Novel approach to solve singularly perturbed boundary value problems with negative shift parameter

Abstract

Singularly perturbed boundary value problems with negative shift parameter are special types of differential difference equations whose solution exhibits boundary layer behaviour. A simple but novel numerical method is developed to approximate the numerical solution of the problems of these types. The method gives accurate solutions for ℎ ≥ in the inner region of the boundary layer where other classical numerical methods fail to give smooth solution. The present method is proved to be point-wise uniformly convergent with second order rate of convergence.

A Parameter‐Uniform Numerical Scheme for Solving Singularly Perturbed Parabolic Reaction‐Diffusion Problems with Delay in the Spatial Variable

“…On the interval [0, 1], we have y(x − 1, 0) � ϕ(x − 1, 0) � 0 and on the interval [1,2], we have y(x − 1, 0) � y 0 (x − 1, 0) � 0. Combining these results give y(x − 1, 0) � 0, so that equation (11) becomes zy/zt(x, 0) � g(x, 0) and since g is a smooth function, we have |y(x, 0)| ≤ C on zR, which implies that…”

“…For instance, Cimen and Amiraliyev [10] solved singularly perturbed delay diferential equation with layer behavior by constructing a fnite diference scheme using the ftted mesh method and obtained an essentially frst-order parameter-uniformly convergent result. Duressa [11] treated a singularly perturbed problem by constructing a numerical scheme using a ftting parameter introduced in a fnite diference approximation. Kumar et al [12] solved singularly perturbed problems involving a term with a large negative shift by developing an exponentially ftted numerical scheme using Numerov's method.…”

Second-order robust finite difference method for singularly perturbed Burgers' equation

“…In this section, two numerical examples have been considered. Their exact solutions are not available, so that the errors are evaluated by using the double mesh principle [1], [5], [14], [20], [24] given…”

A tension spline fitted numerical scheme for singularly perturbed reaction-diffusion problem with negative shift

“…Various research works are available in the literature to address the aforementioned limitations. For instance, Duressa [6] constructed a numerical method for singularly perturbed differential equation involving small delay by introducing a fitting parameter applying the finite difference approximation. Swamy et al [7] treated singularly perturbed problems by replacing the original second order with a first order neutral form of delay differential equation and applied the Trapezoidal rule.…”