Higher-Order Uniformly Convergent Numerical Scheme for Singularly Perturbed Differential Difference Equations with Mixed Small Shifts

Abstract

This paper deals with numerical treatment of singularly perturbed differential difference equations involving mixed small shifts on the reaction terms. The highest-order derivative term in the equation is multiplied by a small perturbation parameter <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mi>ε</a:mi> </a:math> taking arbitrary values in the interval <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M2"> <c:mfenced open="(" close="]" separators="|"> <c:mrow> <c:mn>0,1</c:mn> </c:mrow> </c:mfenced> </c:math> . For small values of <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" id="M3"> <h:mi>ε</h:mi> </h:math> , the solution of the problem exhibits exponential boundary layer on the left or right side of the domain and the derivatives of the solution behave boundlessly large. The terms having the shifts are treated using Taylor’s series approximation. The resulting singularly perturbed boundary value problem is solved using exponentially fitted operator FDM. Uniform stability of the scheme is investigated and analysed using comparison principle and solution bound. The formulated scheme converges uniformly with linear order before Richardson extrapolation and quadratic order after Richardson extrapolation. The theoretical analysis of the scheme is validated using numerical test examples for different values of <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" id="M4"> <j:mi>ε</j:mi> </j:math> and mesh number <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" id="M5"> <l:mi>N</l:mi> </l:math> .

Topics & Concepts