One-dimensional Time-fractional Diffusion Equation Research Articles

On the Inverse Problem of Time-Dependent Coefficient in a Time Fractional Diffusion Problem by Newly Defined Monic Laquerre Wavelets

Abstract The primary aim of this research is to establish the time-dependent diffusion coefficient in a one-dimensional time fractional diffusion equation in Caputo sense by means of newly defined Monic Laquerre wavelets (MLW) and collocation points. We first give the definition of MLW by taking Monic Laquerre's polynomials into account. Later, time fractional diffusion problem is reduced into a system of ordinary fractional and algebraic equations by utilizing MLW. The residual power series method (RPSM) and the overdetermined data are applied to this system to determine the solution and the unknown time-dependent coefficient together in series form. In the end, illustrative examples are presented to show the stability and accuracy of the proposed wavelet method for the inverse problem of determining unknown time-dependent coefficient in fractional diffusion problems. The reliability of the proposed algorithm for the inverse problems is supported by high degree of accuracy in the given examples.

Uniqueness of the potential in a time-fractional diffusion equation

Abstract This article concerns the uniqueness of an inverse coefficient problem of identifying a spatially varying potential in a one-dimensional time-fractional diffusion equation. The input sources are given by a complete system in L 2 ⁢ ( 0 , 1 ) {L^{2}(0,1)} , and measurements are observed at the end point of the spatial interval. Firstly, we provide the positive lower bound of the Green function for the differential operator with different boundary conditions. Then, based on the positive lower bound estimation of the Green function, the relationship between the Green function, the solution of the forward problem, and the potential, such measurements uniquely determine the potential on the entire interval under different boundary conditions.

Uniqueness for an inverse coefficient problem for a one-dimensional time-fractional diffusion equation with non-zero boundary conditions

We consider initial boundary value problems for one-dimensional diffusion equation with time-fractional derivative of order which are subject to non-zero Neumann boundary conditions. We prove the uniqueness for an inverse coefficient problem of determining a spatially varying potential and the order of the time-fractional derivative by Dirichlet data at one end point of the spatial interval. The imposed Neumann conditions are required to be within the correct Sobolev space of order α. Our proof is based on a representation formula of solution to an initial boundary value problem with non-zero boundary data. Moreover, we apply such a formula and prove the uniqueness in the determination of boundary value at another end point by Cauchy data at one end point.

Simultaneous uniqueness for an inverse problem in a time-fractional diffusion equation

This article investigates the uniqueness of identifying the fractional-order, potential, and Robin coefficient simultaneously in one-dimensional time-fractional diffusion equation with non-homogeneous boundary condition. By using one boundary measurement, we prove that the fractional-order, potential on the entire interval, and Robin coefficient are determined simultaneously from asymptotic properties of the Mittag-Leffler function and the Marchenko’s uniqueness theorem.

Recovering the Initial Distribution for a Time-Fractional Diffusion Equation

We consider the inverse problem of reconstructing the initial condition of a one-dimensional time-fractional diffusion equation from measurements collected at a single interior location over a finite time-interval. The method relies on the eigenfunction expansion of the forward solution in conjunction with a Tikhonov regularization scheme to control the instability inherent in the problem. We show that the inverse problem has a unique solution provided exact data is given, and prove stability results regarding the regularized solution. Numerical realization of the method and illustrations using a finite-element discretization are given at the end of this paper.

Inverse source problem for a time-fractional diffusion equation with nonlocal boundary conditions

In this paper, an inverse problem of determining a time-dependent source term in a one-dimensional time-fractional diffusion equation from the energy measurement is studied. This problem is obtained from a classical diffusion problem by replacing the time derivative with a fractional derivative. The well-posedness of the inverse problem is shown by using eigenfunction expansion of a non-self adjoint spectral problem along the generalized Fourier method.

Compact Crank–Nicolson and Du Fort–Frankel Method for the Solution of the Time Fractional Diffusion Equation

In this paper, two compact implicit finite difference methods are developed and analyzed for solving the one-dimensional time fractional diffusion equation. The temporal derivative is approximated by using Grünwald–Letnikov formula. Compact finite difference approximation is used for the second-order derivative in space. The local truncation errors are discussed. The stability analysis and the convergence of the proposed methods are investigated by means of Fourier series method. A comparison between the results of these methods and the exact solution is made. Numerical tests are given to verify the feasibility and accuracy of the methods.

Robin coefficient identification for a time-fractional diffusion equation

This paper is devoted to a Robin coefficient identification problem for one-dimensional time-fractional diffusion equation. Based on the separation of variables, we transform the identification problem into a nonlinear Volterra integral equation of the first kind with the Robin coefficient and the Dirichlet data on a part of boundary as unknown functions. Then, we use a boundary element method to discrete the first kind integral equation and obtain a nonlinear system of algebraic equations. The conjugate gradient method is applied to solve a regularized optimization problem and finally, we obtain a suitable regularized approximation to the Robin coefficient. Three numerical examples are provided to show the effectiveness and robustness of the proposed method.

Stable numerical solution to a Cauchy problem for a time fractional diffusion equation

Abstract In this paper, we consider a Cauchy problem of one-dimensional time fractional diffusion equation for determining the Cauchy data at x =1 from the Cauchy data at x =0. Based on the separation of variables and Duhamel's principle, we transform the Cauchy problem into a first kind Volterra integral equation with the Neumann data as an unknown function and then show the ill-posedness of problem. Further, we use a boundary element method combined with a generalized Tikhonov regularization to solve the first kind integral equation. The generalized cross validation choice rule is applied to find a suitable regularization parameter. Three numerical examples are provided to show the effectiveness and robustness of the proposed method.

An inverse problem for a one-dimensional time-fractional diffusion problem

We study an inverse problem of recovering a spatially varying potential term in a one-dimensional time-fractional diffusion equation from the flux measurements taken at a single fixed time corresponding to a given set of input sources. The unique identifiability of the potential is shown for two cases, i.e. the flux at one end and the net flux, provided that the set of input sources forms a complete basis in L2(0, 1). An algorithm of the quasi-Newton type is proposed for the efficient and accurate reconstruction of the coefficient from finite data, and the injectivity of the Jacobian is discussed. Numerical results for both exact and noisy data are presented.

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.