Document Type : Final File

Authors

Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, P. O. Box 41335-19141, Rasht, Islamic Republic of Iran

Abstract

Compact finite difference scheme is applied for a partial integro-differential equation with a weakly singular kernel. The product trapezoidal method is applied for discretization of the integral term. The order of accuracy in space and time is , where . Stability and convergence in  norm are discussed through energy method. Numerical examples are provided to confirm the theoretical prediction and to show that the combination of the compact finite difference approximation and product trapezoidal method give an efficient method for solving a partial integro-differential equation.

Keywords

Jangveladze T., Kiguradze Z. and Neta B. Numerical Solutions of Three Classes of Nonlinear Parabolic Integro-Differential Equations, Academic Press, New York., (2016).
2. Dumitru B., Darzi R. and Agheli B. New study of weakly singular kernel fractional fourth- order partial integro-differential equations based on the optimum q-homotopic analysis method, J. Comput & Appl. Math., 320: 193-201 (2017).
3. Babaaghaie A. and Maleknejad K. Numerical solutions of nonlinear two-dimensional partial Volterra integro-differential equations by Haar wavelet, J. Comput & Appl. Math., 317: 643-651 (2017).
4. Canuto C. and Quarteroni A. Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comp., 38: 67-86 (1982).
5. Lakestani M., Nemati Saray B. and Dehghan M. Numerical solution for the weakly singular Fredholm integro-differential equations using Legendre multiwavelets, J. Comput. & Appl. Math., 235(11): 3291-3303 (2011).
6. Tahami M., Hemmat A. A. and Yousefi SA. Numerical solution of two-dimensional first kind Fredholm integral equations by using linear Legendre wavelet, Int. J. Wavel. Multi. Resolu.& Inf. Proce., 14(01): 1-20 (2016).
7. Lopez-Marcos J.C. A difference scheme for a nonlinear partial integro-differential equation, SIAM J. Numer. Anal., 27: 20–31(1990).
8. Xu D. On the discretization in time for a partial integro-differential equations with a weakly singular kernel I: smooth initial data, Appl. Math. Comput. 58: 1–27 (1993).
9. Xu D. On the discretization in time for a partial integro-differential equations with a weakly singular kernel II: nonsmooth initial data, Appl. Math. Comput., 58: 29–60 (1993).
10. Singh S., Patel V. K., Singh V. K. and Tohidi E. Numerical solution of nonlinear weakly singular partial integro-differential equation via operational matrices, Appl. Math. & Comput., 298: 310-321 (2017).
11. Fakhar-Izadi F. and Dehghan M. Space–time spectral method for a weakly singular parabolic partial integro-differential equation on irregular domains, Comput. & Math. with Appl., 67(10): 1884–1904 (2014).
12. Tang T. A finite difference scheme for a partial integro-differential equations with a weakly singular kernel, Appl. Numer. Math., 11: 309–319 (1993).
13. Chen H. and Xu D. A compact difference scheme for an evolution equation with a weakly singular kernel, Numer. Math. Theory Methods Appl., 4: 559–572(2012).
14. Luo M., Xu D. and Limei L. A compact difference scheme for a partial integro- differential equation with a weakly singular kernel, Appl. Math. Modelling., 39(2): 947-954 (2015).
15. Spotz W. F. PhD Thesis, University of Texas at Austin, Austin, TX. (1995).