The Operational matrices with respect to generalized Laguerre polynomials and their applications in solving linear differential equations with variable coefficients

Subject Areas : Applied Mathematics

Z Kalateh Bojdi ¹ , S Ahmadi-Asl ² , A Aminataei ³

1 - Department of Mathematics, Birjand University, Birjand, Iran.
2 - Department of Mathematics, Birjand University, Birjand, Iran.
3 - Faculty of Mathematics, K. N. Toosi University of Technology, P.O. Box 16315-1618, Tehran, Iran.

Received: 2015-09-20 Accepted : 2015-09-20 Published : 2015-12-01

Keywords:

References:

[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 Clis, 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 Dierential Equations,
Springer-Verlag, 1992.
[14] R.P. Agraval, D.O. Oregan, Odinary and Partial Dierential Equations,
Springer, 2009.
[15] A.C. King, J. Bilingham, S.R. Otto, Dierential Equations (Linear,
Nonlinear, Integral, Partial), Cambridge University, 2003.
[16] A.M. Wazwaz, The combined Laplace transform-Adomian decomposition
method for handling nonlinear Volterra integro-dierential 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 dierence 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 dierential-
dierence equations, Comput. Appl. Math., 186 (2006) 349-364.
[21] M. Sezer, M. Gulsu, Polynomial solution of the most general linear
Fredholm integro-dierential-dierence 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 dierence equations in terms of Taylor polynomials, Int. J.
Comput. Math., 82 (2005) 629-642.
[23] K. Maleknejad, F. Mirzaee, Numerical solution of integro-dierential
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 dierential equations, Computing,
27 (1981) 15-25.
[26] E.L. Ortiz, On the numerical solution of nonlinear and functional
dierential equa- tions with the Tau method, in: Numerical Treatment
of Dierential Equations in Applications, in: Lecture Notes in Math., 679
(1978) 127-139.
[27] H. Danfu, S. Xufeng, Numerical solution of integro-dierential 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 dierential equations, Int. J.
Math. Math. Sci., Article ID 673085, 11P, 2011.
[31] M. Sezer, A.A. Dascioglu, Taylor polynomial solutions of general linear
dierential-dierence equations with variable coecients, Appl. Math.
Comput., 174 (2006) 1526-1538.
[32] T. Akkaya, S. Yalcinbas, Boubaker polynomial approach for solving high-
order linear dierential-dierence 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 dierential-dierence 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 dierential equations,
Appl. Math. Comput., 142 (2003) 155-165.
[38] N. Kurt, M. Sezer, Polynomial solution of high-order linear Fredholm
integro-dierential 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.