L-Stable Block Hybrid Second Derivative Algorithm for Parabolic Partial Differential Equations

Fidele Fouogang Ngwane, Samuel Nemsefor Jator

Department of Mathematics and Statistics, Austin Peay State University, Clarksville, USA.
Department of Mathematics, USC Salkehatchie, Allendale, USA.

DOI: 10.4236/ajcm.2014.42008                

Abstract

An L-stable block method based on hybrid second derivative algorithm (BHSDA) is provided by a continuous second derivative method that is defined for all values of the independent variable and applied to parabolic partial differential equations (PDEs). The use of the BHSDA to solve PDEs is facilitated by the method of lines which involves making an approximation to the space derivatives, and hence reducing the problem to that of solving a time-dependent system of first order initial value ordinary differential equations. The stability properties of the method is examined and some numerical results presented.

Keywords

Hybrid Second Derivative Method; Off-Step Point; Parabolic; Partial Differential Equations

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Lambert, J.D. (1991) Numerical Methods for Ordinary Differential Systems. John Wiley, New York.
[2]	Vigo-Aguiar, J. and Ramos, H. (2007) A family of A-Stable Collocation Methods of Higher Order for Initial-Value Problems. IMA Journal of Numerical Analysis, 27, 798-817. http://dx.doi.org/10.1093/imanum/drl040
[3]	Brugnano, L. and Trigiante, D. (1998) Solving Differential Problems by Multistep Initial and Boundary Value Methods. Gordon and Breach Science Publishers, Amsterdam.
[4]	Cash, J.R. (1984) Two New Finite Difference Schemes for Parabolic Equations. SIAM Journal of Numerical Analysis, 21, 433-446. http://dx.doi.org/10.1137/0721032
[5]	Enright, W.H. (2000) Continuous Numerical Methods for ODEs with Defect Control. Journal of Computational and Applied Mathematics, 125, 159-170. http://dx.doi.org/10.1016/S0377-0427(00)00466-0
[6]	Hairer, E. and Wanner, G. (1996) Solving Ordinary Differential Equations II. Springer, New York. http://dx.doi.org/10.1007/978-3-642-05221-7
[7]	Henrici, P. (1962) Discrete Variable Methods in ODEs. John Wiley, New York.
[8]	Butcher, J.C. (1987) The Numerical Analysis of Ordinary Differential Equations, Runge-Kutta and General Linear Methods. Wiley, New York.
[9]	Fatunla, S.O. (1991) Block Methods for Second Order IVPs. International Journal of Computational Mathematics, 41, 55-63. http://dx.doi.org/10.1080/00207169108804026
[10]	Jator, S.N. (2010) On the Hybrid Method with Three-Off Step Points for Initial Value Problems. International Journal of Mathematical Education in Science and Technology, 41, 110-118 http://dx.doi.org/10.1080/00207390903189203
[11]	Onumanyi, P., Sirisena, U.W. and Jator, S.N. (1999) Continuous Finite Difference Approximations for Solving Differential Equations. International Journal of Computational Mathematics, 72, 15-27. http://dx.doi.org/10.1080/00207169908804831
[12]	Onumanyi, P., Awoyemi, D.O., Jator, S.N. and Sirisena, U.W. (1994) New Linear Mutlistep Methods with Continuous Coefficients for First Order Initial Value Problems. Journal of the Nigerian Mathematics Society, 37-51.