Problem Of Time-fractional Diffusion Equation Research Articles

Iteration regularization method for a sideways problem of time-fractional diffusion equation

Iteration regularization method for a sideways problem of time-fractional diffusion equation

On a backward problem for inhomogeneous time-fractional diffusion equations

In this paper, we consider a final value problem for time-fractional diffusion equation with inhomogeneous source. The main goal of our paper is to determine an approximated initial data from the observation data at final time by constructing a regularized solution using a mollification method. Under appropriate regularity assumptions of the exact solution, we give convergence rate between the reconstructed solution and the exact one. We also provide a numerical example to illustrate the main results.

The spectral meshless radial point interpolation method for solving an inverse source problem of the time-fractional diffusion equation

In this paper, the spectral meshless radial point interpolation (SMRPI) technique is applied to the inverse source problem of time-fractional diffusion equation in two dimensions. The missing solely time-dependent source is recovered from an additional integral measurement. The applied approach is based on a combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct the shape functions which act as basis functions in the frame of SMRPI. Firstly, we use a difference scheme for the fractional derivative to discretize the governing equation, and we obtain a finite difference scheme with respect to time. Then we use the SMRPI approach to approximate the spatial derivatives. Also, it is proved that the scheme is unconditionally stable with respect to time in space H1. Consequently, when the input data is contaminated with noise, we use the Tikhonov regularization method in order to obtain a stable solution. Numerical results show that the solution is accurate for exact data and stable for noisy data.

An iterative method for an inverse source problem of time-fractional diffusion equation

We propose an iteration method to recover a space-dependent source for a time-fractional diffusion equation from the final measurement. Based on the conditional stability of the inverse problem, we prove the convergence of the iterative regularization method under the a priori parameter choice rule and the a posteriori parameter choice rule, respectively. Numerical examples in one dimension and two dimension are given to validate the effectiveness of the presented method.

Generalized Tikhonov method for the final value problem of time-fractional diffusion equation

ABSTRACTThis paper investigates the final value problem for a time-fractional diffusion equation. We determine the initial data from a noisy final data. A generalized Tikhonov regularization method is proposed to deal with this ill-posed problem, and then the convergence rates is derived by the a-priori and a-posteriori choice rules of regularization parameter. A numerical example is presented to show that this method is stable and feasible.

Numerical computation for backward time-fractional diffusion equation

Based on kernel-based approximation technique, we devise in this paper an efficient and accurate numerical scheme for solving a backward problem of time-fractional diffusion equation (BTFDE). The kernels used in the approximation are the fundamental solutions of the time-fractional diffusion equation which can be expressed in terms of the M-Wright functions. To stably and accurately solve the resultant highly ill-conditioned system of equations, we successfully combine the standard Tikhonov regularization technique and the L-curve method to obtain an optimal choice of the regularization parameter and the location of source points. Several 1D and 2D numerical examples are constructed to demonstrate the superior accuracy and efficiency of the proposed method for solving both the classical backward heat conduction problem (BHCP) and the BTFDE.

Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation

Abstract In this paper, some initial-boundary-value problems for the time-fractional diffusion equation are first considered in open bounded n-dimensional domains. In particular, the maximum principle well-known for the PDEs of elliptic and parabolic types is extended for the time-fractional diffusion equation. In its turn, the maximum principle is used to show the uniqueness of solution to the initial-boundary-value problems for the time-fractional diffusion equation. The generalized solution in the sense of Vladimirov is then constructed in form of a Fourier series with respect to the eigenfunctions of a certain Sturm-Liouville eigenvalue problem. For the onedimensional time-fractional diffusion equation $$(D_t^\alpha u)(t) = \frac{\partial } {{\partial x}}\left( {p(x)\frac{{\partial u}} {{\partial x}}} \right) - q(x)u + F(x,t), x \in (0,l), t \in (0,T)$$ the generalized solution to the initial-boundary-value problem with Dirichlet boundary conditions is shown to be a solution in the classical sense. Properties of this solution are investigated including its smoothness and asymptotics for some special cases of the source function.