Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userIdentification of time-varying source term in time-fractional diffusion equations
277
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
This paper is concerned with the inverse problem of determining the time and space dependent source term of diffusion equations with constant-order time-fractional derivative in $(0,2)$. We examine two different cases. In the first one, the source is the product of two spatial and temporal terms, and we prove that both of them can be retrieved by knowledge of one arbitrary internal measurement of the solution for all times. In the second case, we assume that the first term of the product varies with one fixed space variable, while the second one is a function of all the remaining space variables and the time variable, and we show that both terms are uniquely determined by two arbitrary lateral measurements of the solution over the entire time span. These two source identification results boil down to a weak unique continuation principle in the first case and a unique continuation principle for Cauchy data in the second one, that are preliminarily established. Finally, numerical reconstruction of spatial term of source terms in the form of the product of two spatial and temporal terms, is carried out through an iterative algorithm based on the Tikhonov regularization method.
rate research
Read More
In this work, an inverse problem in the fractional diffusion equation with random source is considered. Statistical moments are used of the realizations of single point observation $u(x_0,t,omega).$ We build the representation of the solution $u$ in integral sense, then prove some theoretical results as uniqueness and stability. After that, we establish a numerical algorithm to solve the unknowns, where the mollification method is used.
In this work, an inverse problem in the fractional diffusion equation with random source is considered. The measurements used are the statistical moments of the realizations of single point data $u(x_0,t,omega).$ We build the representation of the solution $u$ in integral sense, then prove that the unknowns can be bounded by the moments theoretically. For the numerical reconstruction, we establish an iterative algorithm with regularized Levenberg-Marquardt type and some numerical results generated from this algorithm are displayed. For the case of highly heterogeneous media, the Generalized Multiscale finite element method (GMsFEM) will be employed.
This paper deals with the investigation of the computational solutions of an unified fractional reaction-diffusion equation, which is obtained from the standard diffusion equation by replacing the time derivative of first order by the generalized fractional time-derivative defined by Hilfer (2000), the space derivative of second order by the Riesz-Feller fractional derivative and adding the function phi(x,t) which is a nonlinear function overning reaction. The solution is derived by the application of the Laplace and Fourier transforms in a compact and closed form in terms of the H-function. The main result obtained in this paper provides an elegant extension of the fundamental solution for the space-time fractional diffusion equation obtained earlier by Mainardi et al. (2001, 2005) and a result very recently given by Tomovski et al. (2011). Computational representation of the fundamental solution is also obtained explicitly. Fractional order moments of the distribution are deduced. At the end, mild extensions of the derived results associated with a finite number of Riesz-Feller space fractional derivatives are also discussed.
Given $(M,g)$, a compact connected Riemannian manifold of dimension $d geq 2$, with boundary $partial M$, 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 solution on a subset of $partial M$ at fixed time. In the flat case where $M$ is a compact subset of $mathbb R^d$, two out the three coefficients $rho$ (weight), $a$ (conductivity) and $q$ (potential) appearing in the equation $rho partial_t^alpha u-textrm{div}(a abla u)+ q u=0$ on $(0,T)times Omega$ are recovered simultaneously.
We investigate diffusion equations with time-fractional derivatives of space-dependent variable order. We examine the well-posedness issue and prove that the space-dependent variable order coefficient is uniquely determined among other coefficients of these equations, by the knowledge of a suitable time-sequence of partial Dirichlet-to-Neumann maps.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
