[Paper Review] Determination of time dependent factors of coefficients in fractional diffusion equations

Kenichi Fujishiro, Yavar Kian|arXiv (Cornell University)|Jan 8, 2015

Advanced Mathematical Modeling in Engineering8 references6 citations

TL;DR

This paper establishes stability estimates for inverse problems involving time-dependent coefficients in time-fractional diffusion equations, proving that the time-dependent source or reaction coefficient can be uniquely and stably reconstructed from pointwise observations of the solution. Using fractional calculus and energy estimates, the authors derive Hölder-type stability bounds under minimal regularity assumptions, extending classical inverse problem theory to fractional PDEs with anomalous diffusion behavior.

ABSTRACT

We consider fractional diffusion equations and study the stability of the inverse problem of determining the time-dependent parameter in a source term or a coefficient of zero-th order term from observations of the solution at one point in a bounded domain.

Motivation & Objective

• To address the inverse problem of identifying time-dependent coefficients in time-fractional diffusion equations from limited pointwise observations.
• To establish uniqueness and stability of the reconstruction for both source term and zeroth-order coefficient problems.
• To extend classical inverse problem theory to the context of fractional-order PDEs modeling anomalous diffusion.
• To provide quantitative stability estimates under minimal regularity assumptions on coefficients and data.

Proposed method

• Formulate two initial-boundary value problems (IBVPs) for time-fractional diffusion equations with Caputo derivatives of order $\alpha \in (0,1)$.
• Use the method of semigroups and fractional integration to prove existence and regularity of solutions in Sobolev spaces $H^2(\Omega)$.
• Apply generalized Gronwall's inequality to control solution norms and derive stability estimates for the inverse problem.
• Employ Sobolev embedding and trace estimates to relate pointwise solution values to global solution norms.
• Derive a priori bounds on the time-dependent coefficient $f(t)$ using the observed solution at a single point $x_0$.
• Utilize the structure of the fractional derivative and coercive estimates to obtain Hölder-type stability for $f(t)$.

Experimental results

Research questions

• RQ1 Can the time-dependent coefficient $f(t)$ in a time-fractional diffusion equation be uniquely determined from pointwise observations of the solution at a single spatial location?
• RQ2 What stability estimates can be established for the inverse problem of recovering $f(t)$ from a single point observation?
• RQ3 How does the regularity of the coefficient $f(t)$ and the data influence the stability of the reconstruction?
• RQ4 What role does the fractional order $\alpha$ play in the stability of the inverse problem?

Key findings

• The inverse problem of recovering the time-dependent coefficient $f(t)$ from pointwise observations at $x_0 \in \overline{\Omega}$ is uniquely solvable under minimal regularity assumptions.
• A Hölder-type stability estimate is established: $\|f\|_{L^\infty(0,T)} \leq C \|\partial_t^\alpha u(x_0, \cdot)\|_{L^\infty(0,T)}$ for the source term problem.
• For the reaction coefficient problem, the stability estimate takes the form $\|f\|_{L^\infty(0,T)} \leq C \|\partial_t^\alpha v(x_0, \cdot)\|_{L^\infty(0,T)}$, with $C$ depending on domain and data.
• The solution $u$ to the IBVP (1.1) satisfies $\partial_t^\alpha u \in L^p(0,T; H^s(\Omega))$ for $s > d/2$, ensuring pointwise trace regularity.
• The proof relies on energy estimates and a generalized Gronwall inequality to control the growth of the difference between solutions.
• The results are robust under minimal smoothness assumptions: $f \in L^\infty(0,T)$, $R \in L^p(0,T; H^2(\Omega))$, and $q \in L^\infty(0,T; H^2(\Omega))$.