Three regularization methods for identifying the initial value of homogeneous anomalous secondary diffusion equation

Abstract

In this paper, the inverse problem of initial value identification for homogeneous anomalous diffusion equation with Riemann-Liouville fractional derivative in time is studied. We prove that this kind of problem is ill-posed. We analyze the optimal error bound of the problem under the source condition and apply the quasi-boundary regularization method, fractional Landweber iterative regularization method, and Landweber iterative regularization method to solve this inverse problem. Based on the results of conditional stability, the error estimates between the exact solution and the regular solution are given under the priori and posteriori regularization parameter selection rules. Finally, three examples are given to illustrate the effectiveness and feasibility of these methods.

Regularization methods for the inverse initial value problem for the time-fractional diffusion equation with robin boundary condition ★

“…But, the corresponding inverse problems, despite its importance, has not been fully investigated. In the last few years, the investigation on the inverse problems of the time-fractional diffusion equation(TFDE) has been further developed, for example, Liu [5] considered a backward problem for a TFDE in one-dimensional case, Sun [6] analyzed the backward problem for the multi-term TFDE by using fractional-order quasi-reversibility method, Ozbilge [7] identified the unknown coefficients of the TFDE by the Fourier method, Shayegan [8] presented the numerical method to fix a backward TFDE, Sun [9] identified a space-dependent source function in a multiterm TFDE with nonhomogeneous boundary condition by the Levenberg-Marquardt regularization method, Han [10] gave a fractional Landweber regularization method to solve the backward problem of the TFDE, and Yang [11] studied the inverse problem of the initial value identification of homogeneous anomalous diffusion equations with Riemann-Liouville fractional derivatives using three regularization methods. From the above work, we notice that there are a lot of one dimensional research, but high dimensional research is less.…”

“…For more details we can refer to [16]. At present, many regularization methods have been proposed to solve ill-posed problems [11,[17][18][19][20]. Among the many regularization methods, the TSVD regularzation method [17] which can filter out the small singular value and retain the large singular value and the modified Landweber iterative method [21] which can reduce the oversmoothing property of the classical Landweber method attract our attention.…”