Uniformly convergent scheme for fourth-order singularly perturbed convection-diffusion ODE.

This paper contemplates a numerical investigation of the convection-diffusion type's fourth-order singularly perturbed linear and nonlinear boundary value problems. First, the considered linear fourth-order differential equation is converted into a strongly/weakly coupled singularly perturbed system (depending on the coefficient of the first-order derivative) of two ordinary differential equations with Dirichlet boundary conditions to solve the problem numerically. One of the equations is free from the perturbation parameter in the system. To obtain the solution for this system, we propose a numerical method of quadratic B -splines on an exponentially graded mesh. Convergence analysis shows that the proposed numerical scheme is second-order uniformly convergent in the discrete maximum norm. The nonlinear differential equation is linearized using the quasilinearization technique, and then the proposed approach is applied to the linearized problem. The theoretical outcomes are validated by executing the proposed method on three test problems. • It is challenging to obtain parameter-uniform results for higher-order singularly perturbed linear and nonlinear problems. • This paper presents a novel idea to resolve the boundary layers efficiently. • We present a numerical method to solve wild-behaving higher-order problems using quadratic spline basis functions. • An exponentially graded mesh is used to resolve the boundary layers. [ABSTRACT FROM AUTHOR]