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.
From: Kuan Xu [view email]
[v1]
Tue, 28 Apr 2026 09:27:11 UTC (1,816 KB)
[v2]
Wed, 29 Apr 2026 11:28:12 UTC (1,821 KB)
[v3]
Tue, 2 Jun 2026 04:31:31 UTC (1,826 KB)
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