ITM Web Conf.
Volume 34, 2020International Conference on Applied Mathematics and Numerical Methods – third edition (ICAMNM 2020)
|Number of page(s)||18|
|Section||Applied Mathematics and Numerical Methods|
|Published online||03 December 2020|
Uniformly convergent numerical scheme for singularly perturbed parabolic delay differential equations
Department of Applied Mathematics, Adama Science and Technology University, Adama, Ethiopia
2 Department of Mathematics, Jimma University, Jimma, Ethiopia
This paper deals with numerical treatment of singularly perturbed parabolic diﬀerential diﬀerence equations having small shifts on the spatial variable. The considered problem contain small perturbation parameter (ε) multiplied on the diﬀusion term of the equation. For small values of ε the solution of the problem exhibits a boundary layer on the left or right side of the spatial domain depending on the sign of the convective term. The terms involving the shifts are approximated using Taylor’s series approximation. The resulting singularly perturbed parabolic partial diﬀerential equation is solved using implicit Euler method in the temporal discretization with exponentially ﬁtted operator ﬁnite diﬀerence method in the spatial discretization. The uniform stability of the scheme investigated using comparison principle and discrete solution bound by constructing barrier function. Uniform convergence analysis has been carried out. The scheme gives second order convergence for the case ε > N−1 and first order convergence for the case ε « N−1, where N is number of mesh interval. Test examples and numerical results are considered for validating the theoretical analysis of the scheme.
Key words: diﬀerential diﬀerence equations / ﬁtted operator scheme / singular perturbation
© The Authors, published by EDP Sciences, 2020
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