Exponentially Convergent Numerical Method for Abstract Cauchy Problem with Fractional Derivative of Caputo Type

Dmytro Sytnyk () and Barbara Wohlmuth ()
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Dmytro Sytnyk: Department of Mathematics, Technical University of Munich, 85748 Garching, Germany
Barbara Wohlmuth: Department of Mathematics, Technical University of Munich, 85748 Garching, Germany

Mathematics, 2023, vol. 11, issue 10, 1-35

Abstract: We present an exponentially convergent numerical method to approximate the solution of the Cauchy problem for the inhomogeneous fractional differential equation with an unbounded operator coefficient and Caputo fractional derivative in time. The numerical method is based on the newly obtained solution formula that consolidates the mild solution representations of sub-parabolic, parabolic and sub-hyperbolic equations with sectorial operator coefficient A and non-zero initial data. The involved integral operators are approximated using the sinc-quadrature formulas that are tailored to the spectral parameters of A , fractional order α and the smoothness of the first initial condition, as well as to the properties of the equation’s right-hand side f ( t ) . The resulting method possesses exponential convergence for positive sectorial A , any finite t , including t = 0 and the whole range α ∈ ( 0 , 2 ) . It is suitable for a practically important case, when no knowledge of f ( t ) is available outside the considered interval t ∈ [ 0 , T ] . The algorithm of the method is capable of multi-level parallelism. We provide numerical examples that confirm the theoretical error estimates.

Keywords: inhomogeneous Cauchy problem; Caputo fractional derivative; sub-parabolic problem; sub-hyperbolic problem; mild solution; numerical method; contour integration; exponential convergence; parallel algorithm (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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