Abstract
In this paper, a singularly perturbed second-order ordinary differential equation with discontinuous source term subject to mixed-type boundary conditions is considered. A fitted nonpolynomial spline method is suggested. The stability and parameter uniform convergence of the proposed method are proved. To validate the applicability of the scheme, two model problems are considered for numerical experimentation and solved for different values of the perturbation parameter, ε , and mesh size, h . The numerical results are tabulated in terms of maximum absolute errors and rate of convergence, and it is observed that the present method is more accurate and ε -uniformly convergent for h ≥ ε where the classical numerical methods fail to give good result and it also improves the results of the methods existing in the literature.
Highlights
Singular perturbation problems (SPPs) are differential equations with a small positive parameter multiplying the highest derivative term
To validate the applicability of the method, two model problems are considered for numerical experimentation for different values of the perturbation parameter and mesh points
The numerical results are tabulated in terms of maximum absolute errors, numerical rate of convergence, and uniform errors and compared with the results of the previously developed numerical methods existing in the literature (Tables 2 and 4)
Summary
Singular perturbation problems (SPPs) are differential equations with a small positive parameter multiplying the highest derivative term. Examples of SPPs include the Navier-Stokes equation of fluid flow at high Reynolds number, the equation governing flow in porous media, the driftdiffusion equation of semiconductor devices, physics and mathematical models of liquid crystal material, and the convection-diffusion and reaction-diffusion equations to mention but a few [1, 2]. Such equations typically exhibit solutions with layers, which cause severe computational difficulties for standard numerical methods. This gives rise to an interior layer in the exact solution of the problem, in addition to the boundary layer at the outflow boundary point
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