NON-STANDARD FINITE DIFFERENCE METHOD FOR NUMERICAL SOLUTION OF SECOND ORDER LINEAR FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS

Subject Areas : International Journal of Mathematical Modelling & Computations

Pramod Kumar Pandey ¹

1 - Dyal Singh College (University of Delhi) India Department of Mathematics

Received: 2016-04-15 Accepted : 2016-04-15 Published : 2015-03-21

Keywords:

Abstract :

In this article we have considered a non-standard finite difference method for the solution of second order Fredholm integro differential equation type initial value problems. The non-standard finite difference method and the composite trapezoidal quadrature method is used to transform the Fredholm integro-differential equation into a system of equations. We have also developed a numerical method for the numerical approximation of the derivative of the solution of the problems. The numerical results in experiment on some model problems show the simplicity and efficiency of the method. Numerical results showed that the proposed method is convergent and at least second order of accurate.

References:

Delves, L. M. and Mohamed, J. L., Computational Methods for Integral Equations. Cambridge University Press, Cambridge (1985).

Liz, E. and Nieto, J. J., Boundary value problems for second order integro-differential equations of Fredholm type. J. Comput. Appl. Math. 72, 215-225 (1996).

Zhao, J. and Corless, R.M. Corless, Compact finite difference method has been used for integro-differential equations. Appl. Math. Comput.,

: 271-288 (2006).

Chang, S.H., On certain extrapolation methods for the numerical solution of integro-differential equations. J. Math. Comp.,

: 165-171 (1982).

Yalcinbas, S., Taylor polynomial solutions of nonlinear Volterra-Fredholm integral equations.Appl. Math. Comput., 127: 195-206 (2002).

Phillips, D.L., A technique for the numerical solution of certain integral equations of the first kind, J. Assoc. Comput. Mach, 9, 84–96

(1962).

Tikhonov, A.N., On the solution of incorrectly posed problem and the method of regularization. Soviet Math, 4, 1035–1038 (1963).

He, J.H., Variational iteration method for autonomous ordinary differential systems. Appl. Math. Comput., 114(2/3), 115–123 (2000).

Wazwaz, A.M., A reliable modification of the Adomian decomposition method. Appl. Math. Comput., 102, 77–86 (1999).

Saadati, R.,Raftari, B., Abibi, H., Vaezpour , S.M. and Shakeri, S., A Comparison Between the Variational Iteration Method and Trapezoidal Rule for Solving Linear Integro-Differential Equations. World Applied Sciences Journal, 4: 321-325 (2008).

Hu, S., Wan, Z. and Khavanin, M., On the existence and uniqueness for nonlinear integro - differential equations. Jour. Math Phy. Sci., 21, no. 2, 93 - 103 (1987).

Hairer, E., Nørsett, S. P. and Wanner, G., Solving Ordinary Differential Equations I Nonstiff Problems (Second Revised Edition). Springer-Verlag New York, Inc. New York, NY, USA (1993).

Van Niekerk, F. D., Rational one step method for initial value problem. Comput. Math. Applic. Vol.16, No.12, 1035-1039 (1988).

Pandey, P. K., Nonlinear Explicit Method for first Order Initial Value Problems. Acta Technica Jaurinensis, Vol. 6, No. 2, 118-125 (2013).

Ramos, H., A non-standard explicit integration scheme for initial value problems. Applied Mathematics and Computation. 189, no.1,710-718 (2007).

Jain, M.K., Iyenger, S. R. K. and Jain, R. K., Numerical Methods for Scientific and Engineering Computation {(2/e)}. Willey Eastern Limited, New Delhi, (1987).

Lambert, J. D., Numerical Methods for Ordinary Differential Systems. John Wiley, England, 1991.

Pandey, P. K. and Jaboob, S. S. A., Explicit Method in Solving Ordinary Differential Equations of the Second Order”. Int. J. of Pure and Applied Mathematics, vol. 76, no.2, pp.233-239 (2012).

Saadatmandia, A. and Dehghanb, M., Numerical solution of the higher-order linear Fredholm integro-differential-difference equation with variable coefficients. Computers & Mathematics with Applications, Vol. 59, Issue 8, 2996–3004 (2010).

Jaradat, H. M., Awawdeh, F., Alsayyed, O., Series Solutions to the High-order Integro-differential Equations. Analele Universitatii Oradea Fasc. Matematica, Tom XVI, pp. 247-257 (2009).