Inverse source problem for a distributed-order time fractional diffusion equation

Cheng, Xiaoliang; Yuan, Lele; Liang, Kewei

doi:10.1515/jiip-2019-0006

Cited by 12 publications

(5 citation statements)

References 30 publications

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“…where P Tr = [τ , α]. In the current work, an alternative technique is used for the estimation of unknown parameters in the CGIM based on [20]. The CGIM algorithm can be found in [24] in detail.…”

Section: Conjugate Gradient Inverse Methodsmentioning

confidence: 99%

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Estimating Relaxation Time and Fractionality Order Parameters in Fractional Non-Fourier Heat Conduction Using Conjugate Gradient Inverse Approach in Single and Three-Layer Skin Tissues

et al. 2021

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In this work, the relaxation parameter (τ) and fractionality order (α) in the fractional single phase lag (FSPL) non-Fourier heat conduction model are estimated by employing the conjugate gradient inverse method (CGIM). Two different physics of skin tissue are chosen as the studied cases; single and three-layer skin tissues. Single-layer skin is exposed to laser radiation having the constant heat flux of Qin. However, a heat pulse with constant temperature is imposed on the three-layer skin. The required inputs for the inverse problem in the fractional diffusion equation are chosen from the outcomes of the dual phase lag (DPL) theory. The governing equations are solved numerically by utilizing implicit approaches. The results of this study showed the efficiency of the CGIM to estimate the unknown parameters in the FSPL model. In fact, obtained numerical results of the CGIM are in excellent compatibility with the FSPL model.

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Section: Conjugate Gradient Inverse Methodsmentioning

confidence: 99%

“…They investigated the stability and well pose of the inverse problem. Cheng et al [20] obtained the space-dependent source term in the time-fractional diffusion equation based on Caputo derivative. They used the CGM method to solve the inverse problem.…”

Section: Introductionmentioning

confidence: 99%

Estimating Relaxation Time and Fractionality Order Parameters in Fractional Non-Fourier Heat Conduction Using Conjugate Gradient Inverse Approach in Single and Three-Layer Skin Tissues

et al. 2021

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“…For timefractional inverse problems, see [30,22,2,44,54,52,55,45,19,58,33,57]; for fractional Calderon problems, see [17,46,11,26,15]. Furthermore, if we extend the assumption (∆x) 2 ∝ t α to a more general case (∆x) 2 ∝ F (t), the multi-term fractional diffusion equations and even the distributed-order differential equations will be generated, see [47,31,12] for details.…”

Section: Previous Literaturementioning

confidence: 99%

Reconstruction of the time-dependent source term in a stochastic fractional diffusion equation

Лю

Wen

Zhang³

2019

Preprint

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In this work, an inverse problem in the fractional diffusion equation with random source is considered. Statistical moments are used of the realizations of single point observation u(x 0 , t, ω). We build the representation of the solution u in integral sense, then prove some theoretical results as uniqueness and stability. After that, we establish a numerical algorithm to solve the unknowns, where the mollification method is used.

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“…They adopted an algorithm based on the CG method and an early stopping rule under the discrepancy principle. For other types of inverse problems of DOTFDEs (such as teh estimation of diffusion or potential coefficients, source terms, initial and boundary conditions, and domain geometry), we refer to [35][36][37][38].…”

Section: Introductionmentioning

confidence: 99%

Solving Inverse Problem of Distributed-Order Time-Fractional Diffusion Equations Using Boundary Observations and L2 Regularization

2023

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This article investigates the inverse problem of estimating the weight function using boundary observations in a distributed-order time-fractional diffusion equation. We propose a method based on L2 regularization to convert the inverse problem into a regularized minimization problem, and we solve it using the conjugate gradient algorithm. The minimization functional only needs the weight to have L2 regularity. We prove the weak closedness of the inverse operator, which ensures the existence, stability, and convergence of the regularized solution for the weight in L2(0,1). We propose a weak source condition for the weight in C[0,1] and, based on this, we prove the convergence rate for the regularized solution. In the conjugate gradient algorithm, we derive the gradient of the objective functional through the adjoint technique. The effectiveness of the proposed method and the convergence rate are demonstrated by two numerical examples in two dimensions.

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Inverse source problem for a distributed-order time fractional diffusion equation

Cited by 12 publications

References 30 publications

Estimating Relaxation Time and Fractionality Order Parameters in Fractional Non-Fourier Heat Conduction Using Conjugate Gradient Inverse Approach in Single and Three-Layer Skin Tissues

Estimating Relaxation Time and Fractionality Order Parameters in Fractional Non-Fourier Heat Conduction Using Conjugate Gradient Inverse Approach in Single and Three-Layer Skin Tissues

Reconstruction of the time-dependent source term in a stochastic fractional diffusion equation

Solving Inverse Problem of Distributed-Order Time-Fractional Diffusion Equations Using Boundary Observations and L2 Regularization

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