Two regularization methods for identifying the source term of Caputo-Hadamard time-fractional diffusion equation

Article Mathematics, Applied

Mathematical analysis and numerical methods for Caputo-Hadamard fractional diffusion-wave equations

Caixia Ou et al.

Summary: This paper investigates the mathematical analysis and numerical methods for Caputo-Hadamard fractional diffusion-wave equations with initial singularity. The analytical solution is obtained by adopting the modified Laplace transform and the finite Fourier sine transform, and the regularity and logarithmic decay of the solution are researched. The numerical methods for the problems under the singularity hypothesis are then studied.

APPLIED NUMERICAL MATHEMATICS (2022)

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Article Mathematics, Applied

The fractional Landweber method for identifying the space source term problem for time-space fractional diffusion equation

Fan Yang et al.

Summary: This paper deals with the inverse problem of identifying the source term of a time-fractional nonhomogeneous diffusion equation with a fractional Laplacian in a non-local boundary. By converting the problem into solving the first kind of Fredholm integral equation and using the fractional Landweber method, a regularized solution is obtained with proven convergence rates. The proposed method is shown to be efficient and stable through several numerical examples.

NUMERICAL ALGORITHMS (2021)

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Article Engineering, Multidisciplinary

Three Landweber iterative methods for solving the initial value problem of time-fractional diffusion-wave equation on spherically symmetric domain

Fan Yang et al.

Summary: This paper investigates the inverse problem of identifying the initial value of time-fractional diffusion wave equation on a spherically symmetric region, obtaining the exact solution through variable separation and Mittag-Leffler functions and using three different Landweber iterative methods for solving. The effectiveness of the regularization methods is demonstrated through several numerical examples.

INVERSE PROBLEMS IN SCIENCE AND ENGINEERING (2021)

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Article Mathematics, Interdisciplinary Applications

A fractional Tikhonov regularization method for an inverse backward and source problems in the time-space fractional diffusion equations

Smina Djennadi et al.

Summary: This research focuses on two types of inverse problems for diffusion equations with Caputo fractional derivatives in time and fractional Sturm-Liouville operator for space. The study proves these inverse problems to be ill-posed in the sense of Hadamard when an additional condition at the final time is given. A new fractional Tikhonov regularization method is used for stable solution reconstruction with error estimates obtained between exact and regularized solutions through a-priori and a-posteriori parameter choice rules, along with numerical examples provided for validation.

CHAOS SOLITONS & FRACTALS (2021)

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Article Mathematics, Applied