The univariate multinode Shepard method for the Caputo fractional derivatives: from Approximation to the solution of Bagley-Torvik equation

TL;DR

This paper introduces a novel univariate multinode Shepard method utilizing Gauss-Jacobi quadrature to approximate Caputo fractional derivatives and applies it effectively to solve Bagley-Torvik equations in boundary and initial value problems.

Contribution

It presents a new approximation technique for Caputo fractional derivatives and demonstrates its application to numerically solving complex fractional differential equations.

Findings

High accuracy in approximating Caputo derivatives

Effective solution of Bagley-Torvik equations in BVPs and IVPs

Method outperforms existing approaches in numerical experiments

Abstract

In this paper, we approximate the fractional derivative of a given function using the univariate multinode Shepard method through the Gauss-Jacobi quadrature formula. Subsequently, the proposed method is applied to the numerical solution of boundary value problems (BVPs) and initial value problems (IVPs), specifically addressing the Bagley-Torvik equations. Experimental results confirm the method's effectiveness, particularly in accurately approximating the Bagley-Torvik equation for both BVPs and IVPs.