On Approximate Solutions of Second-Order Linear Partial Differential Equations
Author(s)
Yousry S. Hanna
ABSTRACT
In this paper, a Chebyshev polynomial approximation for the solution of second-order partial differential equations with two variables and variable coefficients is given. Also, Chebyshev matrix is introduced. This method is based on taking the truncated Chebyshev expansions of the functions in the partial differential equations. Hence, the result matrix equation can be solved and approximate value of the unknown Chebyshev coefficients can be found.
In this paper, a Chebyshev polynomial approximation for the solution of second-order partial differential equations with two variables and variable coefficients is given. Also, Chebyshev matrix is introduced. This method is based on taking the truncated Chebyshev expansions of the functions in the partial differential equations. Hence, the result matrix equation can be solved and approximate value of the unknown Chebyshev coefficients can be found.
Cite this paper
Y. Hanna, "On Approximate Solutions of Second-Order Linear Partial Differential Equations," Applied Mathematics, Vol. 3 No. 9, 2012, pp. 1001-1007. doi: 10.4236/am.2012.39148.
Y. Hanna, "On Approximate Solutions of Second-Order Linear Partial Differential Equations," Applied Mathematics, Vol. 3 No. 9, 2012, pp. 1001-1007. doi: 10.4236/am.2012.39148.
References
[1] X. F. Yang, Y. X. Liu and S. Bai, “A Numerical Solution of Second-Order Linear Partial Differential Equations by Differential Transform,” Journal of Applied Mathematics and Computing, Vol. 173, No. 2, 2006, pp. 792-802. doi:10.1016/j.amc.2005.04.015 [2] Luhong Ye and Zhiting Xu, “Oscillation Criteria for Second Order Quasilinear Neutral Delay Differential Equations,” Journal of Applied Mathematics and Computing, Vol. 207, No. 2, 2009, pp. 388-396. doi:10.1016/j.amc.2008.10.051 [3] M. Sezer and M. Kaynak, “Chebyshev Polynomial Solutions of Linear Differential Equations,” International Journal of Mathematical Education in Science & Technology, Vol. 27, No. 4, 1996, pp. 607-618. doi:10.1080/0020739960270414 [4] N. K. Basu, “On Double Chebyshev Series Approximation,” SIAM Journal on Numerical Analysis, Vol. 10, No. 3, 1973, pp. 496-505. doi:10.1137/0710045 [5] F. Cases, “Solution of Linear Partial Differential Equations by Lie Algebraic Methods,” Journal of Computational and Applied Mathematics, Vol. 76, No. 1-2, 1996, pp. 159-170. [6] H. Koroglu, “Chebyshev Series Solution of Linear Fredholm Integrodifferential Equations,” International Journal of Mathematical Education in Science & Technology, Vol. 29, No. 4, 1998, pp. 489-500. doi:10.1080/0020739980290403 [7] V. S. Vladimirov, “A Collection of Problems on the Equations of Mathematical Physics,” Mir Publishers, Moscow, 1986, pp. 139-149.
[1] X. F. Yang, Y. X. Liu and S. Bai, “A Numerical Solution of Second-Order Linear Partial Differential Equations by Differential Transform,” Journal of Applied Mathematics and Computing, Vol. 173, No. 2, 2006, pp. 792-802. doi:10.1016/j.amc.2005.04.015 [2] Luhong Ye and Zhiting Xu, “Oscillation Criteria for Second Order Quasilinear Neutral Delay Differential Equations,” Journal of Applied Mathematics and Computing, Vol. 207, No. 2, 2009, pp. 388-396. doi:10.1016/j.amc.2008.10.051 [3] M. Sezer and M. Kaynak, “Chebyshev Polynomial Solutions of Linear Differential Equations,” International Journal of Mathematical Education in Science & Technology, Vol. 27, No. 4, 1996, pp. 607-618. doi:10.1080/0020739960270414 [4] N. K. Basu, “On Double Chebyshev Series Approximation,” SIAM Journal on Numerical Analysis, Vol. 10, No. 3, 1973, pp. 496-505. doi:10.1137/0710045 [5] F. Cases, “Solution of Linear Partial Differential Equations by Lie Algebraic Methods,” Journal of Computational and Applied Mathematics, Vol. 76, No. 1-2, 1996, pp. 159-170. [6] H. Koroglu, “Chebyshev Series Solution of Linear Fredholm Integrodifferential Equations,” International Journal of Mathematical Education in Science & Technology, Vol. 29, No. 4, 1998, pp. 489-500. doi:10.1080/0020739980290403 [7] V. S. Vladimirov, “A Collection of Problems on the Equations of Mathematical Physics,” Mir Publishers, Moscow, 1986, pp. 139-149.