Inverse source problem of variable-exponent time-fractional diffusion equations
Abstract
This paper investigates an inverse temporal source problem for a variable-exponent time-fractional diffusion equation with time-dependent diffusion coefficients. By the perturbation method, the model is equivalently transformed into a constant-order fractional diffusion equation with a convolution perturbation term. Existence, uniqueness, and energy estimates of strong and weak solutions to the forward problem are established, and for a separable source term the uniqueness and Lipschitz stability of the temporal source function are proved using local integral observations. A numerical reconstruction algorithm combining Tikhonov regularization with the conjugate gradient method is developed with a convergence error estimate, and numerical experiments verify its effectiveness and stability for different variable-exponent functions, noise levels and source functions. Since fractional advection--diffusion models describe the turbulent transport of airborne pollutants, the method is also applicable in atmospheric science for recovering the emission history of a pollution source from sparse measurements.
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BibTeX
@misc{hu2026inverse,
title = {{Inverse source problem of variable-exponent time-fractional diffusion equations}},
author = {Li Hu and Zhiyuan Li and Zhidong Zhang},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01019},
eprint = {2611.01019},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01019},
}
RIS
TY - UNPB
TI - Inverse source problem of variable-exponent time-fractional diffusion equations
AU - Li Hu
AU - Zhiyuan Li
AU - Zhidong Zhang
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01019
UR - https://arxiv.org/abs/2611.01019
AB - This paper investigates an inverse temporal source problem for a variable-exponent time-fractional diffusion equation with time-dependent diffusion coefficients. By the perturbation method, the model is equivalently transformed into a constant-order fractional diffusion equation with a convolution perturbation term. Existence, uniqueness, and energy estimates of strong and weak solutions to the forward problem are established, and for a separable source term the uniqueness and Lipschitz stability of the temporal source function are proved using local integral observations. A numerical reconstruction algorithm combining Tikhonov regularization with the conjugate gradient method is developed with a convergence error estimate, and numerical experiments verify its effectiveness and stability for different variable-exponent functions, noise levels and source functions. Since fractional advection--diffusion models describe the turbulent transport of airborne pollutants, the method is also applicable in atmospheric science for recovering the emission history of a pollution source from sparse measurements.
ER -