A New Approximation Method for the Systems of Nonlinear Fredholm Integral Equations

Salih Yalçınbaş^1, and Kübra Erdem¹

¹Department of Mathematics, Faculty of Science and Arts, Celal Bayar University, Manisa, Turkey

Cite this paper:
Salih Yalçınbaş and Kübra Erdem. A New Approximation Method for the Systems of Nonlinear Fredholm Integral Equations. Applied Mathematics and Physics. 2014; 2(2):40-48. doi: 10.12691/amp-2-2-3

Abstract

In this paper, we present a new approximate method for solving systems of nonlinear Fredholm integral equation. This method is based on, first, differentiating both sides of integral equations n times and then substituting the Taylor series the unknown functions in the resulting equation and later, transforming to a matrix equation. By merging these results, a new system which corresponds to a system of linear algebraic equations is obtained. The solution of this system yields the Taylor coefficients of the solution function. Numerical results and comparisons with the exact solution are included to demostrate the validity and applicability of the technique.

Keywords:
nonlinear Fredholm systems Taylor polynomials and series

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References:

[1]	C.T.H. Baker, A perspective on the numerical treatment of Volterra equations, J. Comput. Appl. Math. 125, 217-249, 2000.
[2]	P. Linz, Analytical and numerical methods for Volterra equations, SIAM, Philadelphia, PA, 1985.
[3]	L.M. Delves, J.L. Mohamed, Computational methods for integral equations, Cambridge University Press, Cambridge, 1985.
[4]	F. Bloom, Asymptotic bounds for solutions to a system of damped integrodifferential equations of electromagnetic theory, J. Math. Anal. Appl. 73, 524-542, 1980.
[5]	K. Holmaker, Global asymptotic stability for a stationary solution of a system of integro-differential equations describing the formation of liver zones, SIAM J. Math. Anal. 24, (1) 116-128, 1993.
[6]	M.A. Abdou, Fredholm–Volterra integral equation of the first kind and contact problem, Appl. Math. Comput. 125, 177-193, 2002.
[7]	L.K. Forbes, S. Crozier and D.M. Doddrell, Calculating current densities and fields produced by shielded magnetic resonance imaging probes, SIAM J. Appl. Math. 57, 2, 401-425, 1997.
[8]	P.J. Houwen, B.P.Sommeijer, Euler–Chebyshev methods for integro-differential equations, Appl. Numer. Math. 24, 203-218, 1997.
[9]	W.H. Enright and M. Hu, Continuous Runge-Kutta methods for neutral Volterra integro-differential equations with delay, Appl. Numer. Math. 24, 175-190, 1997.
[10]	K. Maleknejad, F. Mirzae and S. Abbasbandy, Solving linear integro-differential equations system by using rationalized Haar functions method, Appl. Math. Comput. 155, 317-328, 2004.
[11]	K. Maleknejad and K. M. Tavassoli, Solving linear integro-differential equation system by Galerkin methods with hybrid functions, Appl. Math. Comput. 159, 603-612, 2004.
[12]	R. P. Kanwall and K. C. Liu, A Taylor expansion approach for solving integral equations, Int. J. Math. Educ. Sci. Technol., 20, 3, 411-414, 1989.
[13]	M. Sezer, Taylor polynomial solutions of volterra integral equations, Int. J. Math. Educ. Sci. Technol., 25, 5, 625-633, 1994.
[14]	M. Sezer, A method for the approximate solution of the second-order linear differential equations in terms of Taylor polynomials, Int. J. Math. Educ. Sci. Technol., 27, 6, 821- 834, 1996.
[15]	S. Yalçınbaş and M. Sezer, The approximate solution of high-order linear Volterra Fredholm integro-differential equations in terms of Taylor polynomials, Appl. Math. Comput., 112, 291-308, 2000.
[16]	S. Yalçınbaş, Taylor polynomial solutions of nonlinear Volterra-Fredholm integral equations, Appl. Math. Comput., 127, 195-206, 2002.
[17]	S. Yalçınbaş, A. Şahiner, M. Demirbaş, B. Altınay and S. Kocakuş, The approximate solution of high-order linear differential equation systems with variable coefficients in terms of Taylor polynomials, The third international conference ‘‘Tools for mathematical modelling’’, Saint Petersburg, 18-23 June 2001, 8, 175-188, 2001.
[18]	Yalçınbaş and F. Yeniçerioğlu, The approximate solutions of high-order linear differential equation systems with variable coefficients, Far East Journal of Dynamical Systems, 6, 2, 139-157, 2004.
[19]	A. Akyüz-Daşçıoğlu and M. Sezer, Chebyshev polynomial solutions of systems of higher-order linear Fredholh-Volterra integro-differential equations, J. Franklin Institute 342, 688-701, 2005.
[20]	E. Yusufoğlu (Agadjavov), A homotopy perturbation algorithm to solve a system of Fredholm–Volterra type integral equations, Mathematical and Computer Modelling 47, 1099-1107, 2008.
[21]	S. Yalçınbaş and K. Erdem, Approximate solutions of nonlinear Volterra integral equation systems, International Journal of Modern Physics B, 24, 32, 6235-6258, 2010.