Reconstruction and stable recovery of source terms and coefficients appearing in diffusion equations

Kian, Yavar; Yamamoto, Masahiro

doi:10.1088/1361-6420/ab2d42

Cited by 40 publications

(47 citation statements)

References 35 publications

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“…We remark that the well-posdness for problem (1.4) with non-homogenous boundary conditions is meaningful also for other mathematical problems such as optimal control problems (see e.g., [26]) or inverse problems (see e.g., [3,7,11,12,15]).…”

Section: Motivations and A Short Bibliographical Reviewmentioning

confidence: 90%

Well-Posedness for Weak and Strong Solutions of Non-Homogeneous Initial Boundary Value Problems for Fractional Diffusion Equations

Kian

Yamamoto

2021

Fract Calc Appl Anal

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We study the well-posedness for initial boundary value problems associated with time fractional diffusion equations with non-homogenous boundary and initial values. We consider both weak and strong solutions for the problems. For weak solutions, we introduce a definition of solutions which allows to prove the existence of solution to the initial boundary value problems with non-zero initial and boundary values and non-homogeneous source terms lying in some negative-order Sobolev spaces. For strong solutions, we introduce an optimal compatibility condition and prove the existence of the solutions. We introduce also some sharp conditions guaranteeing the existence of solutions with more regularity in time and space.

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Section: Motivations and A Short Bibliographical Reviewmentioning

confidence: 90%

Well-Posedness for Weak and Strong Solutions of Non-Homogeneous Initial Boundary Value Problems for Fractional Diffusion Equations

Kian

Yamamoto

2021

Fract Calc Appl Anal

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“…At time t j + 1 , j = 1, 2, … , N, the difference Equations (13), (8), and (14) can be written as a (M + 1) × (M + 1) linear system of equations:…”

Section: Forward Problemmentioning

confidence: 99%

“…Huntul et al 7 determined an additive space and timewise coefficients in the heat equation. Kian and Yamamoto 8 investigated the inverse problem for recovering the time‐ and space‐dependent sources for classical diffusion equation.…”

Section: Introductionmentioning

confidence: 99%

Recovery of timewise‐dependent heat source for hyperbolic PDE from an integral condition

Huntul

Tamsir

2020

Math Methods in App Sciences

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The inverse problem of recovering the timewise-dependent heat source coefficient along with the temperature in a second-order hyperbolic equation with mixed derivative and with initial and Neumann boundary conditions and integral measurement is, for the first time, numerically investigated. The inverse problem considered in this paper has a unique solution. However, it is an ill-posed problem by being sensitive to noise. The one-dimensional inverse problem is discretized using the FDM and recast as a nonlinear least-squares minimization of Tikhonov regularization function. Numerically, this is effectively solved using the MATLAB subroutine lsqnonlin. The present numerical results demonstrate that accurate and stable approximate solutions have been attained.

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“…Inverse problems for subdiffusion are of relative recent nature, initiated by the pioneering work [4] for recovering the diffusion coefficient from lateral Cauchy data (in the one-dimensional case) using Sturm-Liouville theory; see the work [19] for an overview. The inverse potential problem for the model (1.1) has also been analyzed in several works [18,20,21,30,42]. Miller and Yamamoto [30] proved the unique recovery from data on a space-time subdomain, using an integral transformation.…”

Section: Introductionmentioning

confidence: 99%

An inverse potential problem for subdiffusion: stability and reconstruction*

Jin

2020

Inverse Problems

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In this work, we study the inverse problem of recovering a potential coefficient in the subdiffusion model, which involves a Djrbashian-Caputo derivative of order α ∈ (0, 1) in time, from the terminal data. We prove that the inverse problem is locally Lipschitz for small terminal time, under certain conditions on the initial data. This result extends the result in [6] for the standard parabolic case to the fractional case. The analysis relies on refined properties of two-parameter Mittag-Leffler functions, e.g., complete monotonicity and asymptotics. Further, we develop an efficient and easy-toimplement algorithm for numerically recovering the coefficient based on (preconditioned) fixed point iteration and Anderson acceleration. The efficiency and accuracy of the algorithm is illustrated with several numerical examples.

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Reconstruction and stable recovery of source terms and coefficients appearing in diffusion equations

Cited by 40 publications

References 35 publications

Well-Posedness for Weak and Strong Solutions of Non-Homogeneous Initial Boundary Value Problems for Fractional Diffusion Equations

Well-Posedness for Weak and Strong Solutions of Non-Homogeneous Initial Boundary Value Problems for Fractional Diffusion Equations

Recovery of timewise‐dependent heat source for hyperbolic PDE from an integral condition

An inverse potential problem for subdiffusion: stability and reconstruction*

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