[Paper Review] An inverse problem for distributed order time-fractional diffusion equations

Zhiyuan Li, Fujishiro, Kenichi+1 more|arXiv (Cornell University)|Jul 9, 2017

Fractional Differential Equations Solutions6 references3 citations

TL;DR

This paper investigates the inverse problem of identifying the weight function μ(α) in a distributed order time-fractional diffusion equation from a single point observation u(x₀,t) on (0,T). By employing eigenfunction expansion and Laplace transform techniques, the authors establish a Harnack-type inequality in the frequency domain and prove the uniqueness of μ under the assumption that μ is a finite oscillatory function, thereby resolving the inverse problem for the first time in this setting.

ABSTRACT

This paper deals with the distributed order time-fractional diffusion equations with non-homogeneous Dirichlet (Nuemann) boundary condition. We first prove the wellposedness of the weak solution to the initial boundary value problem for the distributed order time-fractional diffusion equation by means of eigenfunction expansion, which ensure that the weak solution has the classical derivatives. We next give a Harnack type inequality of the solution in the frequency domain under the Laplace transform, from which we further show a uniqueness result for an inverse problem in determining the weight function in the distributed order time derivative from point observation.

Motivation & Objective

• To establish the well-posedness of the weak solution to the initial-boundary value problem for distributed order time-fractional diffusion equations with non-homogeneous Dirichlet or Neumann boundary conditions.
• To investigate the inverse problem of determining the weight function μ(α) in [0,1] from a single point observation u(x₀,t) on (0,T).
• To prove the uniqueness of the solution to the inverse problem under the assumption that μ is a finite oscillatory function.
• To develop a Harnack-type inequality for the solution in the Laplace frequency domain to support the uniqueness argument.

Proposed method

• Utilizes eigenfunction expansion to derive a representation formula for the weak solution and establish its convergence and regularity.
• Applies the Laplace transform to convert the time-fractional PDE into an elliptic equation in the frequency domain.
• Establishes a Harnack-type inequality for the transformed solution to control the behavior of the solution in the frequency domain.
• Employs a contradiction argument based on the strong maximum principle for elliptic equations to prove uniqueness of μ.
• Imposes the condition that μ is a finite oscillatory function to ensure the applicability of the Laplace transform-based argument.
• Uses the fact that equal Laplace transforms at a single point x₀ imply equality of the transformed solutions, leading to μ = μ̃.

Experimental results

Research questions

• RQ1 Can the weight function μ(α) in a distributed order time-fractional diffusion equation be uniquely determined from a single point observation u(x₀,t) on (0,T)?
• RQ2 What conditions on μ(α) ensure the uniqueness of the inverse problem in the distributed order setting?
• RQ3 How can the Laplace transform be used to analyze the inverse problem for time-fractional diffusion equations with distributed order derivatives?
• RQ4 Does a Harnack-type inequality in the frequency domain help establish uniqueness for inverse problems in fractional diffusion?
• RQ5 Can the strong maximum principle be applied to the transformed elliptic system to derive uniqueness results?

Key findings

• The weak solution to the initial-boundary value problem for the distributed order time-fractional diffusion equation exists and is well-posed, with classical derivatives, under the given assumptions.
• A Harnack-type inequality is established for the solution in the Laplace frequency domain, which is crucial for the uniqueness proof.
• Uniqueness of the weight function μ(α) is proven for μ in the admissible set of finite oscillatory functions, based on a single point observation u(x₀,t) on (0,T).
• The proof relies on contradiction: if μ ≠ μ̃, then the Laplace transforms of the solutions would differ at x₀, contradicting the observation data.
• The method extends to both Dirichlet and Neumann boundary conditions via similar arguments involving the Laplace transform and strong maximum principle.
• The result remains open for general μ not in the finite oscillatory class, particularly when data is restricted to (0,T) rather than (0,∞).