Source Term In Space Research Articles

Simultaneous inversion of a fractional order and a space source term in an anomalous diffusion model

Abstract This paper is devoted to recovering simultaneously the fractional order and the space-dependent source term from partial Cauchy’s boundary data in a multidimensional time-fractional diffusion equation. The uniqueness of the inverse problem is obtained by employing analytic continuation and the Laplace transform. Then a modified non-stationary iterative Tikhonov regularization method with a regularization parameter chosen by a sigmoid-type function is used to find a stable approximate solution for the source term and the fractional order. Numerical examples in one-dimensional and two-dimensional cases are provided to illustrate the efficiency of the proposed algorithm.

Recovering the space source term for the fractional-diffusion equation with Caputo\u2013Fabrizio derivative

This article is devoted to the study of the source function for the Caputo–Fabrizio time fractional diffusion equation. This new definition of the fractional derivative has no singularity. In other words, the new derivative has a smooth kernel. Here, we investigate the existence of the source term. Through an example, we show that this problem is ill-posed (in the sense of Hadamard), and the fractional Landweber method and the modified quasi-boundary value method are used to deal with this inverse problem and the regularized solution is also obtained. The convergence estimates are addressed for the regularized solution to the exact solution by using an a priori regularization parameter choice rule and an a posteriori parameter choice rule. In addition, we give a numerical example to illustrate the proposed method.

Identifying the space source term problem for time-space-fractional diffusion equation

In this paper, we consider an inverse source problem for the time-space-fractional diffusion equation. Here, in the sense of Hadamard, we prove that the problem is severely ill-posed. By applying the quasi-reversibility regularization method, we propose by this method to solve the problem (1.1). After that, we give an error estimate between the sought solution and regularized solution under a prior parameter choice rule and a posterior parameter choice rule, respectively. Finally, we present a numerical example to find that the proposed method works well.

The fractional Landweber method for identifying the space source term problem for time-space fractional diffusion equation

This paper is devoted to solve an inverse problem for identifying the source term of a time-fractional nonhomogeneous diffusion equation with a fractional Laplacian in a non-local boundary. Based on the expression of the solution for the direct problem, the inverse problem for searching the space source term is converted into solving the first kind of Fredholm integral equation. The conditional stability for the inverse source problem is investigated. The fractional Landweber method is used to deal with this inverse problem and the regularized solution is also obtained. Furthermore, the convergence rates for the regularized solution can be proved by using an a priori regularization parameter choice rule and an a posteriori parameter choice rule. Several numerical examples are given to show the proposed method is efficient and stable.

Identifying the space source term problem for a generalization of the fractional diffusion equation with hyper-Bessel operator

In this paper, we consider an inverse problem of identifying the source term for a generalization of the time-fractional diffusion equation, where regularized hyper-Bessel operator is used instead of the time derivative. First, we investigate the existence of our source term; the conditional stability for the inverse source problem is also investigated. Then, we show that the backward problem is ill-posed; the fractional Landweber method and the fractional Tikhonov method are used to deal with this inverse problem, and the regularized solution is also obtained. We present convergence rates for the regularized solution to the exact solution by using an a priori regularization parameter choice rule and an a posteriori parameter choice rule. Finally, we present a numerical example to illustrate the proposed method.

Identifying a fractional order and a space source term in a time-fractional diffusion-wave equation simultaneously

This paper is devoted to identify simultaneously a fractional order and a space source term in a multi-dimensional time-fractional diffusion-wave equation by the final time measurement data. We prove the uniqueness and stability of the inverse problem. Then we use the Levenberg–Marquardt method to recover the fractional order and the space source term simultaneously, and propose a finite dimensional approximation algorithm based on the linear finite element basis to find a regularized solution. Numerical examples in one-dimensional and two-dimensional cases are provided to illustrate the efficiency of the proposed algorithm.

Radionuclide transport in shallow groundwater

Radioactive waste management is a strategic activity that is adopted in the nuclear field to safeguard the operating staff, the population and the environment from radiological risks. Studies that support the environmental assessment of a nuclear site offer insight into the understanding of the transport of radionuclides in environmental matrices (e.g. soil, groundwater, surface water, etc.). These studies, that involve the migration of a radiological source in the environment, are conducted to identify how to approach the safety assessment of current or future nuclear facilities. This work analyzes the source term as the key point in the modeling of the transport of radionuclides in groundwater and soil. A methodological approach, which focuses on the dynamic of the source term in space and time, was applied to model the transport of radionuclides. This approach can be used to plan reliable environmental monitoring networks. The analysis was performed at the nuclear site of Saluggia, Vercelli (Italy).