Numerical integration method for two-parameter singularly perturbed time delay parabolic problem

Abstract

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This study presents an (ε, μ)−uniform numerical method for a two-parameter singularly perturbed time-delayed parabolic problems. The proposed approach is based on a fitted operator finite difference method. The Crank–Nicolson method is used on a uniform mesh to discretize the time variables initially. Subsequently, the resulting semi-discrete scheme is further discretized in space using Simpson's 1/3rd rule. Finally, the finite difference approximation of the first derivatives is applied. The method is unique in that it is not dependent on delay terms, asymptotic expansions, or fitted meshes. The fitting factor's value, which is used to account for abrupt changes in the solution, is calculated using the theory of singular perturbations. The developed scheme is demonstrated to be second-order accurate and uniformly convergent. The proposed method's applicability is validated by three model examples, which yielded more accurate results than some other methods found in the literature.

Keywords

• singularly perturbed problems
• delay parabolic differential equation
• numerical integration
• Simpson's 1/3rd rule
• parameter-uniform