Inverse Problem For Parabolic Equation Research Articles

An inverse problem for pseudoparabolic equation: existence, uniqueness, stability, and numerical analysis

In this paper, we study an inverse problem for a linear third-order pseudoparabolic equation. The investigated inverse problem consists of determining a space-dependent coefficient of the right-hand side of a pseudoparabolic equation. As an additional information a final overdetermination condition is considered. Under the suitable conditions on the data of the problem, the unique solvability of the considered inverse problem is established. The stability of solutions is also proved. The established results are also true for inverse problems for parabolic equations, which could be obtained as a regularization of the studied pseudoparabolic equation. In addition, the pseudoparabolic problem is discretized using the cubic B-spline functions and recast as a nonlinear least-squares minimization of the Tikhonov regularization function. Numerically, this is effectively solved using the MATLAB subroutine lsqnonlin. Both exact and noisy data are inverted. Numerical results for three benchmark test examples are presented and discussed. Moreover, the von Neumann stability analysis is also discussed.

On an Iterative Method of Solving Direct and Inverse Problems for Parabolic Equations

On an Iterative Method of Solving Direct and Inverse Problems for Parabolic Equations

Nonlinear Inverse Problems for Parabolic Equations with Time–Dependent Coefficients. Reduction to Nonlocal Problems with Samarskii–Ionkin Type Conditions

Nonlinear Inverse Problems for Parabolic Equations with Time–Dependent Coefficients. Reduction to Nonlocal Problems with Samarskii–Ionkin Type Conditions

On the asymptotic behavior of inverse problems for parabolic equation

On the asymptotic behavior of inverse problems for parabolic equation

A new algorithm to determine the creation or depletion term of parabolic equations from boundary measurements

We propose a robust numerical method to find the coefficient of the creation or depletion term of parabolic equations from the measurement of the lateral Cauchy information of their solutions. Most papers in the field study this nonlinear and severely ill-posed problem using optimal control. The main drawback of this widely used approach is the need of some advanced knowledge of the true solution. In this paper, we propose a new method that opens a door to solve nonlinear inverse problems for parabolic equations without any initial guess of the true coefficient. This claim is confirmed numerically. The key point of the method is to derive a system of nonlinear elliptic equations for the Fourier coefficients of the solution to the governing equation with respect to a special basis of L2. We then solve this system by a predictor–corrector process, in which our computation to obtain the first and second predictors is effective. The desired solution to the inverse problem under consideration follows.

Recovery of non-smooth radiative coefficient from nonlocal observation by diffusion system

Abstract The heat conduction process in composite medium can be modeled by a parabolic equation with discontinuous radiative coefficient. To detect the composite medium characterized by such a non-smooth coefficient from measurable information about the heat distribution, we consider a nonlinear inverse problem for parabolic equation, with the average measurement of temperature field in some time interval as the inversion input. We firstly establish the uniqueness for this nonlinear inverse problem, based on the property of the direct problem and the known uniqueness result for linear inverse source problem. To solve the inverse problem from a nonlinear operator equation, the differentiability and the tangential condition of this nonlinear map is analyzed. An iterative process called two-point gradient method is proposed by minimizing data-fit term and the penalty term alternatively, with rigorous convergence analysis in terms of the tangential condition. Numerical simulations are presented to illustrate the effectiveness of the proposed method.

About one approach to solving the inverse problem for parabolic equation

About one approach to solving the inverse problem for parabolic equation

About one approach to solving the inverse problem for parabolic equation

About one approach to solving the inverse problem for parabolic equation

Ritz–Galerkin method for solving an inverse problem of parabolic equation with moving boundaries and integral condition

ABSTRACTIn this paper, we consider an inverse problem for parabolic equation with moving boundaries and integral condition. Existence and uniqueness are proved using tool from double-layer heat potential and Volterra integral equations of the second kind. For computation part, the Ritz–Galerkin method is first implemented to get an approximate solution of intermediate function H(x, t), and then based on relationships between H(x, t) and , respectively, numerical solutions of and u(x, t) are obtained. Finally, numerical examples are presented to show the validity and applicability of our method.

