A New Method for Solving Multi-Dimensional Fredholm Integral Equations and Its Convergence Analysis
الموضوعات : مجله بین المللی ریاضیات صنعتی
N. Mahmoodi ‎Darani‎
1
(Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran.)
الکلمات المفتاحية: Multi-dimensional linear and nonlinear, Fredholm Integral Equations, Multi-dimensional expansion, Convergence analysis, Boundary conditions,
ملخص المقالة :
In this paper, we focus on obtaining an approximate solution for multi-dimensional Fredholm integral equations of second kind. An expansion method is used for treatment multi-dimensional Fredholm integral equation of second kind. This method reduces multi-dimensional integral equation to a partial differential equation. After constructing boundary conditions, this partial differential equation reduces to algebraic equation that can be solved easily with any of the usual methods. Furthermore some theorems are proved for convergence analysis. Finally, for showing the efficiency of the method we use some numerical examples
[1] M. A. Abdou, A. A.Badr, M. Basseem, On a method for solving a two dimensional nonlinear integral equation of the second kind, J. Comp. Appl. Math.nm 235 (2011) 3589-3598.
[2] M. A. Abdou, W. G. El-Sayed, E. l. Deebs, A Solution of a nonlinear integral equation, J. Appl. Math. Comp. 160 (2005) 1-14.
[3] K. E. Atkinson, A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind, SIAM, Philadelphia, PA (1976).
[4] Z. Avazzadeh, M. Heydari, Chebyshev polynomial for solving two dimensional linear and nonlinear integral equations of the second kind, J. Comput. Appl. Math. 31 (2010) 127-142.
[5] L. M. Delves, J. L. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, New York, (1985).
[6] O. Diekman, Thresholds and traveling waves for the geographical spread of infection, J. Math. Biol. 6 (1978) 109-130.
[7] M. A. FariborziAraghi, S. Noeiaghdam, Validation of Numerical Algorithms: Stochastic Arithmetic, EntekhabBartar Publisher, Iran (2021) ISBN: 978-622-6498-09-8.
[8] M. A. Golberg, Numerical Solution of Integral Equations, Plenum Press, New York (1990).
[9] L. Hacia, On approximate solution for integral equations of mixed type, ZAMM. Z. Angew. Math.Mech. 76 (1996) 415-416.
[10] L. Hacia, On approximate solving of the fourier problems,Demonstration Math 12 (1979) 913-922.
[11] P. Huabsomboon, B. Novaprateep, H. Kaneko, On Taylor-series expansion methods for second kind integral equations, J. Comput. Appl. Math. 234 (2010) 1466-1472.
[12] P. J. Kauthen, Continuous time collocation methods for Volterra-Fredholm integral equations, Numer. Math. 56 (1989) 409-424.
[13] F. Ling, F. R. Lin, A fast numerical solution method for two-dimensional Fredholm integral equations of the second kind based on piecewise polynomial interpolation, Journal of Applied. Mathematics Computer 216 (2010) 3073-3080.
[14] k. Maleknejad, N. Aghazadeh, Numerical solution of volterra integral equations of the second kind with convolution kernel by using Taylor series expansion method, Appl. Math. Comput. 161 (2005) 915-922.
[15] N. Mikaeilvand, S. Noeiaghdam, Mean value theorem for integrals and its application on numerically solving of Fredholm integral equation of second kind with Toeplitz plus Hankel Kernel, Int. J. Industrial Mathematics, (2014).
[16] A. S. Noeiaghdam, M. A. FariborziAraghi, S. Abbasbandy, Valid implementation of Sinccollocation method to solve the fuzzy Fredholm integral equation, Journal of Computational and Applied Mathematics (2020) 112632.
[17] B. G. Pachpatte, On mixed VolterraFredholm type integral equations, Indian J. Pure Appl. Math. 17 (1986) 488-496.
[18] K. Ren, B. Zhang, H. Qiao, A simple Taylorseries expansion method for a class of second kind integral equation, J. Comput. Appl. Math. 110 (1999) 15-24.
