An Approach to Generalized High-Order Fredholm Integro-Differential Equations via Fractional Cubic Spline

Abstract

 This paper presents an efficient numerical technique for the approximate solution of multi-term integro-differential equations using a new form of fractional cubic spline (FCS). The proposed method is derived from a fractional boundary condition and a fractional continuity condition on a fractional spline, expressed in matrix form. A detailed convergence analysis is provided, and sufficient conditions for stability and error bounds of the method are established. Moreover, several numerical examples are solved, and the results are compared with exact solutions and methods from previous papers, presented in tables and figures. These comparisons demonstrate the efficacy and suitability of the suggested fractional spline scheme for solving integro-differential problems, achieving higher accuracy with fewer grid points. All computational results were obtained using Python 3.12.4.