TY  - JOUR
T1  - A generalized operational method for solving integro–partial differential equations based on Jacobi polynomials
AU  - Borhanifar, Abdollah
AU  - Sadri, Khadijeh
PY  - 2016
DA  - April
JF  - Hacettepe Journal of Mathematics and Statistics
PB  - Hacettepe University
WT  - DergiPark
SN  - 2651-477X
SP  - 311
EP  - 335
VL  - 45
IS  - 2
LA  - en
AB  - In this paper, a numerical method is developed for solving linear andnonlinear integro-partial differential equations in terms of the two variables Jacobi polynomials. First, some properties of these polynomialsand several theorems are presented then a generalized approach implementing a collocation method in combination with two dimensionaloperational matrices of Jacobi polynomials is introduced to approximate the solution of some integro–partial differential equations withinitial 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 matricesof integration and product to achieve the best approximation of thetwo dimensional integro–differential equations. Numerical results aregiven to confirm the reliability of the proposed method for solving theseequations.
KW  - Best approximation
KW  - Collocation method
KW  - Integro–partial differential equations
KW  - Operational matrix
KW  - Shifted Jacobi polynomials
CR  - Bhrawy, A. H. An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system, Appl. Math. Comput. 247, 30–46, 2014.
CR  - 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.
CR  - 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.
CR  - 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.
CR  - 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.
CR  - 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.
CR  - Borhanifar, A. and Abazari, R. Exact solutions for nonlinear Schrödinger equations by
differential transformation method, App. Math. Computing. 35 (1), 37–51, 2011.
CR  - 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.
CR  - 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.
CR  - 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.
CR  - 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.
CR  - 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.
CR  - 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.
CR  - 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.
CR  - 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.
CR  - 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.
CR  - 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.
CR  - 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.
CR  - 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
CR  - 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.
CR  - 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.
CR  - 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.
CR  - Ma, J. Finite element methods for partial Volterra integro–differential equations on two–
dimensional unbounded spatial domains, Appl. Math. Comput. 186, 598–609, 2007.
CR  - Szegö, G. Orthogonal Polynomials (American Mathematical Society, Providence, Rhode
Island, 1939).
CR  - 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.
CR  - 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.
UR  - https://dergipark.org.tr/en/pub/hujms/issue//524218
L1  - https://dergipark.org.tr/en/download/article-file/644566
ER  -