Computational methods for reaction–diffusion problems for fourth order ordinary differential equations with a small parameter at the highest derivative

Abstract

In this paper basically-asymptotic numerical methods for solving singularly perturbed two-point boundary value problems for fourth order ordinary differential equations of the form−εyiv(x)+b(x)y′′(x)+c(x)y(x)=f(x),x∈D:=(0,1),y(0)=p,y′(0)=q,y′′(0)=r,y′′(1)=s,is considered. Here a prime “′” denotes a differentiation with respect to x, b(x), c(x) and f(x) are smooth functions, b(x)⩾β>0, 0⩾c(x)⩾−γ, γ>0 and 0<ε≪1. The above boundary value problem is transformed into an equivalent weakly coupled system of two first order ordinary differential equations subject to suitable initial conditions and one second order singularly perturbed ordinary differential equations subject to suitable boundary conditions. In order to solve this system three computational methods are suggested in this paper. In these methods, first we find a zero order asymptotic approximation of the solution of the weakly coupled system. Then the system is decoupled by replacing the first component of the solution by its zero order asymptotic approximation of the solution in the second order equation. Then the second order equation is solved separately by three methods namely fitted operator method, fitted mesh method and boundary value technique. Error estimates are derived and examples are provided to illustrate the methods.