Approximation of Fractional Derivatives Via Gauss Integration

TL;DR

This paper presents new numerical methods using Gauss integration techniques to approximate fractional derivatives, focusing on inverse Laplace transforms with error bounds and demonstrating effectiveness through numerical examples.

Contribution

It introduces modified Gauss integration and Gauss-Laguerre integration methods for fractional derivatives, including error analysis and practical numerical implementations.

Findings

Effective approximation of fractional derivatives using MGI and GLI methods.

Error bounds established for the proposed numerical solutions.

Numerical examples demonstrate accuracy and applicability.

Abstract

In this paper approximations of three classes of fractional derivatives (FD) using modified Gauss integration (MGI) and Gauss-Laguerre integration (GLI) are considered. The main solutions of these fractional derivatives depend on inverse of Laplace transforms, which are handled by these procedures. In the modified form of integration the weights and nodes are obtained by means of a difference equation, which gives a proper approximation form for the inverse of Laplace transform and hence the fractional derivatives. Theorem is established to indicate the boundary of the error of the solutions. Numerical examples are given to illuminate the results of the application of these methods.