Asymptotic numerical methods for singularly perturbed fourth-order ordinary differential equations of reaction-diffusion type

Singularly perturbed boundary value problems (BVPs) for fourth-order ordinary differential equations (ODES) with a small positive parameter multiplying the highest derivative of the form − yiv(x)+b(x)y″(x)−c(x)y(x)=−f(x), x D (0,1), y(0)=p, y(1)=q, y″(0)=−r, y″(1)=−, 0≤ 1, are 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 e multiplying the highest derivative, and suitable boundary conditions. In this paper, computational methods for solving this system are presented. In these methods, 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 fitted operator method, fitted mesh method, and boundary value technique. Error estimates are derived and examples are provided to illustrate the methods.