Abstract
This paper is devoted to identify a space-dependent source function in a multiterm time-fractional diffusion equation with nonhomogeneous boundary condition from a part of noisy boundary data. The well-posedness of a weak solution for the corresponding direct problem is proved by the variational method. We firstly investigate the uniqueness of an inverse initial problem by the analytic continuation technique and the Laplace transformation. Then, the uniqueness of the inverse source problem is derived by employing the fractional Duhamel principle. The inverse problem is solved by the Levenberg-Marquardt regularization method, and an approximate source function is found. Numerical examples are provided to show the effectiveness of the proposed method in one- and two-dimensional cases.
Highlights
It is well known that the standard diffusion equation has been used to describe the Gaussian process of particle motion
It is not hard to find that the solutions of initial boundary value problem (IBVP) (50) and (51) are independent of the source term f, and the inverse source problem becomes more precise, i.e., can we determine f ðxÞ from uf ðx, tÞ for ðx, tÞ ∈ Γ × I uniquely
We investigate an inverse space-dependent source problem of a multiterm time-fractional diffusion equation with nonhomogeneous boundary condition in a general domain
Summary
It is well known that the standard diffusion equation has been used to describe the Gaussian process of particle motion. We investigate an inverse space-dependent source problem in a multiterm time-fractional diffusion equation with nonhomogeneous boundary condition. Zhang et al [11] proved a uniqueness result to inverse the space-dependent source term in one-dimensional case by using one-point Cauchy data and provided an efficient numerical method. Yan et al [13] studied an inverse spatialdependent source problem by noisy boundary data in a time-fractional diffusion-wave equation and carried out the numerical inversion experiments by a nonstationary iterative Tikhonov regularization method. Jiang et al [14] established a weak unique continuation property for time-fractional diffusion-advection equations and studied an inverse problem on determining the spatial component in the source term by interior measurements.
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