A matrix method for system of integro-differential equations by using generalized Laguerre polynomials

Document Type : Research Article

Authors

• M. Matinfar
• A. Riahifar

Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran.

10.22067/ijnao.v6i2.45227

Abstract

The purpose of this research is to present a matrix method for solving system of linear Fredholm integro-differential equations(FIDEs) of the second kind on unbounded domain with degenerate kernels in terms of generalized Laguerre polynomials(GLPs). The method is based on the approximation of the truncated generalized Laguerre series. Then the system of (FIDEs) along with initial conditions are transformed into the matrix equations, which corresponds to a system of linear algebraic equations with the unknown generalized Laguerre coefficients. Combining these matrix equations and then olving the system yields the generalized Laguerre coefficients of the solution function. In addition, several numerical examples are given to demonstrate the validity, efficiency and applicability of the technique.

Keywords

• Systems of linear Fredholm integro-dierential equations
• Un- bounded domain
• Generalized Laguerre polynomials
• Operational matrix of integration

20.1001.1.24236977.2016.6.2.6.0

---

References

1. Akhavan, S. Numerical solution of singular Fredholm integro-differential equations of the second kind via Petrov-Galerkin method by using Legendre multiwavelet, Journal of mathematics and computer science. 9 (2014), 321-331.

2. Bhrawy, A.H., Alghamdi, M.A. and Taha, T.M. A new modified gen-eralized Laguerre operational matrix of fractional integration for solving fractional di_erential equations on the half line, Advances in Difference

Equations. (2012), 0:179. doi: 10.1186/1687-1847-2012-179.

3. Bhrawy, A.H., Al-Zahrani, A.A., Alhamed, Y.A. and Baleanu, D. A new generalized Laguerre-Gauss collocation scheme for numerical solution of generalized fractional pantograph equations, Romanian Reports Of Physics. 59 (7-8) (2014), 646-657.

4. Bhrawy, A.H., Baleanu, D., Assas, L.M. and Tenreiro Machado, J.A. On a generalized Laguerre operational matrix of fractional integration, Mathematical Problems in Engineering. (2013), Article ID 569286, 7 pages.

5. Bhrawy, A.H. and Taha, T.M. An operational matrix of fractional integration of the Laguerre polynomials and its application on a semi-infinite interval, Mathematical Sciences. (2012), 6:41. doi: 10.1186/2251-7456-6-41.

6. Bhrawy, A.H., Tharwat, M.M. and Alghamdi, M.A. A new operational matrix of fractional integration for shifted jacobi polynomials, Bulletin of the Malaysian Mathematical Sciences Society. 37 (4) (2014), 983-995.

7. De Bonis, M.C. and Mastroianni, M.G. Nystrom method for systems of integral equations on the real semiaxis, IMA Journal of Numerical Analysis. 29 (2009), 632-650.

8. Funaro, D. Polynomial Approximations of Di_erential Equations, Springer Verlag, 1992.

9. Maalek Ghaini, F.M., Tavassoli Kajani, F. and Ghasemi, M. Solving boundary integral equation using Laguerre polynomials, World Applied Sciences Journal. 7 (1) (2009), 102-104.

10. Mastroianni, G. and Milovanovic, G.V. Some numerical methods for second kind Fredholm integral equations on the real semiaxis, IMA Journal of Numerical Analysis. 29 (2009), 1046-1066.

11. Muskhelishvili, N.I. Singular integral equations, Noordhoff, Holland, 1953.

12. Nik Long, N. M. A., Eshkuvatov, Z. K., Yaghobifar, M. and Hasan, M. Numerical solution of infinite boundary integral equation by using Galerkin method with Laguerre polynomials, World Academy of Science Engineering

and Technology. 47 (2008), 334-337.

13. Sanikidze, D.G. On the numerical solution of a class of singular integral equations on an infinite interval, Differential Equations. 41 (9) (2005), 1353-1358.

14. Shen, J., Tang, T. and Wang, L.L. Spectral Methods Algorithms, Analysis and Applications, Springer, 2011.

15. Shen, J. and Wang, L.L. Some Recent Advances on Spectral Methods for Unbounded Domains, J. Commun. comput. Phys. 5 (2009), 195-241.

16. Sloan, I.H. Quadrature methods for integral equations of the second kind over infinite intervals, Mathematics of Computation. 36 (154) (1981), 511-523.

17. Volterra, V. Theory of functionnals of integro-differential equations, Dover, New York, 1959.