Some Higher-Degree Lacunary Fractional Splines in the Approximation of Fractional Differential Equations

Abstract

In this article, we begin by introducing two classes of lacunary fractional spline functions by using the Liouville–Caputo fractional Taylor expansion. We then introduce a new higher-order lacunary fractional spline method. We not only derive the existence and uniqueness of the method, but we also provide the error bounds for approximating the unique positive solution. As applications of our fundamental findings, we offer some Liouville–Caputo fractional differential equations (FDEs) to illustrate the practicability and effectiveness of the proposed method. Several recent developments on the the theory and applications of FDEs in (for example) real-life situations are also indicated.