An efficient solution technique for fractional integro-differential equations modeling chemical phenomena

Abstract

Fractional integro-differential equations serve as powerful mathematical tools that model complex real-world phenomena with memory effects and non-local dependencies, providing superior accuracy over traditional integer-order models across diverse fields, including physics, chemistry, biology, engineering, and finance. In this work, we propose an efficient numerical method for approximating the solution of second-order fractional integro-differential equations. The approach reformulates the original problem into a real-valued functional using the least-squares technique. Minimizing this functional then yields the numerical solution to the considered problem. We assess the accuracy of the proposed method by applying it to several benchmark test problems and comparing the results with those of existing techniques, including the Haar wavelet method, the fractional iteration method, the \(C^3\)-spline method, and the reproducing kernel least-squares method. The findings indicate that the proposed approach achieves higher accuracy than these existing methods. To demonstrate the broad applicability of the proposed approach, we solve the well-known Bagley-Torvik equation, can be regarded as a special case of the our proposed problem. Furthermore, we have conducted a thorough analysis of error estimation and the existence and uniqueness of the solution, thereby providing strong theoretical support for our numerical strategy.