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

Subject Areas : Linear and multilinear algebra; matrix theory

Z. Kalateh Bojdi ¹ (Department of Mathematics, Birjand University, Birjand, Iran)
S. Ahmadi-Asl ² (Department of Mathematics, Birjand University, Birjand, Iran)
A. Aminataei ³ (Faculty of Mathematics, K. N. Toosi University of Technology, P.O. Box 16315-1618, Tehran, Iran)

Received: 2013-01-01 Accepted : 2013-01-01 Published : 2013-06-01

Keywords:

References:

[1] R.P. Agraval, and D.O. Oregan, Ordinary and Partial Dierential Equations, Springer, 2009.
[2] A. Aminataei, and S.S. Hussaini, The comparison of the stability of decomposition method with numerical methods of equation solution, Appl. Math. Comput. 186 (2007), pp. 665{669.
[3] A. Aminataei, and S.S. Hussaini, The barrier of decomposition method, Int. J. Contemp. Math. Sci. 5 (2010), pp. 2487{2494.
[4] R. Askey, Orthogonal Polynomials and Special Functions, SIAM-CBMS, Philadelphia, 1975.
[5] T. Akkaya, and S. Yalcinbas, Boubaker polynomial approach for solving high-order linear differential-difference equations, AIP Conference Proceedings of 9th international conference on mathematical problems in engineering, 56 (2012), PP. 26-33.
[6] G. Ben-yu, The State of Art in Spectral Methods. Hong Kong University, 1996.
[7] J.P. Boyd, Chebyshev and Fourier Spectral Methods, Dover Publications, Inc, New York, 2000.
[8] C. Canuto, M.Y. Hussaini, A. Quarteroni, and T.A. Zang, Spectral Method in Fluid Dynamics, Prentice Hall, Engelwood Clis, NJ, 1984.
[9] C. Canuto, M.Y. Hussaini, A. Quarteroni, and T.A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer-Verlag, 2006.
[10] H. Danfu, and S. Xufeng, Numerical solution of integro-dierential equations by using CAS wavelet operational matrix of integration, Appl. Math. Comput. 194 (2007), pp. 460-466.
[11] C.F. Dunkl, and Y. Xu, Orthogonal Polynomials of Several Variables, Cambridge University Press, 2001.
[12] K. Erdem, and S. Yalcinbas, Bernoulli polynomial approach to high-order linear differential-difference equations, AIP Conference Proceedings of Numerical Analysis and Applied Mathematics, 73 (2012), 360-364.
[13] M.R. Eslahchi, and M. Dehghan, Application of Taylor series in obtaining the orthogonal operational matrix, Computers and Mathematics with Applications, 61 (2011), PP. 2596-2604.
[14] D. Funaro, Polynomial Approximations of Dierential Equations, Springer-Verlag, 1992.
[15] W. Gautschi, Orthogonal Polynomials (Computation and Approximation), Oxford University Press, 2004.
[16] D. Gottlieb, and S.A. Orszag,Numerical Analysis of Spectral Methods: Theory and Applications, SIAM-CBMS, Philadelphia, 1977.
[17] M. Gulsu, and M. Sezer, A method for the approximate solution of the high-order linear difference equations in terms of Taylor polynomials, Int. J. Comput. Math. 82 (2005), pp. 629-642.
[18] M. Gulsu, M. Sezer, and Z. Guney, Approximate solution of general high-order linear non-homogenous difference equations by means of Taylor collocation method, Appl. Math. Comput. 173 (2006), pp. 683-693.
[19] M. Gulsu, and M. Sezer, A Taylor polynomial approach for solving differential-difference equations, Comput. Appl. Math. 186 (2006), pp. 349-364.
[20] J.S. Hesthaven, S. Gottlieb, and D. Gottlieb, Spectral Methods for Time-Dependent Problems, Cambridge University, 2009.
[21] C.H. Hsiao, Hybrid function method for solving Fredholm and Volterra integral equations of the second kind, Comput. Appl. Math. 230 (2009), pp. 59-68.
[22] A. Imani, A. Aminataei, and A. Imani, Collocation method via Jacobi polynomials for solving nonlinear ordinary differential equations, Int. J. Math. Math. Sci., Article ID 673085, 11P, 2011.
[23] F. Khellat, S. A. Youse, The linear Legendre wavelets operational matrix of integration and its application, J. Frank. Inst. 343 (2006), PP. 181-190.
[24] A.C. King, J. Bilingham, and S.R. Otto, Differential Equations (Linear, Nonlinear, Integral, Partial), Cambridge University, 2003.
[25] E.L. Ortiz, and L. Samara, An operational approach to the Tau method for the numerical solution of nonlinear differential equations, Computing, 27 (1981), pp. 15-25.
[26] E.L. Ortiz, On the numerical solution of nonlinear and functional differential equations with the Tau method, in: Numerical Treatment of Differential Equations in Applications, in: Lecture Notes in Math. 679 (1978), pp. 127-139.
[27] F. Marcellan, and W.V. Assche, Orthogonal Polynomials and Special Functions (a Computation and Applications), Springer-Verlag Berlin Heidelberg, 2006.
[28] K. Maleknejad, and F. Mirzaee, Numerical solution of integro-differential equations by using rationalized Haar functions method, Kyber. Int. J. Syst. Math. 35 (2006), pp. 1735-1744.
[29] M. Razzaghi, and Y. Ordokhani, Solution of nonlinear Volterra Hammerstein integral equations via rationalized Haar functions, Math. Prob. Eng. 7 (2001), PP. 205-219.
[30] M. Razzaghi, and S.A. Youse, Legendre wavelets method for the nonlinear Volterra-Fredholm integral equations, Math. Comput. Simul. 70 (2005), pp. 1-8.
[31] M.H. Reihani, and Z. Abadi, Rationalized Haar functions method for solving Fredholm and Volterra integral equations, Comput. Appl. Math. 200 (2007), pp. 12-20.
[32] M. Sezer, and A.A. Dascioglu, Taylor polynomial solutions of general linear dierential-dierence equations with variable coefficients, Appl. Math. Comput. 174 (2006), pp. 1526-1538.
[33] M. Sezer, and M. Gulsu, Polynomial solution of the most general linear Fredholm integro-differential-difference equation by means of Taylor matrix method, Int. J. Complex Variables. 50 (2005), pp. 367-382.
[34] J. Shen, T. Tang, and L.L. Wang, Spectral Methods Algorithms, Analysis and Applications, Springer, 2011.
[35] L.N. Trefethen, Spectral Methods in Matlab, SIAM, Philadelphia, PA, 2000.

[36] A.M. Wazwaz, The combined Laplace transform-Adomian decomposition method for handling nonlinear Volterra integro-differential equations, Appl. Math. Comput. 216 (2010), pp. 1304-1309.