[Paper Review] Initial-boundary value problem for distributed order time-fractional diffusion equations

Zhiyuan Li, Yavar Kian+1 more|arXiv (Cornell University)|Sep 20, 2017

Fractional Differential Equations Solutions12 references3 citations

TL;DR

This paper establishes the existence, uniqueness, and continuous dependence of weak solutions to initial-boundary value problems for distributed order time-fractional diffusion equations with general weight functions. Using Laplace transform techniques and spectral analysis, it proves analyticity in time of the solution under suitable source term conditions, extending regularity results beyond single-order fractional models.

ABSTRACT

We examine initial-boundary value problems for diffusion equations with distributed order time-fractional derivatives. We prove existence and uniqueness results for the weak solution to these systems, together with its continuous dependency on initial value and source term. Moreover, under suitable assumption on the source term, we establish that the solution is analytic in time.

Motivation & Objective

• To establish the existence and uniqueness of weak solutions for distributed order time-fractional diffusion equations with general weight functions.
• To prove continuous dependence of the solution on the initial data and source term.
• To investigate the regularity of the solution, particularly its analyticity in time under suitable source term assumptions.
• To extend the theoretical framework for distributed order fractional diffusion beyond existing results on convergence or formal solutions.

Proposed method

• Formalizing the problem as an abstract evolutionary integro-differential equation in a Hilbert space setting.
• Applying the Laplace transform with respect to time to characterize the weak solution as the inverse Laplace transform of a resolvent-type equation.
• Using spectral theory of the elliptic operator $\mathcal{A}$ to analyze the resolvent and derive decay estimates for the solution kernel.
• Employing estimates involving the weight function $\mu(\alpha)$ and the sine integral $\sin(\pi\alpha)$ to control the behavior of the resolvent in the Laplace domain.
• Applying Rolle’s theorem and asymptotic analysis to derive bounds on integrals involving $r^\alpha$ and $\mu(\alpha)$, ensuring convergence of the inverse Laplace transform.
• Establishing the analyticity of the solution in time by proving that the Laplace transform of the solution decays faster than any polynomial at infinity.

Experimental results

Research questions

• RQ1 Under what conditions does a weak solution exist for the distributed order time-fractional diffusion equation with Dirichlet boundary conditions?
• RQ2 How does the solution depend continuously on the initial data and source term in the $L^2$-framework?
• RQ3 Can the solution be shown to be analytic in time, and under what assumptions on the source term?
• RQ4 What is the role of the weight function $\mu(\alpha)$ in determining the regularity and decay properties of the solution?

Key findings

• The weak solution to the distributed order time-fractional diffusion equation exists and is unique in the $L^2(0,T;H^1_0(\Omega))$ framework under standard assumptions on the coefficients and $\mu\in L^\infty(0,1)$.
• The solution depends continuously on the initial data $u_0$ and the source term $F$, with stability estimates uniform in the weight function $\mu$.
• If the source term $F$ is analytic in time, then the solution $u$ is analytic in time on $[0,T]$.
• The Laplace transform of the solution satisfies a resolvent-type equation whose kernel decays faster than any polynomial at infinity, implying analyticity of $u$ in time.
• The proof relies on a key estimate: $\int_{a_n}^{\infty} \frac{\Phi_n(r)}{r} dr \leq \frac{C}{\lambda_n}$ for large $n$, which controls the inverse Laplace transform and ensures regularity.
• The analysis shows that the solution's decay and regularity are governed by the lower bound of $\mu(\alpha)$ near $\alpha_0 \in (0,1)$, with $\mu(\alpha_0) > 0$ ensuring sufficient growth of the moment generating function.