The Operational matrices with respect to generalized Laguerre polynomials and their applications in solving linear di erential equations with variable coecients

Document Type: Research Articles

Authors

1 دانشگاه بیرجند

2 دانشگاه خواجه نصیر الدین توسی تهران

Abstract

In this paper, a new and ecient approach based on operational matrices with respect to the gener-
alized Laguerre polynomials for numerical approximation of the linear ordinary di erential equations
(ODEs) with variable coecients is introduced. Explicit formulae which express the generalized La-
guerre expansion coecients for the moments of the derivatives of any di erentiable function in terms
of the original expansion coecients of the function itself are given in the matrix form. The main
importance of this scheme is that using this approach reduces solving the linear di erential equations
to solve a system of linear algebraic equations, thus greatly simplify the problem. In addition, several
numerical experiments are given to demonstrate the validity and applicability of the method.

[1] W. Gautschi, Orthogonal Polynomials (Computation and
Approximation), Oxford University Press, 2004.
[2] C.F. Dunkl, Y. Xu, Orthogonal Polynomials of Several Variables,
Cambridge University Press, 2001.
[3] F. Marcellan, W.V. Assche, Orthogonal Polynomials and Special
Functions (a Computation and Applications), Springer-Verlag Berlin
Heidelberg, 2006.
[4] R. Askey, Orthogonal Polynomials and Special Functions, SIAM-CBMS,
Philadelphia, 1975.
[5] D. Gottlieb, S.A. Orszag, Numerical Analysis of Spectral Methods: Theory
and Applications, SIAM-CBMS, Philadelphia, 1977.
[6] J.P. Boyd, Chebyshev and Fourier Spectral Methods, Dover Publications,
Inc, New York, 2000.
[7] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods:
Fundamentals in Single Domains, Springer-Verlag, 2006.
[8] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Method in
Fluid Dynamics, Prentice Hall, Engelwood Cli s, NJ, 1984.
[9] L.N. Trefethen, Spectral Methods in Matlab, SIAM, Philadelphia, PA,
2000.
[10] J.S. Hesthaven, S. Gottlieb, D. Gottlieb, Spectral Methods for Time-
Dependent Problems, Cambridge University, 2009.
[11] G. Ben-yu, The State of Art in Spectral Methods. Hong Kong University,
1996.

[12] J. Shen, T. Tang, L.L. Wang, Spectral Methods Algorithms, Analysis and
Applications, Springer, 2011.
[13] D. Funaro, Polynomial Approximations of Di erential Equations,
Springer-Verlag, 1992.
[14] R.P. Agraval, D.O. Oregan, Odinary and Partial Di erential Equations,
Springer, 2009.
[15] A.C. King, J. Bilingham, S.R. Otto, Di erential Equations (Linear,
Nonlinear, Integral, Partial), Cambridge University, 2003.
[16] A.M. Wazwaz, The combined Laplace transform-Adomian decomposition
method for handling nonlinear Volterra integro-di erential equations,
Appl. Math. Comput., 216 (2010) 1304-1309.
[17] A. Aminataei, S.S. Hussaini, The comparison of the stability of
decomposition method with numerical methods of equation solution, Appl.
Math. Comput., 186 (2007) 665-669.
[18] A. Aminataei, S.S. Hussaini, The barrier of decomposition method, Int.
J. Contemp. Math. Sci., 5 (2010) 2487-2494.
[19] M. Gulsu, M. Sezer, Z. Guney, Approximate solution of general high-order
linear non-homogenous di erence equations by means of Taylor collocation
method, Appl. Math. Comput., 173 (2006) 683-693.
[20] M. Gulsu, M. Sezer, A Taylor polynomial approach for solving di erential-
di erence equations, Comput. Appl. Math., 186 (2006) 349-364.
[21] M. Sezer, M. Gulsu, Polynomial solution of the most general linear
Fredholm integro-di erential-di erence equation by means of Taylor
matrix method, Int. J. Complex Variables., 50 (2005) 367-382.
[22] M. Gulsu, M. Sezer, A method for the approximate solution of the high-
order linear di erence equations in terms of Taylor polynomials, Int. J.
Comput. Math., 82 (2005) 629-642.
[23] K. Maleknejad, F. Mirzaee, Numerical solution of integro-di erential
equations by using rationalized Haar functions method, Kyber. Int. J.
Syst. Math., 35 (2006) 1735-1744.
[24] M.H. Reihani, Z. Abadi, Rationalized Haar functions method for solving
Fredholm and Volterra integral equations, Comput. Appl. Math, 200
(2007) 12-20.

[25] E.L. Ortiz, L. Samara, An operational approach to the Tau method for
the numer- ical solution of nonlinear di erential equations, Computing,
27 (1981) 15-25.
[26] E.L. Ortiz, On the numerical solution of nonlinear and functional
di erential equa- tions with the Tau method, in: Numerical Treatment
of Di erential Equations in Applications, in: Lecture Notes in Math., 679
(1978) 127-139.
[27] H. Danfu, S. Xufeng, Numerical solution of integro-di erential equations
by using CAS wavelet operational matrix of integration, Appl. Math.
Comput., 194 (2007) 460-466.
[28] C.H. Hsiao, Hybrid function method for solving Fredholm and Volterra
integral equations of the second kind, Comput. Appl. Math., 230 (2009)
59-68.
[29] M. Razzaghi, S.A. Youse , Legendre wavelets method for the nonlinear
Volterra- Fredholm integral equations, Math. Comput. Simul., 70 (2005)
1-8.
[30] A. Imani, A. Aminataei, A. Imani, Collocation method via Jacobi
polynomials for solving nonlinear ordinary di erential equations, Int. J.
Math. Math. Sci., Article ID 673085, 11P, 2011.
[31] M. Sezer, A.A. Dascioglu, Taylor polynomial solutions of general linear
di erential-di erence equations with variable coecients, Appl. Math.
Comput., 174 (2006) 1526-1538.
[32] T. Akkaya, S. Yalcinbas, Boubaker polynomial approach for solving high-
order linear di erential-di erence equations, AIP Conference Proceedings
of 9th international conference on mathematical problems in engineering,
56 (2012) 26-33.
[33] K. Erdem, S. Yalcinbas, Bernoulli polynomial approach to high-order
linear di erential-di erence equations, AIP Conference Proceedings of
Numerical Analysis and Applied Mathematics, 73 (2012) 360-364.
[34] M.R. Eslahchi, M. Dehghan, Application of Taylor series in obtaining
the orthogonal operational matrix, Computers and Mathematics with
Applications, 61 (2011) 2596-2604.
[35] M. Razzaghi, Y. Ordokhani, Solution of nonlinear Volterra Hammerstein
integral equations via rationalized Haar functions, Math. Prob. Eng., 7
(2001) 205-219.

[36] F. Khellat, S. A. Youse , The linear Legendre wavelets operational matrix
of integration and its application, J. Frank. Inst., 343 (2006) 181-190.
[37] C. Kesan, Taylor polynomial solutions of linear di erential equations,
Appl. Math. Comput., 142 (2003) 155-165.
[38] N. Kurt, M. Sezer, Polynomial solution of high-order linear Fredholm
integro-di erential equations with constant coecients, J. Frank. Inst.,
345 (2008) 839-850.
[39] A. Golbabai, M. Javidi, Application o f homotopy perturbation method
for solving eighth-order boundary value problems, Appl. Math. Comput.,
213 (2007) 203-214.