# Solving linear two-dimensional Fredholm integral equations system by triangular functions

## DOI:

https://doi.org/10.14419/ijamr.v5i4.6174## Published:

2016-10-25## Keywords:

Two-dimensional Fredholm integral equations system of the second kind (2D-FIES-2), Two m-sets of one-dimensional triangular functions (1D-TFs), Twodimensional triangular functions (2D-TFs).## Abstract

In this paper, we intend to offer a numerical method to solve linear two-dimensional Fredholm integral equations system of the second kind. This method converts the given two-dimensional Fredholm integral equations system into a linear system of algebraic equations by using twodimensional triangular functions. Moreover, we prove the convergence of the method. Finally the proposed method is illustrated by two examples and also results are compared with the exact solution by using computer simulations.

## References

[1] H. Almasieh, M. Roodaki, Triangular functions method for the solution of Fredholm integral equations system, Ain Shams Engineering Journal, (2012), 3, 411- 416.

[2]E. Babolian, Z. Masouri and S. Hatamzadeh-Varmazyar, A direct method for numerically solving integral equations system using orthogonal triangular functions, Int. J. Industrial Math. (2009), 2, 135-145.

[3]E. Babolian, R. Mokhtari , M. Salmani, Using direct method for solving variational problems via triangular orthogonal functions, Appl Math Comput (2007) , 191, 206-17.

[4]E. Babolian, Z. Masouri, S. Hatamzadeh-Varmazyar, Numerical solution of nonlinear Volterra-Fredholm integro-differential equations via direct method using triangular functions, Comput Math Appl (2009), 58, 239-47.

5]A. Deb, G. Sarkar, A. Sengupta, Triangular orthogonal functions for the analysis of continuous time systems. New Delhi: Elsevier, (2007).

[6]A. Deb, A. Dasgupta, G. Sarkar, A new set of orthogonal functions and its application to the analysis of dynamic systems. J Frank Inst (2006), 343, 1- 26.

[7]K. Maleknejad, Z. Jafari Behbahani, Applications of two-dimensional triangular functions for solving nonlinear class of mixed Volterra Fredholm integral equations, Math Comput Model (2012), 55, 1833- 44.

[8]F. Mirzaee, M. Komak Yari, E. Hadadiyan, Numerical solution of two-dimensional fuzzy Fredholm integral equations of the second kind using triangular functions, Beni-Suef university journal of basic and appl. (2015), 1-10.

[9] X. Lan, Variational iteration method for solving integral equations, Comput. Math Appl. (2007), 54, 1071- 78.

[10] F.G. Tricomi, Integral equations, Dover Publications, New York, (1982)

[11] W. Xie, F. Lin, A fast numerical solution method for two dimensional Fredholm integral.equations of the second kind, Appl Numer Math, (2009), 59, 1709- 19.