Non-Standard Difference Method for Numerical Solution of Linear Fredholm Integro-Differential Type Two-Point Boundary Value Problems

In this article we have considered Fredholm integro-differential equation type second-order boundary value problems and proposed a rational difference method for numerical solution of the problems. The composite trapezoidal quadrature and non-standard difference method are used to convert Fredholm integro-differential equation into a system of equations. 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.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Pandey, P. (2015) Non-Standard Difference Method for Numerical Solution of Linear Fredholm Integro-Differential Type Two-Point Boundary Value Problems. Open Access Library Journal, 2, 1-10. doi: 10.4236/oalib.1101465.

 [1] Delves, L.M. and Mohamed, J.L. (1985) Computational Methods for Integral Equations. Cambridge University Press, Cambridge. http://dx.doi.org/10.1017/CBO9780511569609 [2] Liz, E. and Nieto, J.J. (1996) Boundary Value Problems for Second Order Integro-Differential Equations of Fredholm Type. Journal of Computational and Applied Mathematics, 72, 215-225. http://dx.doi.org/10.1016/0377-0427(95)00273-1 [3] Zhao, J. and Corless, R.M. (2006) Compact Finite Difference Method Has Been Used for Integro-Differential Equations. Applied Mathematics and Computation, 177, 271-288. http://dx.doi.org/10.1016/j.amc.2005.11.007 [4] Ortiz, E.L. and Samara, L. (1981) An Operational Approach to the Tau Method for the Numerical Solution of Nonlinear Differential Equations. Computing, 27, 15-25. http://dx.doi.org/10.1007/BF02243435 [5] Chang, S.H. (1982) On Certain Extrapolation Methods for the Numerical Solution of Integro-Differential Equations. Mathematics of Computation, 39, 165-171. http://dx.doi.org/10.1090/S0025-5718-1982-0658220-4 [6] Yalcinbas, S. (2002) Taylor Polynomial Solutions of Nonlinear Volterra-Fredholm Integral Equations. Applied Mathematics and Computation, 127, 195-206. http://dx.doi.org/10.1016/S0096-3003(00)00165-X [7] Phillips, D.L. (1962) A Technique for the Numerical Solution of Certain Integral Equations of the First Kind. Journal of the ACM, 9, 84-96. http://dx.doi.org/10.1145/321105.321114 [8] Tikhonov, A.N. (1963) On the Solution of Incorrectly Posed Problem and the Method of Regularization. Soviet Mathematics, 4, 1035-1038. [9] He, J.H. (2000) Variational Iteration Method for Autonomous Ordinary Differential Systems. Applied Mathematics and Computation, 114, 115-123. http://dx.doi.org/10.1016/S0096-3003(99)00104-6 [10] Wazwaz, A.M. (1999) A Reliable Modification of the Adomian Decomposition Method. Applied Mathematics and Computation, 102, 77-86. http://dx.doi.org/10.1016/S0096-3003(98)10024-3 [11] Saadati, R., Raftari, B., Abibi, H., Vaezpour, S.M. and Shakeri, S. (2008) A Comparison between the Variational Iteration Method and Trapezoidal Rule for Solving Linear Integro-Differential Equations. World Applied Sciences Journal, 4, 321-325. [12] Hu, S., Wan, Z. and Khavanin, M. (1987) On the Existence and Uniqueness for Nonlinear Integro-Differential Equations. Journal of Mathematical and Physical Sciences, 21, 93-103. [13] Hairer, E., Nørsett, S.P. and Wanner, G. (1993) Solving Ordinary Differential Equations I Nonstiff Problems (Second Revised Edition). Springer-Verlag, New York. [14] Van Niekerk, F.D. (1988) Rational One Step Method for Initial Value Problem. Computers & Mathematics with Applications, 16, 1035-1039.http://dx.doi.org/10.1016/0898-1221(88)90259-3 [15] Pandey, P.K. (2013) Nonlinear Explicit Method for First Order Initial Value Problems. Acta Technica Jaurinensis, 6, 118-125. [16] Ramos, H. (2007) A Non-Standard Explicit Integration Scheme for Initial Value Problems. Applied Mathematics and Computation, 189, 710-718. http://dx.doi.org/10.1016/j.amc.2006.11.134 [17] Jain, M.K., Iyenger, S.R.K. and Jain, R.K. (1987) Numerical Methods for Scientific and Engineering Computation. Willey Eastern Limited, New Delhi. [18] Lambert, J.D. (1991) Numerical Methods for Ordinary Differential Systems. Wiley, England. [19] Pandey, P.K. (2013) A Non-Classical Finite Difference Method for Solving Two Point Boundary Value Problems. Pacific Journal of Science and Technology, 14, 147-152. [20] Varga, R.S. (2000) Matrix Iterative Analysis, Second Revised and Expanded Edition. Springer-Verlag, Heidelberg. http://dx.doi.org/10.1007/978-3-642-05156-2 [21] Henrici, P. (1982) Discrete Variable Methods in Ordinary Differential Equations. John Wiley and Sons, New York. [22] Volkov, Y.S. and Miroshnichenko, V.L. (2009) Norm Estimates for the Inverses of Matrices of Monotone Type and Totally Positive Matrics. Siberian Mathematical Journal, 50, 982-987. http://dx.doi.org/10.1007/s11202-009-0108-2 [23] Varah, J.M. (1975) A Lower Bound for the Smallest Singular Value of a Matrix. Linear Algebra and Its Applications, 11, 3-5. http://dx.doi.org/10.1016/0024-3795(75)90112-3 [24] Ahlberg, J.H. and Nilson, E.N. (1963) Convergence Properties of the Spline Fit. Journal of the Society for Industrial and Applied Mathematics, 11, 95-104. http://dx.doi.org/10.1137/0111007 [25] Horn, R.A. and Johnson, C.R. (1990) Matrix Analysis. Cambridge University Press, New York. [26] Shaw, R.E., Garey, L.E. and Lizotte, D.J. (2001) A Parrllel Numerical Algorithm for Fredholm Integro-Differential Two Point Boundary Value Problems. International Journal of Computer Mathematics, 77, 305-318. http://dx.doi.org/10.1080/00207160108805067