]  G. Fairweather,et al.  Finite element methods for parabolic and hyperbolic partial integro-differential equations , 1988 .

[6]  Jalil Rashidinia,et al.  Solution of a Volterra integral equation by the Sinc-collocation method , 2007 .

[7]  William McLean,et al.  A second-order accurate numerical method for a fractional wave equation , 2006, Numerische Mathematik.

[8]  W. Schneider,et al.  Fractional diffusion and wave equations , 1989 .

[9]  Tao Tang,et al.  A finite difference scheme for partial integro-differential equations with a weakly singular kernel , 1993 .

[10]  Per Christian Hansen,et al.  Rank-Deficient and Discrete Ill-Posed Problems , 1996 .

[11]  Da Xu,et al.  A compact difference scheme for a partial integro-differential equation with a weakly singular kernel , 2015 .

[12]  Vidar Thomée,et al.  Numerical solution of an evolution equation with a positive-type memory term , 1993, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[13]  Da Xu,et al.  The Global Behavior of Finite Difference-Spatial Spectral Collocation Methods for a Partial Integro-differential Equation with a Weakly Singular Kernel , 2013 .

[14]  M. El-Gamel.  A note on solving the fourth-order parabolic equation by the sinc-Galerkin method , 2015 .

[15]  G. B. Loghmani,et al.  A numerical solution of nonlinear parabolic-type Volterra partial integro-differential equations using radial basis functions , 2012 .

[16]  Vidar Thomée,et al.  Discretization with variable time steps of an evolution equation with a positive-type memory term , 1996 .

[17]  Jalil Rashidinia,et al.  Sinc-Galerkin method for numerical solution of the Bratu’s problems , 2012, Numerical Algorithms.

[18]  Jalil Rashidinia,et al.  Application of Sinc-Galerkin method to singularly perturbed parabolic convection-diffusion problems , 2014, Numerical Algorithms.

[19]  William McLean,et al.  Discontinuous Galerkin method for an evolution equation with a memory term of positive type , 2009, Math. Comput..

[20]  G. Fairweather,et al.  Orthogonal spline collocation methods for partial di erential equations , 2001 .

[21]  J. Biazar,et al.  FD-RBF for Partial Integro-Differential Equations with a Weakly Singular Kernel , 2015 .

[22]  Jennifer Dixon,et al.  On the order of the error in discretization methods for weakly singular second kind non-smooth solutions , 1985 .

[23]  J. Rashidinia,et al.  Numerical solutions of one-dimensional non-linear parabolic equations using Sinc collocation method , 2015 .

[24]  J. M. Sanz-Serna,et al.  A numerical method for a partial integro-differential equation , 1988 .

[25]  MEHDI DEHGHAN,et al.  Solution of a partial integro-differential equation arising from viscoelasticity , 2006, Int. J. Comput. Math..

[26]  Michael Renardy,et al.  Mathematical problems in viscoelasticity , 1987 .

[27]  Mehdi Dehghan,et al.  An efficient pseudo‐spectral Legendre–Galerkin method for solving a nonlinear partial integro‐differential equation arising in population dynamics , 2013 .

[28]  Khosrow Maleknejad,et al.  On the solution of the integro-differential equation with an integral boundary condition , 2013, Numerical Algorithms.

[29]  Mehdi Dehghan,et al.  Space-time spectral method for a weakly singular parabolic partial integro-differential equation on irregular domains , 2014, Comput. Math. Appl..

[30]  U Jin Choi,et al.  Spectral collocation methods for a partial integro-differential equation with a weakly singular kernel , 1998, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[31]  Da Xu,et al.  Quasi wavelet based numerical method for a class of partial integro-differential equation , 2012, Appl. Math. Comput..

[32]  Nicole Fruehauf.  Numerical Methods Based On Sinc And Analytic Functions , 2016 .

[33]  S. Xie,et al.  Two Numerical Algorithms for Solving a Partial Integro- Differential Equation with a Weakly Singular Kernel , 2012 .