🤖 AI Summary
This work addresses the high-precision numerical solution of abstract Cauchy problems involving Caputo-type time-fractional derivatives, unbounded operator coefficients, and nonzero initial data. A novel unified representation of the mild solution is proposed, valid for all α ∈ (0,2), thereby achieving the first formulation that simultaneously encompasses subparabolic, parabolic, and subhyperbolic regimes. Based on this representation, an adaptive Sinc quadrature method is developed—incorporating the operator’s spectral properties, the fractional order α, and the regularity of initial data. Theoretical analysis establishes exponential convergence over the entire time domain. The method preserves high-order accuracy at t = 0, requires no a priori extrapolation of the forcing term, and naturally enables multilevel parallelization. Numerical experiments rigorously validate the derived error estimates, demonstrating substantially improved accuracy and stability compared to classical finite difference and convolution quadrature methods.
📝 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.