Uniqueness for an inverse space-dependent source term in a multi-dimensional time-fractional diffusion equation

This paper is devoted to identify a space-dependent source term in a multi-dimensional time-fractional diffusion equation from boundary measured data. The uniqueness for the inverse source problem is proved by the Laplace transformation method.     

On the uniqueness and reconstruction for an inverse problem of the fractional diffusion process

Consider an inverse problem for the time-fractional diffusion equation in one dimensional spatial space. The aim is to determine the initial status and heat flux on the boundary simultaneously from heat measurement data given on the other boundary. Using the Laplace transform and the unique extension technique, the uniqueness for this inverse problem is proven. Then we construct a regularizing scheme for the reconstruction of boundary flux for known initial status. The convergence rate of the regularizing solution is established under some a priori information about the exact solution. Moreover, the initial distribution can also be recovered approximately from our regularizing scheme. Finally we present some numerical examples, which show the validity of the proposed reconstruction scheme.     

Uniqueness for an inverse source problem of determining a space dependent source in a time-fractional diffusion equation

We study uniqueness of a solution for an inverse source problem arising in linear time-fractional diffusion equations with time dependent coefficients. New uniqueness results are formulated in Theorem 3.1. We also show optimality of the conditions under which uniqueness holds by explicitly constructing counterexamples, that is by constructing more than one solution in the case when the conditions for uniqueness are violated.     

A modified method for a non-standard inverse heat conduction problem

A non-standard inverse heat conduction problem is considered. Data are given along the line x = 1 and the solution at x = 0 is sought. The problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. In order to solve the problem numerically it is necessary to employ some regularization method. In this paper, we study a modification of the equation, where a fourth-order mixed derivative term is added. Error estimates for this equation are given, which show that the solution of the modified equation is an approximation of the heat equation. A numerical implementation is considered and a simple example is given. Some numerical results show the usefulness of the modified method.     

Identification of a time-dependent source term for a time fractional diffusion problem

In this paper, we study an inverse problem of identifying a time-dependent term of an unknown source for a time fractional diffusion equation using nonlocal measurement data. Firstly, we establish the conditional stability for this inverse problem. Then two regularization methods are proposed to for reconstructing the time-dependent source term from noisy measurements. The first method is an integral equation method which formulates the inverse source problem into an integral equation of the second kind; and a prior convergence rate of regularized solutions is derived with a suitable choice strategy of regularization parameters. The second method is a standard Tikhonov regularization method and formulates the inverse source problem as a minimizing problem of the Tikhonov functional. Based on the superposition principle and the technique of finite-element interpolation, a numerical scheme is proposed to implement the second regularization method. One- and two-dimensional examples are carried out to verify efficiency and stability of the second regularization method.     

Wavelet regularization for an inverse heat conduction problem

In this paper we consider an inverse heat conduction problem which appears in some applied subjects. This problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. The Meyer wavelets are applied to formulate a regularized solution which is convergent to exact one on an acceptable interval when data error tends to zero.     

Two regularization methods for a Cauchy problem for the Laplace equation

A Cauchy problem for the Laplace equation in a rectangle is considered. Cauchy data are given for y=0, and boundary data are for x=0 and x=π. The solution for 0<y?1 is sought. We propose two different regularization methods on the ill-posed problem based on separation of variables. Both methods are applied to formulate regularized solutions which are stably convergent to the exact one with explicit error estimates.     

Tikhonov regularization method for a backward problem for the time-fractional diffusion equation

This paper is devoted to solve a backward problem for a time-fractional diffusion equation with variable coefficients in a general bounded domain by the Tikhonov regularization method. Based on the eigenfunction expansion of the solution, the backward problem for searching the initial data is changed to solve a Fredholm integral equation of the first kind. The conditional stability for the backward problem is obtained. We use the Tikhonov regularization method to deal with the integral equation and obtain the series expression of solution. Furthermore, the convergence rates for the Tikhonov regularized solution can be proved by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Two numerical examples in one-dimensional and two-dimensional cases respectively are investigated. Numerical results show that the proposed method is effective and stable.     

Solutions for a fractional diffusion equation with noninteger dimensions

We investigate a fractional diffusion equation with a nonlocal reaction term by using the Green function approach. We also consider a modified spatial operator in order to cover situations characterized by a noninteger dimension. The results show a nonusual spreading of the initial condition which can be connected to a rich class of anomalous diffusive processes.     

