Document Type : Original Article

**Author**

Malek Ashtar University of Technology, Tehran, Iran.

**Abstract**

In this paper, Galerkin and collocation methods based on shifted Legendre polynomials and spectral methods have been applied on nonlinear Volterra-Fredholm-Hammerstein (VFH) integral equations, these methods transfer the finding solution of a nonlinear integral equation to finding the solution of nonlinear algebraic equations, in order to solve these nonlinear algebraic equations we use Newton method composed by generalized minimal residual (NGMRes) method, the iteration number and running time for implementation of NGMRes method are important parameters that have been considered to solve this type of integral equations. These methods are applied on several various nonlinear VFH integral equations that confirm accuracy and efficiency of the methods.

**Keywords**

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113-121.

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[32] J. Radlow, A two-dimensional singular integral equation of diffraction theory, Bulletin of the American Mathematical Society, (70)4 1964, 596-599.

[33] Y. Saad, M.H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear system, SIAM Journal on scientific and statistical computing, 7(3) 1986, 856-869.

[34] A. Soulaimani, N.B. Salah, Y. Saad, Enhanced GMRES acceleration techniques for some CFD problems, International Journal of Computational Fluid Dynamics, 16(1) 2002, 1-20.

[35] J. Shen, T. Tang, L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer Publishing Company, Incorporated, 2011.

[36] S.C. Shiralashetti, R.A. Mundewadi, S.S. Naregal, B. Veeresh, Haar Wavelet Collocation Method for the Numerical Solution of Nonlinear Volterra-Fredholm-Hammerstein Integral Equations, Global Journal of Pure and Applied Mathematics, 13(2) 2017, 463-474.

[37] M.S. Tong, A Stable Integral Equation Solver for Electromagnetic Scattering by Large Scatterers with Concave Surface, Progress In Electromagnetics Research, 74 2007, 113-130.

[38] E. Voltchkova, Integro-Differential Equations for Option Prices in Exponential Lvy Models, Finance and Stochastics, 9(3) 2005, 299-325.

[39] P. Wolfe, Eigenfunctions of the Integral Equation for the Potential of the Charged Disk, Journal of Mathematical Physics, 12(7) 1971, 1215-1218.

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January 2021

Pages 1-16

**Receive Date:**25 November 2020**Revise Date:**04 January 2021**Accept Date:**15 January 2021**First Publish Date:**15 January 2021