[Paper Review] A uniqueness result for time-fractional diffusion equations with space-dependent variable order

Yavar Kian, Éric Soccorsi+1 more|arXiv (Cornell University)|Jan 15, 2017

Fractional Differential Equations Solutions13 references3 citations

TL;DR

This paper establishes the uniqueness of space-dependent variable-order coefficients in time-fractional diffusion equations by proving that these coefficients can be uniquely identified from a sequence of partial Dirichlet-to-Neumann maps over time. The approach relies on well-posedness analysis and inverse problem techniques for variable-order fractional PDEs.

ABSTRACT

We investigate diffusion equations with time-fractional derivatives of space-dependent variable order. We examine the well-posedness issue and prove that the space-dependent variable order coefficient is uniquely determined among other coefficients of these equations, by the knowledge of a suitable time-sequence of partial Dirichlet-to-Neumann maps.

Motivation & Objective

• To address the inverse problem of identifying variable-order coefficients in time-fractional diffusion equations with space-dependent orders.
• To establish the well-posedness of the forward problem for such equations.
• To determine whether the variable-order coefficient can be uniquely reconstructed from boundary measurements.
• To analyze the structural properties of the Dirichlet-to-Neumann map in the context of variable-order fractional operators.

Proposed method

• Formalizing the time-fractional diffusion equation with space-dependent variable order using Caputo-type fractional derivatives.
• Defining the forward problem in a bounded domain with appropriate boundary and initial conditions.
• Constructing a sequence of partial Dirichlet-to-Neumann maps over time to encode boundary measurements.
• Applying uniqueness arguments based on the analyticity and monotonicity properties of the Dirichlet-to-Neumann map.
• Using energy estimates and weak solution theory to ensure well-posedness of the forward problem.
• Establishing the uniqueness of the variable-order coefficient by comparing solutions generated from different orders.

Experimental results

Research questions

• RQ1 Can the space-dependent variable-order coefficient in a time-fractional diffusion equation be uniquely determined from boundary measurements?
• RQ2 How does the variable order affect the behavior of the Dirichlet-to-Neumann map in the context of inverse problems?
• RQ3 What conditions ensure the well-posedness of the forward problem for variable-order time-fractional diffusion equations?
• RQ4 Is the sequence of partial Dirichlet-to-Neumann maps sufficient to reconstruct the variable-order coefficient uniquely?
• RQ5 How does the variable-order structure influence the identifiability of the coefficient from boundary data?

Key findings

• The space-dependent variable-order coefficient in the time-fractional diffusion equation is uniquely identifiable from a time-sequence of partial Dirichlet-to-Neumann maps.
• The forward problem for the variable-order time-fractional diffusion equation is well-posed under standard assumptions on the domain and coefficients.
• The uniqueness result holds under the condition that the variable-order coefficient belongs to a suitable class of measurable and bounded functions.
• The analysis relies on the analytic properties of the Dirichlet-to-Neumann map and its dependence on the variable-order parameter.
• The inverse problem is uniquely solvable when the boundary data sequence is sufficiently rich to encode the spatial variation of the order.
• The method does not require full boundary measurements, only partial Dirichlet-to-Neumann data over time.