Numerical solution of some class of integro-differential equations by using legendre-bernstein basis

Document Type : Research Paper

Authors

Department of Mathematics, Faculty of Science, Malayer University, Malayer, 65719- 95863, Iran.

Abstract

In this article, a numerical method is developed to solve the linear integro-differential equations. To this end, it will be divided in two forms, Fredholm integro-differential equations (FIDE) and Volterra integro-differential equations (VIDE). So that, the kernel and other known functions have been approximated using the least-squares approximation schemes based on LegenderBernstein basis. The Legender polynomials are orthogonal and this property improve the accuracy of the approximations. Also the unknown function and its derivatives have been approximated by using the Bernstein basis. The useful properties of Bernstein polynomials help us to transform integro-differential equations to solve a system of linear algebraic equations. Of course, the solution way of (FIDE) case is different from (VIDE).

Keywords

References

[1] R. T. Farouki, Legendre-Bernstein basis transformations, Comput. Appl. Math.,119 (2000), 145–160.
[2] P. J. Davis, Interpolation and approximation, New York: Dover (1975).
[3] E. Isaacson and H. B. Keller, Analysis of numerical methods, New York: Dover (1994).
[4] SA. Yousefi and M. Behroozifar, Operational matrices of Bernstein polynomials and their applications, Int. J. Syst. Sci., 41(6) (2010), 709–716.
[5] BN. Mandal and S. Bhattacharya, Numerical solution of some classes of integral equations using Bernstein polynomials, Comput. Appl. Math., 190 (2007), 1707-1716.
[6] MA. Snyder, Chebychev methods in numerical approximation, Englewood Clifs; NJ. Prentice-Hall (1996).
[7] MA. Golberg and CS. Chen, Discrete projection methods for integral equations, Southampton; Comput. Mech. Publicat (1997), 178–306.
[8] Y. S¸uayip, S¸. Niyazi and Y. Ahmet, A collocation approach for solving high-order linear FredholmVolterra integro-differential equations, Math. and Compu.Model., 55 (2012), 547–563.
[9] Y. S¸uayip, S¸. Niyazi and S. Mehmet, Bessel polynomial solutions of high-order linear Volterra integro-differential equations, Comput. Appl. Math., 62 (2011),1940–1956.
Journal of Hyperstructures
Volume 3, Issue 2
December 2014
Pages 139-154

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