Müntz-Legendre Matrix Method to solve Delay Fredholm Integro-Differential Equations with constant coefficients

Abstract

In this study, we present the M¨untz-Legendre matrix method to solve the linear delay Fredholm integro-differential equationswith constant coeffcients. By using this method, we obtain the approximate solutions in form of the M¨untz-Legendre polynomials. Themethod reduces the problem to a system of the algebraic equations by means of the required matrix relations of the solution form. Bysolving this system, the approximate solution is obtained. Also, an error estimation scheme based the residual function is presented forthe method and the approximate solutions are improved by this error estimation. Finally, the method will be illustrated on the examples

Keywords

• Linear delay Fredholm integro-differential equation,M¨untz-Legendre polynomials,residual function

M ¨untz-legendre matrix method to solve the delay fredholm integro-differential equations with constant coefficients

Öz

References

1. A. Saadatmandi, M. Dehghan, Numerical solution of the higher-order linear Fredholm integro-differential–difference equation with variable coefficients, Comput. Math. Appl. 59 (2010) 2996–3004.
2. N. S¸ahin, S¸. Y¨uzbas¸ı, M. Sezer, A Bessel Polynomial approach for solving general linear Fredholm integro-differential-difference equations, Int. J. Comput. Math., 88(14) (2011) 3093-3111.
3. A. Aky¨uz-Das¸cıoglu M. Sezer, Chebyshev polynomial solutions of systems of higher-order linear Fredholm–Volterra integro- differential equations, J. Franklin Inst. 342 (2005) 688–701.
4. N. Kurt, M. Sezer, Polynomial solution of high-order linear Fredholm integro-differential equations with constant coefficients, J. Frankin Inst. 345 (2008), 839–850.
5. Kajani M.T., Ghasemi M. & Babolian E., Numerical solution of linear integro-differential equation by using sine–cosine wavelets, Appl. Math. Comput. 180 (2006) 569–574.
6. Yusufo˘glu E., Improved homotopy perturbation method for solving Fredholm type integro-differential equations, Chaos Solitons Fractals 41 (2009) 28–37.
7. Darania P., Ebadian A., A method for the numerical solution of the integro-differential equations, Appl. Math. Comput. 188 (2007) 657–668.
8. Y¨uksel G., Y¨uzbas¸ı S¸. & Sezer M., A Chebyshev Method for a class of high-order linear Fredholm integro-differential equations, J.Adv. Res. Appl. Math., 4(1) (2012) 49-67.

9. Shahmorad S., Numerical solution of general form linear Fredholm.Volterra integro differantial equations by the tau method with an error estimation, Appl. Math. Comput. 167 (2005) 1418-1429.
10. F.A. Oliveira, Collacation and residual correction,Numerische Mathematik, 36, 27-31, 1980.
11. ˙I. C¸ elik, Collocation method and residual correction using Chebyshev series, Applied Mathmematics and Computation, 174, 910- 920, 2006. [12] S¸. Y¨uzbas¸ı,
12. An efficient algorithm for solving multi-pantograph equation systems, Computers and Mathematics with
13. Applications, 64, 589-603, 2012.