Legendre Approximation for Solving Linear HPDEs and Comparison with Taylor and Bernoulli Matrix Methods

Affiliation(s)

Department of Applied Mathematics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran.

Department of Applied Mathematics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran.

ABSTRACT

The aim of this study is to give a Legendre polynomial approximation for the solution of the second order linear hyper-bolic partial differential equations (HPDEs) with two variables and constant coefficients. For this purpose, Legendre matrix method for the approximate solution of the considered HPDEs with specified associated conditions in terms of Legendre polynomials at any point is introduced. The method is based on taking truncated Legendre series of the functions in the equation and then substituting their matrix forms into the given equation. Thereby the basic equation reduces to a matrix equation, which corresponds to a system of linear algebraic equations with unknown Legendre coefficients. The result matrix equation can be solved and the unknown Legendre coefficients can be found approximately. Moreover, the approximated solutions of the proposed method are compared with the Taylor [1] and Bernoulli [2] matrix methods. All of computations are performed on a PC using several programs written in MATLAB 7.12.0.

The aim of this study is to give a Legendre polynomial approximation for the solution of the second order linear hyper-bolic partial differential equations (HPDEs) with two variables and constant coefficients. For this purpose, Legendre matrix method for the approximate solution of the considered HPDEs with specified associated conditions in terms of Legendre polynomials at any point is introduced. The method is based on taking truncated Legendre series of the functions in the equation and then substituting their matrix forms into the given equation. Thereby the basic equation reduces to a matrix equation, which corresponds to a system of linear algebraic equations with unknown Legendre coefficients. The result matrix equation can be solved and the unknown Legendre coefficients can be found approximately. Moreover, the approximated solutions of the proposed method are compared with the Taylor [1] and Bernoulli [2] matrix methods. All of computations are performed on a PC using several programs written in MATLAB 7.12.0.

Cite this paper

E. Tohidi, "Legendre Approximation for Solving Linear HPDEs and Comparison with Taylor and Bernoulli Matrix Methods,"*Applied Mathematics*, Vol. 3 No. 5, 2012, pp. 410-416. doi: 10.4236/am.2012.35063.

E. Tohidi, "Legendre Approximation for Solving Linear HPDEs and Comparison with Taylor and Bernoulli Matrix Methods,"

References

[1] B. Bulbul and M. Sezer, “Taylor Polynomial Solution of Hyperbolic Type Partial Differential Equations with Constant Coefficients,” International Journal of Computer Mathematics, Vol. 88, No. 3, 2011, pp. 533-544. doi:10.1080/00207161003611242

[2] E. Tohidi, and M. Shirazian, “Numerical Solution of Linear HPDEs via Bernoulli Operational Matrix of Differentiation and Comparison with Taylor Matrix Method,” Mathematical Science Letters, Accepted.

[3] M. Dehghan and A. Shokri, “A Numerical Method for Solving the Hyperbolic Telegraph Equation,” Numerical Methods for Partial Differential Equations, Vol. 24, No. 4, 2008, pp. 1080-1093. doi:10.1002/num.20306

[4] A. Ashyralyev and M. E. Koksal, “On the Numerical Solution of Hyperbolic PDE with Variable Space Operator,” Numerical Methods for Partial Differential Equations, Vol. 25, No. 5, 2009, pp. 1086-1099. doi:10.1002/num.20388

[5] F. Gao and C. Chi, “Unconditionally Stable Difference Schemes for a One-Space-Dimensional Linear Hyperbolic Equation,” Applied Mathematics and Computation, Vol. 187, No. 2, 2007, pp. 1272-1276. doi:10.1016/j.amc.2006.09.057

[6] R. K. Mohanty and M. K. Jain, “An Unconditionally Stable Alternating Direction Implicit Scheme for the Two Space Dimensional Linear Hyperbolic Equation,” Numerical Methods for Partial Differential Equations, Vol. 17, No. 6, 2001, pp. 684-688. doi:10.1002/num.1034

[7] R. K. Mohanty, M. K. Jain and U. Arora, “An Unconditionally Stable Adi Method for the Linear Hyperbolic Equation in Three Space Dimensions,” International Journal of Computer Mathematics, Vol. 79, No. 1, 2002, pp. 133-142. doi:10.1080/00207160211918

[8] R. K. Pandy, N. Kumar, A. Bhardwaj and G. Dutta, “Solution of Lane-Emden Type Equations Using Legendreoperational Matrix of Differentiation,” Applied Mathematics and Computation, Vol. 218, No. 14, pp. 76297637.

