A Numerical Scheme for Solving Fourth-Order Convection–Reaction–Diffusion Problems with Boundary Layers

Abstract

This paper presents a numerical approach for decoupling singularly perturbed boundary value problems involving fourth-order ordinary differential equations, characterized by a small positive parameter ϵ multiplying the highest derivative. Such equations arise in various engineering and physics applications, including the modeling of diffusing chemical species, viscous flows with convection and diffusion, and heat transfer in electronic chips or microfluidic channels. We focus on problems with Lidstone boundary conditions and demonstrate how the fourth-order equation can be decomposed into a system of two second-order problems—one independent of $\epsilon$, and the other singularly perturbed with ϵ multiplying the highest derivative. These problems often exhibit boundary layers, where the solution undergoes rapid changes near the domain boundaries. Numerical solutions to such higher-order problems are typically more challenging than those for lower-order ones. To address this, we propose a linear finite element method combined with a Shishkin mesh to accurately resolve boundary layers. We prove that the solution obtained from the decoupled second-order system is equivalent to that of the original fourth-order problem. The proposed method is direct and highly accurate, with computational time increasing linearly with the number of grid points.