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Two methods of solving the inverse heat conduction problem with employment of the discrete Fournier transform are presented in this article. The first one operates similarly to the SVD algorithm and consists in reducing the number of components of the discrete Fournier transform which are taken into account to determine the solution to the inverse problem. The second method is related to the regularization of the solution to the inverse problem in the discrete Fournier transform domain. Those methods were illustrated by numerical examples. In the first example, an influence of the boundary conditions disturbance by a random error on the solution to the inverse problem (its stability) was examined. In the second example, the temperature distribution on the inner boundary of the multiply connected domain was determined. Results of calculations made in both ways brought very good outcomes and confirm the usefulness of applying the discrete Fournier transform to solving inverse problems. [ABSTRACT FROM AUTHOR]

In this paper, a concentration identification problem of the time fractional inhomogeneous diffusion equation (TFIDE) is investigated. The TFIDE is obtained from the classical diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative of order. Our purpose is to recover the solute concentration from source measurement and boundary data. We show that this problem is severely ill-posed and further apply a convolution-type regularization method to solve it. Convergence estimates are given under a priori bound assumptions for the exact solution. Finally, we provide two numerical examples to confirm our theoretical analysis and show that the corresponding numerical method works effectively. [ABSTRACT FROM PUBLISHER]

This paper considers the boundary-value inverse heat conduction problem with steady boundary. To solve this problem, different approaches based on the Laplace and Fourier transforms are proposed. Application of the Laplace transform makes it possible to obtain an operator equation describing the explicit dependence of the desired boundary-value function on the initial data at the other boundary. This approach to solving the problem is used for the first time. The method based on the direct and inverse Fourier transforms with respect to the time variable provides stable solutions. The estimation of error of these solutions is the best with respect to the order. The range of application of each of the proposed methods and the stability of the solutions obtained by these methods was evaluated by a computational experiment. [ABSTRACT FROM AUTHOR]

In this article, we consider a classical ill-posed problem which is the backward problem for parabolic equation with time-dependent coefficient [image omitted] Applying the method of Fourier transform and the modified quasi-boundary value method, we shall give a regularization for the problem. A sharp error estimate between the regularized solution and the exact solution is given in this article. A numerical experiment is provided illustrating the main results. [ABSTRACT FROM AUTHOR]

In this article we consider the numerical problem of shape reconstruction of an unknown characteristic source inside a domain. We consider a steady-state conductivity problem modelled by the Poisson equation, where the heat source is the non-homogeneous characteristic function. A well-known result, back to 1938, by Novikov (Novikov, Sur le probleme inverse du potentiel, Dokl. Akad. Nauk 18 (1938), pp. 165-168) says that star-shaped sources can be reconstructed uniquely from the Cauchy boundary data (see also the work of Isakov e.g. (Isakov, Inverse problems for partial diferential equations, Applied Mathematical Sciences, Vol. 127, Springer-Verlag, New York, 1998). Here we consider the reciprocity functional that maps harmonic functions to their integral in the unknown characteristic support. We connect the uniqueness result with the recovery of a function from a certain knowledge of the Fourier coefficients, by taking harmonic monomials as test functions. We also establish a numerical method that consists in an algebraic non-linear system of equations, leading to an approximation of the radial function that defines the boundary of the unknown source. Simulations showing the performance of the numerical method are presented. [ABSTRACT FROM AUTHOR]