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.
Cite this paper
Pandey, P. K. (2015). Non-Standard Difference Method for Numerical Solution of Linear Fredholm Integro-Differential Type Two-Point Boundary Value Problems. Open Access Library Journal, 2, e1465. doi: http://dx.doi.org/10.4236/oalib.1101465.
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
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
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
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
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
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
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.
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.
Hairer, E., N？rsett, S.P. and Wanner, G. (1993) Solving Ordinary Differential
Equations I Nonstiff Problems (Second Revised Edition). Springer-Verlag, New
York.
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
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.
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
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
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
