Volume 35, Issue 1
Implicit Finite Difference Scheme for Singularly Perturbed Burger-Huxley Equations

J. Part. Diff. Eq., 35 (2022), pp. 87-100.

Published online: 2021-10

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• Abstract

In this paper, an implicit finite difference scheme is presented to solve one dimensional unsteady singularly perturbed Burger-Huxley equation. The quadratically convergent quasilinearization technique is used to linearize the nonlinear term of the equation. The innovative significance of this paper is the procedure to consider initial guesses in order to start the quasilinearization technique. This basic initial guessing causes to produce a more accurate solutions with the small iteration number for the problem under consideration. The derivatives are replaced by finite difference approximation, then we obtain the two-level time direction and the three-term recurrence relation in the spatial direction. The convergence analysis of the proposed method has been established. Numerical experiments were conducted to support the theoretical results. Further, the result shows that the proposed method gives a more accurate solution with a higher rate of convergence than some existing methods.

• Keywords

Singularly perturbed Burger-Huxley equation higher-order accurate solution.

65M06, 65M12, 65M15

maashookoo.reemii@gmail.com (Masho Jima Kabeto)

gammeef@gmail.com (Gemechis File Duressa)

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@Article{JPDE-35-87, author = {Jima Kabeto , Masho and Duressa , Gemechis File}, title = {Implicit Finite Difference Scheme for Singularly Perturbed Burger-Huxley Equations}, journal = {Journal of Partial Differential Equations}, year = {2021}, volume = {35}, number = {1}, pages = {87--100}, abstract = {

In this paper, an implicit finite difference scheme is presented to solve one dimensional unsteady singularly perturbed Burger-Huxley equation. The quadratically convergent quasilinearization technique is used to linearize the nonlinear term of the equation. The innovative significance of this paper is the procedure to consider initial guesses in order to start the quasilinearization technique. This basic initial guessing causes to produce a more accurate solutions with the small iteration number for the problem under consideration. The derivatives are replaced by finite difference approximation, then we obtain the two-level time direction and the three-term recurrence relation in the spatial direction. The convergence analysis of the proposed method has been established. Numerical experiments were conducted to support the theoretical results. Further, the result shows that the proposed method gives a more accurate solution with a higher rate of convergence than some existing methods.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v35.n1.6}, url = {http://global-sci.org/intro/article_detail/jpde/19911.html} }
TY - JOUR T1 - Implicit Finite Difference Scheme for Singularly Perturbed Burger-Huxley Equations AU - Jima Kabeto , Masho AU - Duressa , Gemechis File JO - Journal of Partial Differential Equations VL - 1 SP - 87 EP - 100 PY - 2021 DA - 2021/10 SN - 35 DO - http://doi.org/10.4208/jpde.v35.n1.6 UR - https://global-sci.org/intro/article_detail/jpde/19911.html KW - Singularly perturbed KW - Burger-Huxley equation KW - higher-order KW - accurate solution. AB -

In this paper, an implicit finite difference scheme is presented to solve one dimensional unsteady singularly perturbed Burger-Huxley equation. The quadratically convergent quasilinearization technique is used to linearize the nonlinear term of the equation. The innovative significance of this paper is the procedure to consider initial guesses in order to start the quasilinearization technique. This basic initial guessing causes to produce a more accurate solutions with the small iteration number for the problem under consideration. The derivatives are replaced by finite difference approximation, then we obtain the two-level time direction and the three-term recurrence relation in the spatial direction. The convergence analysis of the proposed method has been established. Numerical experiments were conducted to support the theoretical results. Further, the result shows that the proposed method gives a more accurate solution with a higher rate of convergence than some existing methods.

Masho Jima Kabeto & Gemechis File Duressa. (2021). Implicit Finite Difference Scheme for Singularly Perturbed Burger-Huxley Equations. Journal of Partial Differential Equations. 35 (1). 87-100. doi:10.4208/jpde.v35.n1.6
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