Regularized Lagrange Principle and Pontryagin Maximum Principle in Optimal Control and Inverse Problems

Abstract Regularized Lagrange principles in non-iterative and iterative forms for the “simplest” convex programming problem in a Hilbert space with operator constraint-equality are formulated. On their basis regularized Lagrange principle and Pontryagin maximum principle in iterative forms with stopping rules are derived in optimal control problem and in inverse problem for parabolic equation.

Inverse Problems for Parabolic Equation with Discontinuous Coefficients

Abstract We consider the reaction-diffusion equation with discontinuities in the diffusion coefficient and the potential term. We start by deriving the Carleman estimate for the discontinuous reaction-diffusion operator which is deployed in the inverse problems of finding the stability result of the two discontinuous coefficients from the internal observations of the given parabolic equation.

Numerical Solution to a Class of Inverse Problems for Parabolic Equation

A class of inverse problems for parabolic equation is considered. In particular, boundary value problems with nonlocal conditions are reduced to such class of problems. The proposed numerical approach is based on the method of lines to reduce the problem to a system of ordinary differential equations. To solve this system, an analog of the transfer method for boundary conditions is applied. The results of numerical experiments are given.

Computational experiment on the numerical solution of some inverse problems of mathematical physics

In this article the computational experiment on the numerical solution of the most popular linear inverse problems for equations of mathematical physics are presented. The discretization of retrospective inverse problem for parabolic equation is performed using difference scheme with non-positive weight multiplier. Similar difference scheme is also used for the numerical solution of Cauchy problem for two-dimensional Laplace equation. The results of computational experiment, performed on model problems with exact solution, including ones with randomly perturbed input data are presented and discussed.

On some Inverse Problem for a Parabolic Equation with a Parameter

Today inverse problems of mathematical physics play an important role in science and applications [1]. Coefficient inverse problems for parabolic equations are problems of finding solutions of differential equation with one (or more) unknown coefficients. An inverse problem for a parabolic equation with a parameter is investigated. The parameter has the same dimension as the spatial variable. Inverse problems with unknown parameter arise in various problems: in studying boundaryvalue problems for mixed-type equations and equation systems [2, 3]; in solving various inverse problems [4–7]; in studying boundary-value problems for equation systems with small parameters [8, 9].

Inverse problems for parabolic equations with interior degeneracy and Neumann boundary conditions

Abstract We consider a parabolic problem with degeneracy in the interior of the spatial domain and we focus on the well-posedness of the problem and on inverse source problems. The novelties of the present paper are two. First, the degeneracy point is in the interior of the spatial domain. Second, we consider Neumann boundary conditions so that no previous result can be adapted to this situation.

Analytical Methods of the Theory of Inverse Problems for Parabolic Equations

We obtain new presentations for solutions and coefficients of some equations of mathematical physics which are used for studying multidimensional inverse problems. Bibliography: 13 titles.

An approach to numerical solution of some inverse problems for parabolic equations

The inverse problems for a parabolic equation with an unknown source (space or time dependent) on the right-hand side are considered. The numerical method is based on the method of lines. The results of numerical experiments on test problems are given.

An inverse problem for a parabolic equation in a free-boundary domain degenerating at the initial time moment

We consider an inverse problem for a one-dimensional parabolic equation with unknown time-dependent major coefficient in a domain whose unknown boundary weakly degenerates at the initial time moment. The conditions for existence and uniqueness of the classical solution of the problem are established.

On some classes of inverse problems for parabolic equations

Abstract We study solvability of inverse problems of finding the right-hand side together with a solution itself for vector-valued parabolic equations. The usual boundary conditions are supplemented with the overdetermination conditions which are the values of a solution on some system of surfaces.

Uniqueness classes in inverse problems for parabolic equations with several unknown coefficients

Uniqueness classes in inverse problems for parabolic equations with several unknown coefficients