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Robust numerical method for singularly perturbed differential equations with large delay

Abdulla Murad Ibrahim, Duressa Gemechis File, Debela Habtamu Garoma

Demonstratio Mathematica

2021

Vol. 54, nr 1

576--589

In this paper, a singularly perturbed differential equation with a large delay is considered. The considered problem contains a large delay parameter on the reaction term. The solution of the problem exhibits the interior layer due to the delay parameter and the strong right boundary layer due to the small perturbation parameter ε. The resulting singularly perturbed problem is solved using the fitted non-polynomial spline method. The stability and parameter uniform convergence of the proposed method is proved. To validate the applicability of the scheme, two model problems of the variable coefficient are considered for numerical experimentation.

The finite difference method on adaptive mesh for singularly perturbed nonlinear 1D reaction diffusion boundary value problems

Duru Hakkı, Güneş Baransel

Journal of Applied Mathematics and Computational Mechanics

2020

Vol. 19, nr 4

45--56

In this paper, we study singularly perturbed nonlinear reaction-diffusion equations. The asymptotic behavior of the solution is examined. The difference scheme which is accomplished by the method of integral identities with using of interpolation quadrature rules with weight functions and remainder term integral form is established on adaptive mesh. Uniform convergence and stability of the difference method are discussed in the discrete maximum norm. The discrete scheme shows that orders of convergent rates are close to 2. An algorithm is presented, and some problems are solved to validate the theoretical results.