Asymptotic numerical method for boundary value problems for singularly perturbed fourth-order ordinary differential equations with a weak interior layer

Singularly perturbed two-point boundary value problems (SPBVPs) of convection–diffusion type for fourth-order ordinary differential equations (ODEs) with a small positive parameter multiplying the highest derivative with a discontinuous source term is considered. The given fourth-order BVP is transformed into a system of weakly coupled systems of two second-order ODEs, one without the parameter and the other with the parameter ε multiplying the highest derivative, and suitable boundary conditions. In this paper, a computational method for solving this system is presented. In this method, we first find the zero-order asymptotic approximation expansion 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 expansion of the solution in the second equation. Then the second equation is solved by the numerical method which is constructed for this problem which involves Shishkin mesh.