Fitted tension spline method for singularly perturbed parabolic problem with a large temporal lag

Articles in Press

Document Type : Research Article

Authors

• Sisay Ketema Tesfaye ¹
• Tekle Gemechu Dinka ¹
• Gemechis File Duressa ²
• Mesfin Mekuria Woldaregay ¹

¹ Department of Applied Mathematics, School of Applied Natural Sciences, Adama Science and Technology University, Po. Box No. 1888, Adama, Ethiopia.

² Department of Mathematics, College of Natural Sciences, Jimma University, Jimma, Ethiopia.

10.22067/ijnao.2024.86799.1388

Abstract

We develop a fitted tension spline numerical scheme for singularly perturbed parabolic problem with a large temporal lag. A priori bounds and properties of the continuous solution are discussed. Due to the problem's small parameter $\varepsilon$, as a multiple diffusion term, the solution possesses a multi-scale character in the boundary layer region, which it exhibits on right side of the domain. This results in a challenging duty to solve such problems analytically or using classical numerical methods. Classical numerical methods cause spurious nonphysical oscillations unless an unacceptable number of mesh points is considered, which requires a large computational cost. To handle this difficulty, the method comprises the Crank-Nicolson method in the temporal direction and the fitted spline method in the spatial direction on uniform meshes. The stability of the method is studied using the discrete maximum principle and discrete solution bounds. We proved that the proposed scheme is uniformly convergent, with an order one in the space and an order two in the time directions. Two numerical examples are considered to validate the efficiency and applicability of the proposed scheme. Further, the boundary layer behavior of the solutions are given graphically.

Keywords

• Singularly perturbed
• Delay differential equation
• Exponentially fitted finite difference
• Tension spline
• Uniform convergence

Main Subjects

• Computational Mathematics

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