We are pleased to introduce some papers of ICST's Laboratory of Environmental Science which recently accepted to journals.
In this paper, a backward diffusion problem for a space-fractional diffusion equation (SFDE) with nonlinear source in a strip is investigated.
This problem is obtained from the classical diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order α ∈ (0,2]. We show that such a problem is severely ill-posed and further propose a new modified regularization method to solve it based on the solution given by the Fourier method. Convergence estimates are presented under a priori bound assumptions for the exact solution. Ourmethod improve some results of a previous paper, including the earlier paper [G.H. Zheng, T. Wei, Two regularization methods for solving a Riesz-Feller space-fractional backward diffusion problem, Inverse Problems 26 (2010), no. 11, 115017] and some other papers. A general case of nonlinear terms for this problem is also considered.
.
On a Riesz–Feller space fractional backward diffusion problem with a nonlinear source
.
In this paper, a backward diffusion problem for a space-fractional diffusion equation with a nonlinear source in a strip is investigated. This problem is obtained from the classical diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order
α ∈ (0
,2]
. A nonlinear problem is severely ill-posed, therefore we propose two new modified regularization solutions to solve it. We further show that the approximated problems are well-posed and their solutions converge if the original problem has a classical solution. In addition, the convergence estimates are presented under a priori bounded assumption of the exact solution. For estimating the error of the proposed method, a numerical example has been implemented.
.
Regularized solution of an inverse source problem for a time fractional diffusion equation
.
In this paper, we study on an inverse problem to determine an unknown source term in a time fractional diffusion equation, whereby the
data are obtained at the later time. In general, this problem is illposed, therefore the Tikhonov regularization method is proposed to solve the problem. In the theoretical results, a priori error estimate between the exact solution and its regularized solutions is obtained. We also propose two methods, a priori and a posteriori parameter choice rules, to estimate 2 the convergence rate of the regularized methods. In addition, the proposed regularized methods have been verified by numerical experiments to estimate the errors between the regularized solutions and exact solutions.
Eventually, from the numerical results it shows that the posteriori parameter choice rule method converges to the exact solution faster than the priori parameter choice rule method.
.
(For full version, please click on title of each paper)
