An inverse source problem for a two terms time-fractional diffusion equation

In this paper, we consider an inverse problem for a linear heat equation involving two time-fractional derivatives, subject to a nonlocal boundary condition. We determine a source term independent of the space variable, and the temperature distribution with an over- determining function of integral type.

Conforming and nonconforming finite element methods for biharmonic inverse source problem

This paper deals with the numerical approximation of the biharmonic inverse source problem in an abstract setting in which the measurement data is finite-dimensional. This unified framework in particular covers the conforming and nonconforming finite element methods (FEMs). The inverse problem is analysed through the forward problem. Error estimate for the forward solution is derived in an abstract set-up that applies to conforming and Morley nonconforming FEMs. Since the inverse problem is ill-posed, Tikhonov regularisation is considered to obtain a stable approximate solution. Error estimate is established for the regularised solution for different regularisation schemes. Numerical results that confirm the theoretical results are also presented.

A noniterative reconstruction method for solving a time-fractional inverse source problem from partial boundary measurements

In this paper, a noniterative method for solving an inverse source problem governed by the two-dimensional time-fractional diffusion equation is proposed. The basic idea consists in reconstructing the geometrical support of the unknown source from partial boundary measurements of the associated potential. A Kohn-Vogelius type shape functional is considered together with a regularization term penalizing the relative perimeter of the unknown set of anomalies. Identifiability result is derived and uniqueness of a minimizer is ensured. The shape functional measuring the misfit between the solutions of two auxiliary problems containing information about the boundary measurements is minimized with respect to a finite number of ball-shaped trial anomalies by using the topological derivative method. In particular, the second-order topological gradient is exploited to devise an efficient and fast noniterative reconstruction algorithm. Finally, some numerical experiments are presented, showing different features of the proposed approach in reconstructing multiple anomalies of varying shapes and sizes by taking noisy data into account .

Uniqueness for Inverse Source Problems of Determining a Space-Dependent Source in Time-Fractional Equations with Non-Smooth Solutions

In this contribution, we investigate an inverse source problem for a fractional diffusion and wave equation with the Caputo fractional derivative of the space-dependent variable order. More specifically, we discuss the uniqueness of a solution when reconstructing a space-dependent source from a time-averaged measurement, or a final in time measurement. Weakly singular solutions are included in the class of admissible solutions. The obtained results are also valid if the order of the fractional derivative is constant.

Numerical Experimental for an Inverse Source Problem Method

This paper examines extensions of an iterative method for inverse evaluation of the source function for two elliptic systems. The method begins with a starting value for the undetermined source. Next, a background field and equations for the error field are obtained. 2-D domains are considered. This method is suitable for Helmholtz and Poisson operators. In the presence of finite-difference grid resolution, a varying amount of boundary data, and methods of filtering the noise in the boundary data and the noise intensity of the boundary data, the performance, accuracy, and iteration count of the algorithm are investigated. Keywords: Source, Inverse Problems, Poisson, Noise, Ill-Posedness, Well-Posed