Approximations for Linear Tenth-order Boundary Value Problems through Polynomial and Non-polynomial Cubic Spline Techniques

Higher order differential equations have always been a tedious problem to solve for the mathematicians and engineers. Different numerical techniques were carried out to obtain numerical approximations to such problems. This research work presented and illustrated a novel numerical technique to approximate the tenth-order boundary value problems (BVPs). The techniques developed in this research were based on the principle of employing non-polynomial cubic spline method (NPCSM) and polynomial cubic spline method (PCSM) along with the decomposition procedure. The decomposition procedure was used to reduce the tenth-order BVPs into the corresponding system of second-order BVPs. The NPCSM and PCSM schemes were constructed for each second-order ordinary differential equation (ODE), whereas the first-order derivatives were approximated by finite central differences. The performance of the new developed numerical techniques was illustrated by numerical tests that involved comparing numerical solutions with the exact solution on a collection of test problems.