Quintic Spline Solutions of Fourth Order Boundary-Value Problems

TL;DR

This work introduces a quintic spline framework for solving fourth-order linear boundary-value problems of the form $(\frac{d^{4}}{dx^{4}}+f(x))y(x)=g(x)$ under two-point boundary data. It constructs a quintic spline with node values and first four derivatives, derives end conditions that guarantee unique solvability and $O(h^{6})$ convergence, and solves the interior values via a sparse linear system $(\mathbf{A}+h^{4}\mathbf{B}\mathbf{F})\mathbf{Y}=\mathbf{C}+\mathbf{e}$ with $\mathbf{e}=O(h^{6})$. The paper validates the method through three numerical examples with analytic solutions, achieving high accuracy not only in function values but also in higher-order derivatives up to $y^{(4)}$, demonstrating the approach’s effectiveness for beam bending-type problems. Overall, this quintic-spline technique offers a high-order, derivative-preserving numerical method for fourth-order boundary-value problems with practical computational efficiency.

Abstract

In this paper Quintic Spline is defined for the numerical solutions of the fourth order linear special case Boundary Value Problems. End conditions are also derived to complete the definition of spline.The algorithm developed approximates the solutions, and their higher order derivatives of differential equations. Numerical illustrations are tabulated to demonstrate the practical usefulness of method.