[Paper Review] Fractional derivative in spaces of continuous functions

Davide Guidetti|arXiv (Cornell University)|Jul 7, 2017

Nonlinear Differential Equations Analysis4 references3 citations

TL;DR

This paper establishes a functional calculus for fractional derivatives in spaces of continuous functions by defining fractional powers of a differential operator B via contour integration. It proves that the fractional integral operator B^{-α} corresponds to the Riemann-Liouville integral of order α, and derives a solution representation for time-fractional evolution equations using the Mittag-Leffler function, providing existence and regularity results for strict solutions in Hölder spaces.

ABSTRACT

This is a first version of a paper concerning abstract evolution equation with fractional time derivatives. Maximal regularity results in spaces of continuous and Hoelder continuous functions are described.

Motivation & Objective

• To develop a rigorous functional calculus for fractional derivatives in the Banach space C([0,T];X) of continuous X-valued functions.
• To characterize the fractional integral operator B^{-α} as the Riemann-Liouville integral of order α.
• To establish the solution theory for time-fractional evolution equations with Caputo-type derivatives using the Mittag-Leffler function.
• To derive sharp regularity conditions for strict solutions in Hölder spaces, linking the regularity of data to the fractional order α and the Hölder exponent γ.
• To extend the theory to elliptic operators on bounded domains with Neumann boundary conditions, proving existence and regularity of solutions to mixed problems.

Proposed method

• Define the operator B on C([0,T];X) as the derivative with zero initial condition, showing it is closed and has resolvent norm bounded by T.
• Use complex contour integration over a sector Γ to define the fractional power B^{-α} via the Dunford-Taylor integral.
• Prove that B^{-α}f(t) equals the Riemann-Liouville fractional integral of order α by residue calculus and the reflection formula for the Gamma function.
• Establish the resolvent identity for B^α, showing that (λ - B^α)^{-1}f(t) = -∫₀ᵗ E_{α,α}(λ(t-s)^α)(t-s)^{α-1}f(s)ds.
• Apply the theory to the time-fractional diffusion equation with Caputo derivative by transforming the problem into a Volterra integral equation.
• Use real interpolation theory and known regularity results for elliptic operators to derive Hölder regularity of solutions in terms of data regularity and α.

Experimental results

Research questions

• RQ1 How can fractional derivatives be rigorously defined in the space of continuous functions with values in a Banach space?
• RQ2 What is the precise relationship between the operator-theoretic fractional integral B^{-α} and the classical Riemann-Liouville integral?
• RQ3 What are the necessary and sufficient conditions on initial data and forcing terms for the existence of a strict solution to a time-fractional PDE in Hölder spaces?
• RQ4 How does the regularity of the solution depend on the fractional order α and the Hölder exponent γ of the data?
• RQ5 Can the solution theory for time-fractional equations be extended to elliptic operators with Neumann boundary conditions on bounded domains?

Key findings

• The fractional integral operator B^{-α} is equivalent to the Riemann-Liouville integral of order α, i.e., B^{-α}f(t) = 1/Γ(α) ∫₀ᵗ (t-s)^{α-1}f(s)ds.
• The resolvent of B^α is given explicitly by (λ - B^α)^{-1}f(t) = -∫₀ᵗ E_{α,α}(λ(t-s)^α)(t-s)^{α-1}f(s)ds.
• For α ∈ (0,2)  \{1}, the operator B^α generates a solution to the time-fractional equation C^αu - e^{iϕ}Δu = f with Neumann boundary conditions.
• A strict solution exists in C([0,T];D(A)) with C^αu and Au bounded in C^γ(Ω̄) if the initial data and forcing term satisfy specific Hölder regularity conditions.
• The regularity threshold is sharp: for α ∈ (1,2), the initial velocity u₁ must lie in C^{2−2(1−β)/α}(Ω̄) with normal derivative zero if the exponent exceeds 1.
• The natural scaling relation between the time regularity β and spatial regularity γ is γ = 2β/α, consistent with the parabolic case α=1.