Fractional calculus via variable-transform-based spectral approximations
This work provides a unifying and numerically stable approach for solving a broad class of fractional calculus problems, which are important in various scientific and engineering domains.
The paper presents a novel framework for constructing spectral approximations to fractional integral operators using variable transforms and Chebyshev polynomials, enabling stable and fast spectral methods for fractional calculus problems that are intractable for existing methods.
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.