[9] J. Biazar and H. Ebrahimi, “An Approximation to the Solution of Hyperbolic Equations by Adomian Decomposition Method and Comparison with Characteristics Method,” Applied Mathematics and Computation, Vol. 163, No. 2, 2005, pp. 633-638. doi:10.1016/j.amc.2004.04.005

[10] X. Yang, Y. Liu and S. Bai, “A Numerical Solution of Second-Order Linear Partial Differential Equations by Differential Transform,” Applied Mathematics and Computation, Vol. 173, No. 2, 2006, pp. 792-802. doi:10.1016/j.amc.2005.04.015

[1] B. Bulbul and M. Sezer, “Taylor Polynomial Solution of Hyperbolic Type Partial Differential Equations with Constant Coefficients,” International Journal of Computer Mathematics, Vol. 88, No. 3, 2011, pp. 533-544. doi:10.1080/00207161003611242

[2] E. Tohidi, and M. Shirazian, “Numerical Solution of Linear HPDEs via Bernoulli Operational Matrix of Differentiation and Comparison with Taylor Matrix Method,” Mathematical Science Letters, Accepted.

[3] M. Dehghan and A. Shokri, “A Numerical Method for Solving the Hyperbolic Telegraph Equation,” Numerical Methods for Partial Differential Equations, Vol. 24, No. 4, 2008, pp. 1080-1093. doi:10.1002/num.20306

[4] A. Ashyralyev and M. E. Koksal, “On the Numerical Solution of Hyperbolic PDE with Variable Space Operator,” Numerical Methods for Partial Differential Equations, Vol. 25, No. 5, 2009, pp. 1086-1099. doi:10.1002/num.20388

[5] F. Gao and C. Chi, “Unconditionally Stable Difference Schemes for a One-Space-Dimensional Linear Hyperbolic Equation,” Applied Mathematics and Computation, Vol. 187, No. 2, 2007, pp. 1272-1276. doi:10.1016/j.amc.2006.09.057

[6] R. K. Mohanty and M. K. Jain, “An Unconditionally Stable Alternating Direction Implicit Scheme for the Two Space Dimensional Linear Hyperbolic Equation,” Numerical Methods for Partial Differential Equations, Vol. 17, No. 6, 2001, pp. 684-688. doi:10.1002/num.1034

[7] R. K. Mohanty, M. K. Jain and U. Arora, “An Unconditionally Stable Adi Method for the Linear Hyperbolic Equation in Three Space Dimensions,” International Journal of Computer Mathematics, Vol. 79, No. 1, 2002, pp. 133-142. doi:10.1080/00207160211918

[8] R. K. Pandy, N. Kumar, A. Bhardwaj and G. Dutta, “Solution of Lane-Emden Type Equations Using Legendreoperational Matrix of Differentiation,” Applied Mathematics and Computation, Vol. 218, No. 14, pp. 76297637.

[9] J. Biazar and H. Ebrahimi, “An Approximation to the Solution of Hyperbolic Equations by Adomian Decomposition Method and Comparison with Characteristics Method,” Applied Mathematics and Computation, Vol. 163, No. 2, 2005, pp. 633-638. doi:10.1016/j.amc.2004.04.005

[10] X. Yang, Y. Liu and S. Bai, “A Numerical Solution of Second-Order Linear Partial Differential Equations by Differential Transform,” Applied Mathematics and Computation, Vol. 173, No. 2, 2006, pp. 792-802. doi:10.1016/j.amc.2005.04.015