A Numerical Scheme to Solve Fourth Order Convection Reaction Diffusion Problems with Boundary Layers

Abstract

This paper presents an approach to decoupling singularly perturbed boundary value prob- lems for fourth-order ordinary differential equations, which feature a small positive parameter ε multiplying the highest derivative. These equations have various real-world applications in engineering and physics, such as describing the behavior of diffusing chemical species, viscous flows with convection and diffusion effects, and heat transfer in microfluidic channels or electronic chips. We focus on Lidstone boundary conditions and demonstrate how to break down fourth-order differential equations into a system of second-order problems—one without the parameter and one with ε multiplying the highest derivative. Singularly per- turbed problems often involve boundary layers where the solution changes rapidly near the boundaries. Numerical solutions for higher-order problems are significantly more challeng- ing than those for lower-order problems. We propose a linear finite element algorithm and incorporate the Shishkin mesh scheme to capture the solution near the boundary layers. We prove that the solution obtained from the second-order system is equivalent to that of the fourth-order problem. Our solver is both direct and highly accurate, with computation time scaling linearly with the number of grid points.