Two-step Bi-basis Hybrid Block Method for Direct Approximation of Fourth Order Ordinary Differential Equations with Intermediate Bi-intra Points

Abstract

Most recently, higher order problems are being addressed by decomposing it into system of lower order problems. However, it was discovered that methods with high order and strong stability were able to approximate the resulting systems accurately as the problems become unstable in the region of the new transformation field. This research actually sought for methods of solution of higher order problems without any need for system transformation. The method is proposed for the direct solution of fourth order ordinary differential equations. The fundamental basis is sought from the combination of Shifted Chebyshev Orthogonal Polynomial and the Hermite Orthogonal Polynomial, these polynomial functions are then used to obtain the method using the concept of interpolation and collocation. The proposed method is found to be consistent and zero-stable, which then implies convergence. From the numerical results obtained, the efficiency of the method was obtained and its superiority strength was also established when comparison was made with existing.