Salih Tatar

An inverse source problem for a one-dimensional space–time fractional diffusion equation

Fractional(nonlocal) diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogues and they are used to model anomalous diffusion, especially in physics. This paper deals with a nonlocal inverse source problem for a one-dimensional space–time fractional diffusion equation where and . At first we define and analyze the direct problem for the space–time fractional diffusion equation. Later we define the inverse source problem. Furthermore, we set up an operator equation and derive the relation between the solutions of the operator equation and the inverse source problem. We also prove some important properties of the operator . By using these properties and analytic Fredholm theorem, we prove that the inverse source problem is well posed, i.e. can be determined uniquely and depends continuously on additional data .

Simultaneous inversion for the exponents of the fractional time and space derivatives in the space-time fractional diffusion equation

Fractional (nonlocal) diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogues and they are used to model anomalous diffusion, especially in physics. This paper is devoted to a nonlocal inverse problem related to the space-time fractional equation . The existence of the solution for the inverse problem is proved by using quasi-solution method which is based on minimizing an error functional between the output data and the additional data. In this context, an input–output mapping is defined and continuity of the mapping is established. The uniqueness of the solution for the inverse problem is also proved by using eigenfunction expansion of the solution and some basic properties of fractional Laplacian. A numerical method based on discretization of the minimization problem, steepest descent method and least squares approach is proposed for the solution of the inverse problem. The numerical method determines the exponents of the fractional time and space derivatives simultaneously. Numerical examples with noise-free and noisy data illustrate applicability and high accuracy of the proposed method.

An inverse problem for a nonlinear diffusion equation with time-fractional derivative

Abstract A nonlinear time-fractional inverse coefficient problem is considered. The unknown coefficient depends on the solution. It is proved that the direct problem has a unique solution. Afterwards the continuous dependence of the solution of the corresponding direct problem on the coefficient is proved. Then the existence of a quasi-solution of the inverse problem is obtained in the appropriate class of admissible coefficients.

Numerical solution of the nonlinear evolutional inverse problem related to elastoplastic torsional problem

Abstract This paper is devoted to the determination of an unknown function that describes elastoplastic properties of a bar under torsion. The mathematical (evolution) model leads to an inverse problem that consists of determining the unknown coefficient , in the nonlinear parabolic equation , , , using measured output data given in the integral form. Existence of a quasi-solution of the considered inverse problem is obtained in the appropriate class of admissible coefficients. The direct problem is solved using a semi-implicit finite difference scheme. The inverse problem is solved using the semi-analytic inversion method (also known the fast algorithm). Finally, some examples are presented related to direct and inverse problems.

Numerical solution of the nonlinear parabolic problem related to inverse elastoplastic torsional problem

In this article, we study a class of direct and inverse coefficient problems defined for a nonlinear parabolic equation. An existence of a quasi-solution of the considered inverse problem is obtained in the appropriate class of admissible coefficients. Direct problem is solved numerically by using semi-implicit finite difference scheme and then some examples are presented related to steady-state solution of the nonlinear direct problem and ill-posedness of the inverse problem.

A modification of the semi-analytic inversion method: determination of the yield stress and a comparison with the parametrization algorithm

Abstract In this study, an effective modification of the semi-analytic inversion method is presented. The semi-analytic inversion method is developed to solve an inverse coefficient problem arising in materials science instead of the parametrization method as a different and stronger method. The inverse coefficient problem is related to reconstruction of the unknown coefficient , , from the nonlinear equation , . The semi-analytic inversion method has some advantages. The first distinguishable feature of this method is that it uses only a few measured output data to determine the whole unknown curve, whereas the parametrization algorithm uses many measured output data for the determination of only some part of the unknown curve. The second distinguishable feature of this method is its well-posedness. In the semi-analytic inversion method, the algorithm for determination of the yield stress, which is one of the main unknowns of the inverse problem, is very complicated. That is why we need to modify this algorithm. The demonstrated numerical results for different engineering materials also show that the modified semi-analytic inversion method allows us to determine the elastoplastic parameters of a kind of engineering materials with high accuracy, even various noise levels.

Quasi-solution approach for a two dimensional nonlinear inverse diffusion problem

This paper is related to a class of inverse problems for identification of the coefficient in a square porous medium. The unknown coefficient depends on the solution and belongs the class of admissible coefficients. Using continuous dependence of solutions on the coefficient convergence in the corresponding direct problems, the existence of the quasi-solution of the inverse problem is obtained in the appropriate class of admissible coefficients.

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

Fractional (nonlocal) diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogs and they are used to model anomalous diffusion, especially in physics. In this paper, we study a backward problem for an inhomogeneous time-fractional diffusion equation with variable coefficients in a general bounded domain. Such a backward problem is of practically great importance because we often do not know the initial density of substance, but we can observe the density at a positive moment. The backward problem is ill-posed and we propose a regularizing scheme by using Tikhonov regularization method. We also prove the convergence rate for the regularized solution by using an a priori regularization parameter choice rule. Numerical examples illustrate applicability and high accuracy of the proposed method.

Analytical solutions of a class of inverse coefficient problems

a b s t r a c t This paper is devoted to some class of inverse coefficient problems. By using a well- known transformation, the inverse problem is transformed to a new problem without the unknown time dependent coefficient. Therefore, the new inverse problem can be solved easily. To show the efficiency of the present method, some examples are presented.

Recovery of the solute concentration and dispersion flux in an inhomogeneous time fractional diffusion equation

Abstract In this paper, we consider recovery of solute concentration and dispersion flux in an inhomogeneous time fractional diffusion equation. We prove that the considered problem is ill-posed, i.e. the solution does not depend continuously on the data. In order to obtain a regularized solution, we propose a truncation regularization method. The convergence estimates are established under some priori bound assumptions for the exact solution. We present three numerical examples to show efficiency of the method.