Fractional calculus via variable-transform-based spectral approximations

TL;DR

This paper introduces a new spectral approximation framework for fractional integral operators using variable-transform-based transplanted Chebyshev polynomials, enabling stable, fast, and versatile fractional calculus methods.

Contribution

It unifies spectral approximation approaches for fractional operators via variable transforms, including algebraic and exponential, with demonstrated numerical stability and broad applicability.

Findings

Spectral approximations are numerically stable and optimal in complexity.

The framework produces versatile spectral methods for various fractional calculus problems.

Numerical examples show the method's effectiveness on intractable problems.

Abstract

We present a novel and unifying framework for constructing spectral approximations to fractional integral operators. These spectral approximations are based on transplanted Chebyshev polynomials, which are obtained by composing Chebyshev polynomials with a variable transform. When an algebraic transform is used, the framework produces spectral approximations based on Jacobi fractional polynomials. When an exponential transform is used, it yields a versatile spectral approximation that is applicable to a much broader class of fractional calculus problems. The construction of such spectral approximations is both numerically stable and optimal in terms of complexity. These spectral approximations lead to stable and fast spectral methods for fractional calculus. The spectral approximation based on the double-exponential transform is demonstrated through extensive numerical examples that are intractable for existing spectral methods.