A uniformly convergent numerical scheme for singularly perturbed parabolic turning point problem

Document Type : Research Article

Authors

1 Department of Applied Mathematics, Adama Science and Technology University, Adama, Ethiopia

2 Department of Mathematics, Jimma University, Jimma, Ethiopia

Abstract

A uniformly convergent numerical scheme is developed for solving a singularly perturbed parabolic turning point problem. The properties of continuous solutions and the bounds of the derivatives are discussed. Due to the presence of a small parameter as a multiple of the diffusion coefficient, it causes computational difficulty when applying classical numerical methods. As a result, the scheme is formulated using the Crank-Nicolson method in the temporal discretization and an exponentially fitted finite difference method in the space on a uniform mesh. The existence of a unique discrete solution is guaranteed by the comparison principle. The stability and convergence analysis of the method are investigated. Two numerical examples are considered to validate the applicability of the scheme. The numerical results are displayed in tables and graphs to support the theoretical findings. The scheme converges uniformly with order one in space and two in time.

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Journal of Mathematical Modeling
Volume 12, Issue 3
September 2024
Pages 501-516

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