A generalized operational method for solving integro–partial differential equations based on Jacobi polynomials

Abstract

In this paper, a numerical method is developed for solving linear and nonlinear integro-partial differential equations in terms of the two variables Jacobi polynomials. First, some properties of these polynomials and several theorems are presented then a generalized approach implementing a collocation method in combination with two dimensional operational matrices of Jacobi polynomials is introduced to approximate the solution of some integro–partial differential equations with initial or boundary conditions. Also, it is shown that the resulted approximate solution is the best approximation for the considered problem. The main advantage is to derive the Jacobi operational matrices of integration and product to achieve the best approximation of the two dimensional integro–differential equations. Numerical results are given to confirm the reliability of the proposed method for solving these equations.

Keywords

• Best approximation,Collocation method,Integro–partial differential equations,Operational matrix,Shifted Jacobi polynomials

References

1. Bhrawy, A. H. An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system, Appl. Math. Comput. 247, 30–46, 2014.
2. Bhrawy, A. H., Abdelkawy, M. A., Zaky, M. A., and Baleanu, D. Numerical simulation of time variable fractional order mobile-immobile advection-dispersion model, Rom. Rep. Phys. 67 (3), 773–791, 2016.
3. Bhrawy, A. H., Alofi, A. S., and Ezz–Eldien, S. S. A quadrature tau method for fractional differential equations with variable coefficients, Appl. Math. Lett. 24, 2146–2152, 2011.
4. Bhrawy, A. H.,Doha, E. H., Baleanu, D., and Ezz–Eldien, S. S. A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion–wave equations, J. Comput. Phys. 293, Issue C, 142–156, 2015.
5. Bhrawy, A. H. and Zaky, M. A. A method based on the Jacobi tau approximation for solving multi–term time–space fractional partial differential equations, J. Comput. Phys. 281, 876– 895, 2015.
6. Bhrawy, A. H., Zaky, M. A., and Baleanu, D. New Numerical Approximations for Space– Time Fractional Burgers’ Equations via a Legendre Spectral–Collocation Method, Rom. Rep. Phys. 67 (2), 340–349, 2015.
7. Borhanifar, A. and Abazari, R. Exact solutions for nonlinear Schrödinger equations by differential transformation method, App. Math. Computing. 35 (1), 37–51, 2011.
8. Borhanifar, A., Jafari, H., and Karimi, S. A. New solitary wave solutions for the bad boussinesq and good boussinesq equations, Numer. Meth. part. Diff. Equ. 25, 1231–1237, 2009.

9. Borhanifar, A. and Kabir, M. M. New periodic and soliton solutions by application of application of exp–function method for nonlinear evolution equation, Comput. Appl. Math. 229, 158–167, 2009.
10. Borhanifar, A., Kabir, M. M., and Vahdat, M. New periodic and soliton wave solutions for the generalized zakharov system and (2 + 1)–dimensional nizhnik-novikov-veselov system, Chaos. Solit. Fract. 42, 1646–1654, 2009.
11. Borhanifar, A. and Sadri, Kh. A new operational approach for numerical solution of generalized functional integro–differential equations, J. Comput. Appl. Math. 279, 80–96, 2015.
12. Borhanifar, A. and Sadri, Kh. Numerical solution for systems of two dimensional integral equations by using Jacobi operational collocation method, Sohag. J. Math. 1, 15–26, 2014.
13. Borhanifar, A. and zamiri, A. Application of ( G0 G )−expansion method for the Zhiber–Shabat equation and other related equations, Math. Comput. Model. 54, 2109–2116, 2011.
14. Doha, E. H., Abd-Elhameed, W. M., and Youssri, Y. H. Efficient spectral–Petrov–Galerkin methods for the integrated forms of third–and fifth–order elliptic differential equations using general parameters generalized Jacobi polynomials, Appl. Math. Comput. 218, 7727–7740, 2012.
15. Doha, E. H. and Bhrawy, A. H. An efficient direct solver for multi dimensional elliptic Robin boundary value problems using a Legendre spectral–Galerkin method, Comput. Math. Appl. 64 (4), 558– 571, 2012.
16. Doha, E. H., Bhrawy, A. H., Abdelkawy, M. A., and Van Gorder, R. A. Jacobi–Gauss– Lobatto collocation method for the numerical solution of 1 + 1 nonlinear Schrödinger equations, J. Comput. Phys. 261, 244–255, 2014.
17. Doha, E. H., Bhrawy, A. H., and Ezz–Eldien, S. S. A new Jacobi operational matrix: An application for solving fractional differential equations, Appl. Math. Model., 36 (10), 4931– 4943, 2012. [18] Doha, E. H., Bhrawy, A. H., and Ezz-Eldien, S. S. Efficient Chebyshev spectral methods for solving multi–term fractional orders differential equations, Appl. Math. Model. 35, 5662– 5672, 2011.
18. Doha, E. H., Bhrawy, A. H., and Hafez, R. M. A Jacobi–Jacobi dual–Petrov–Galerkin method for third–and fifth–order differential equations, Math. Com. Model. 2011; 53, 1820–1832, 2011.
19. Doha, E. H., Bhrawy, A. H., and Hafez, R. M. On Shifted Jacobi Spectral Method For High– Order Multi–Point Boundary Value Problems, Commun. Nonlin. Sci Numer. Simulat. 17 (10), 3802–3810, 2012. 335
20. Guezane-Lakoud, A., Bendjazia, N., and Khaldi, R. Galerkin method applied to telegraph integro-differential equation with a weighted integral condition, Bound. Val. Prob. 102, 1–12, 2013.
21. Hosseini, S. M. and Shahmorad, S. Numerical solution of a class of integro–differential equations by the Tau method with an error estimation, Appl. Math. Comput. 136, 559–570, 2003.
22. Karimi Vanani, S. and Aminataei, A. Tau approximate solution of fractional partial differential equations, Comput. Math. Appl. 62, 1075–1083, 2011. [24] Kreyszig, E. Introduction Functional Analysis with Applications (USA: Wiley, 1978). [25] Luo, M., Xu, D., Li, L., and Yang, X. Quasi wavelet based on numerical method for Volterra integro–differential equations on unbounded spatial domains, J. Appl. Math. Comput. 227, 509–517, 2014.
23. Ma, J. Finite element methods for partial Volterra integro–differential equations on two– dimensional unbounded spatial domains, Appl. Math. Comput. 186, 598–609, 2007.
24. Szegö, G. Orthogonal Polynomials (American Mathematical Society, Providence, Rhode Island, 1939).
25. Tari, A., Rahimi, M. Y., Shahmorad, S., and Talati, F. Development of the Tau Method for the Numerical Solution of Two–dimensional Linear Volterra Integro-differential Equations, Compu. Meth. Appl. Math. 9, 421–435, 2009.
26. Tari, A. and Shahmorad, S. Differential transform method for the system of two dimensional nonlinear Volterra integro–differential equations, J. Comput. Math. Appl. 61, 2621–2629, 2011.