Higher-Order Numerical Solutions of the Fractional Relaxation-Oscillation Equation using Fractional Integration

TL;DR

This paper develops asymptotic expansion formulas for trapezoidal approximations of fractional integrals and applies them to numerically solve fractional relaxation-oscillation equations, enhancing computational methods for fractional differential equations.

Contribution

It introduces new asymptotic expansion formulas for fractional integrals and applies these to improve numerical solutions of fractional relaxation-oscillation equations.

Findings

Derived asymptotic expansion formulas for fractional integrals

Provided high-order numerical solutions for fractional relaxation-oscillation equations

Enhanced accuracy of fractional integral approximations

Abstract

In the present paper we derive the asymptotic expansion formula for the trapezoidal approximation of the fractional integral. We use the expansion formula to obtain approximations for the fractional integral of order $\alpha,1+\alpha,2+\alpha,3+\alpha$ and $4+\alpha$. The approximations are applied for computing the numerical solutions of the fractional relaxation-oscillation equation.