Direct Solution of Third-Order Linear Integro-Differential Equation Using Cubic B-Spline Method

Abstract

Solving Volterra integro-differential equations with cubic B-splines offers a powerful, flexible numerical approach to model complex real-world systems, such as those in population dynamics or material science. The collocation method, based on the cubic B-spline approach, has been developed to solve third-order linear Fredholm and Volterra integro-differential equations. Cubic B-splines are often preferred over other interpolation and approximation techniques due to their smoothness, local control, numerical stability, and computational efficiency. This cubic B-spline method collocates both the solution and its derivatives. Meanwhile, the integral part is approximated using the GaussLegendre quadrature formula. A theoretical convergence analysis of the cubic B-spline method has been conducted and found that the order convergence is second order. The method was applied to six test examples, and the numerical solutions were compared with the corresponding analytical solutions. The maximum absolute error and mean square root error are decreased by increasing the number N. It has been found that the proposed method is efficient and capable of solving third-order linear integro-differential equations for different values of N.