A Uniformly Convergent Collocation Method for Singularly Perturbed Delay Parabolic Reaction-Diffusion Problem

Abstract

In this article, a numerical solution is proposed for singularly perturbed delay parabolic reaction-diffusion problem with mixed-type boundary conditions. The problem is discretized by the implicit Euler method on uniform mesh in time and extended cubic B-spline collocation method on a Shishkin mesh in space. The parameter-uniform convergence of the method is given, and it is shown to be <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mi>ε</a:mi> </a:math> -uniformly convergent of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M2"> <c:mi>O</c:mi> <c:mfenced open="(" close=")"> <c:mrow> <c:mi>Δ</c:mi> <c:mi>t</c:mi> <c:mo>+</c:mo> <c:msup> <c:mrow> <c:mi>N</c:mi> </c:mrow> <c:mrow> <c:mo>−</c:mo> <c:mn>2</c:mn> </c:mrow> </c:msup> <c:msup> <c:mrow> <c:mi mathvariant="normal">ln</c:mi> </c:mrow> <c:mrow> <c:mn>2</c:mn> </c:mrow> </c:msup> <c:mi>N</c:mi> </c:mrow> </c:mfenced> </c:math> , where <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" id="M3"> <h:mi>Δ</h:mi> <h:mi>t</h:mi> </h:math> and <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" id="M4"> <j:mi>N</j:mi> </j:math> denote the step size in time and number of mesh intervals in space, respectively. The proposed method gives accurate results by choosing suitable value of the free parameter <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" id="M5"> <l:mi>λ</l:mi> </l:math> . Some numerical results are carried out to support the theory.

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