Weak unique continuation property and a related inverse source problem for time-fractional diffusion-advection equations

Abstract

In this paper, we first establish a weak unique continuation property for timefractional diffusion-advection equations. The proof is mainly based on the Laplace transform and the unique continuation properties for elliptic and parabolic equations. The result is weaker than its parabolic counterpart in the sense that we additionally impose the homogeneous boundary condition. As a direct application, we prove the uniqueness for an inverse problem on determining the spatial component in the source term in by interior measurements. Numerically, we reformulate our inverse source problem as an optimization problem, and propose an iteration thresholding algorithm. Finally, several numerical experiments are presented to show the accuracy and efficiency of the algorithm.

Inverse problems for a perturbed time fractional diffusion equation with final overdetermination

“…Note that m ≤ 0 implies M k ≥ 0. Thus, due to (18), the properties of g and the monotonicity of E (−z), we obtain…”

“…For instance, it is an extension to the fractional case of the parabolic integro‐differential equation u t =ϰ(Δ u + m ∗Δ u )+ F that describes heat processes with memory . Moreover, it is a generalization of an equation with multiple Caputo derivatives , 0< μ j < β <1, that was studied in previous works . Rewriting the latter equation as defining m as a solution of the Volterra equation of the second kind m + k ∗ m =− k and applying the operator , where is the unity operator, to the Equation , we reach with F = z + m ∗ z .…”

“…25 Moreover, it is a generalization of an equation with multiple Caputo derivatives u + ∑ l =1 b u = Δu + z, 0 < j < < 1, that was studied in previous works. 18,26,27 Rewriting the latter equation as…”

“…Problems with final overdetermination for diffusion equations containing single fractional time (and also space) derivatives were studied in papers (reconstruction of source terms) and Sakamoto and Yamamoto (reconstruction of an initial state). An inverse problem to recover a source term in an equation containing multiple Caputo time derivatives by means of local interior measurements was treated in Jiang et al…”

“…6 Problems with final overdetermination for diffusion equations containing single fractional time (and also space) derivatives were studied in papers [11][12][13][14][15][16] (reconstruction of source terms) and Sakamoto and Yamamoto 17 (reconstruction of an initial state). An inverse problem to recover a source term in an equation containing multiple Caputo time derivatives by means of local interior measurements was treated in Jiang et al 18 Recently, the second author 19 introduced a perturbed time fractional diffusion equation that contains an additional convolution term with a kernel m and generalizes diffusion models with multiple time fractional derivatives. The paper 19 studies reconstruction of m and an order of a derivative from measurements over the time.…”