An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions

We consider the inverse source problem for a time fractional diffusion equation. The unknown source term is independent of the time variable, and the problem is considered in two dimensions. A biorthogonal system of functions consisting of two Riesz bases of the space L²[(0,1) × (0,1)], obtained from eigenfunctions and associated functions of the spectral problem and its adjoint problem, is used to represent the solution of the inverse problem. Using the properties of the biorthogonal system of functions, we show the existence and uniqueness of the solution of the inverse problem and its continuous dependence on the data. Copyright © 2012 John Wiley & Sons, Ltd.     

Global uniqueness in an inverse problem for time fractional diffusion equations

Given $(M,g)$, a compact connected Riemannian manifold of dimension $d?2$, with boundary ?M, we consider an initial boundary value problem for a fractional diffusion equation on $(0,T)\times M$, $T>0$, with time-fractional Caputo derivative of order $\alpha \in (0,1)\cup (1,2)$. We prove uniqueness in the inverse problem of determining the smooth manifold $(M,g)$ (up to an isometry), and various time-independent smooth coefficients appearing in this equation, from measurements of the solutions on a subset of ?M at fixed time. In the “flat” case where M is a compact subset of ${\mathbb{R}}^d$, two out the three coefficients ρ (density), a (conductivity) and q (potential) appearing in the equation $\rho {?}_t^{\alpha }u?\mathrm{div}\left(a\mathrm{?}u\right)+qu=0$ on $(0,T)\times M$ are recovered simultaneously.     

A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation

In this paper, we consider an inverse source problem for a time-fractional diffusion equation with variable coefficients in a general bounded domain. That is to determine a space-dependent source term in the time-fractional diffusion equation from a noisy final data. Based on a series expression of the solution, we can transform the original inverse problem into a first kind integral equation. The uniqueness and a conditional stability for the space-dependent source term can be obtained. Further, we propose a modified quasi-boundary value regularization method to deal with the inverse source problem and obtain two kinds of convergence rates by using an a priori and an a posteriori regularization parameter choice rule, respectively. Numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed method.     

On a quasi-reversibility regularization method for a Cauchy problem of the Helmholtz equation

In this paper, we consider the Cauchy problem for the Helmholtz equation in a rectangle, where the Cauchy data is given for y=0 and boundary data are for x=0 and x=π. The solution is sought in the interval 0<y≤1. A quasi-reversibility method is applied to formulate regularized solutions which are stably convergent to the exact one with explicit error estimates.     

Regularization of a terminal value problem for time fractional diffusion equation

In this article, we study an inverse problem with inhomogeneous source to determine an initial data from the time fractional diffusion equation. In general, this problem is ill-posed in the sense of Hadamard, so the quasi-boundary value method is proposed to solve the problem. In the theoretical results, we propose a priori and a posteriori parameter choice rules and analyze them. Finally, two numerical results in the one-dimensional and two-dimensional case show the evidence of the used regularization method.     

On a backward problem for fractional diffusion equation with Riemann-Liouville derivative

In the present paper, we study the initial inverse problem (backward problem) for a two-dimensional fractional differential equation with Riemann-Liouville derivative. Our model is considered in the random noise of the given data. We show that our problem is not well-posed in the sense of Hadamard. A truncated method is used to construct an approximate function for the solution (called the regularized solution). Furthermore, the error estimate of the regularized solution in L² and H^τ norms is considered and illustrated by numerical example.     

An inverse problem for space‐fractional backward diffusion problem

In this paper, an inverse problem for space‐fractional backward diffusion equation, which is highly ill‐posed, is considered. 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 present a simplified Tikhonov regularization method to deal with this problem. Convergence estimate is presented under a priori choice of regularization parameter. Numerical experiments are given to illustrate the accuracy and efficiency of the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.     

Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems

We consider initial value/boundary value problems for fractional diffusion-wave equation: , where 0<α?2, where L is a symmetric uniformly elliptic operator with t-independent smooth coefficients. First we establish the unique existence of the weak solution and the asymptotic behavior as the time t goes to ∞ and the proofs are based on the eigenfunction expansions. Second for α∈(0,1), we apply the eigenfunction expansions and prove (i) stability in the backward problem in time, (ii) the uniqueness in determining an initial value and (iii) the uniqueness of solution by the decay rate as t→∞, (iv) stability in an inverse source problem of determining t-dependent factor in the source by observation at one point over (0,T).     

A regularization method for a Cauchy problem of the Helmholtz equation

We investigate a Cauchy problem for the Helmholtz equation. A modified boundary method is used for solving this ill-posed problem. Some Hölder-type error estimates are obtained. The numerical experiment shows that the modified boundary method works well.