An inverse coefficient problem with nonlinear parabolic equation

Two main aims of this paper are to develop a numerical method to solve an inverse source problem for parabolic equations and apply it to solve a nonlinear coefficient inverse problem. The inverse source problem in this paper is the problem to reconstruct a source term from external observations. Our method to solve this inverse source problem consists of two stages. We first establish an equation of the derivative of the solution to the parabolic equation with respect to the time variable. Then, in the second stage, we solve this equation by the quasi-reversibility method. The inverse source problem considered in this paper is the linearization of a nonlinear coefficient inverse problem. Hence, iteratively solving the inverse source problem provides the numerical solution to that coefficient inverse problem. Numerical results for the inverse source problem under consideration and the corresponding nonlinear coefficient inverse problem are presented.

Coefficient inverse problems are reformulated to a unified integral differential equation. The presented method of conversion of the considered inverse problems to a unified Volterra integral-differential equation gives an opportunity to distribute the acquired results also to analogous inverse problems for non-linear parabolic equations of different types.

In this paper, modified variational iteration algorithm-II is investigated for finding approximate solutions of nonlinear Parabolic equations. Comparisons of the MVIA-II with trigonometric B-spline collocation method, variational iteration method, homotopy perturbation transform method, Adomian decomposition method, and modified variational iteration method are carried out, which show that the proposed algorithm (MVIA-II) is robust one. Some nonlinear Parabolic equations are given to demonstrate the implementation and accuracy of the MVIA-II.

An inverse coefficient problem is considered for a stationary nonlinear convection- diffusion-reaction equation, in which reaction coefficient has a rather common dependence on substance concentration and on spacial variable. The solvability of the considered nonlinear boundary value problem is proved in a general case. The existence of solutions of the inverse problem is proved for the reaction coefficients, which are defined by the product of two functions. The first function depends on a spatial variable, the second one depends nonlinearly on the solution of the boundary value problem. The mentioned inverse problem consists in reconstructing the first function with the help of additional information provided by the solution of the boundary value problem.

In this paper, our primary focus is on studying the properties of self-similar solutions to the Cauchy problem. We specifically examine the behavior of these solutions in a double nonlinear time-dependent parabolic equation and their absorption in a non-homogeneous medium. Through the research of the topic matter, our aim is to deliver a more thorough comprehension of the finite speed perturbations propagation in the solution of the Cauchy problem for a nonlinear parabolic equation. This establishment of a property is essential to understand the dynamic nature of these equations. Furthermore, we delve into the self-similar analysis of the solution, which allows us to ascertain the condition of Fujita type global solvability for the Cauchy problem in a double nonlinear degenerate-type parabolic equation within a non-homogeneous medium. This analysis provides valuable insights into the behavior and potential solvability of these equations on a global mean. Additionally, we establish estimates for weak solutions depending on the growing density and the value of numerical parameters. By establishing these estimates, we provide a more comprehensive understanding of the behavior of the solutions in different scenarios.

Necessary conditions of optimality for the time optimal problem for nonlinear monotone parabolic equations are presented. For some problems governed by a parabolic variational inequality, these conditions are applied to a penalty problem associated with the inequality. Optimality conditions for the time optimal problem for the variational inequality are derived by passing to the limit in the penalizing term.

Identification of time-dependent convection coefficient in a time-fractional diffusion equation

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.

The Hardy inequality and nonlinear parabolic equations on Carnot groups

: Coefficient Inverse Problems (CIPs) for Partial Differential Equations (PDEs) represent a very important tool for such needs of the Army as imaging of unknown targets hidden in cluttered heterogeneous backgrounds. The goal of this project is the development of globally convergent numerical methods for a wide class of CIPs. These methods are tested on mathematical models of the interest to the Army such as imaging of antipersonnel land mines and targets on battlefields covered by smogs and flames. In our definition global entails: (1) a rigorous convergence analysis that does not depend on the quality of the initial guess, and (2) numerical simulations that confirm the advertised convergence property. A conventional way to solve a CIP is via the minimization of a least squares objective functional. This functional characterizes misfit between the data and the solution of that PDE with a for the unknown coefficient. However, it is well known to researchers working on computations of inverse problems that the phenomenon of multiple local minima of these functionals represents the major obstacle for the development of reliable numerical methods for multidimensional CIPs. This phenomenon in turn is caused by the above mentioned non-linearity and ill-posedness. Because of local minima, one should somehow guess in advance about a good approximation for the solution. Without the availability of a first good guess, however, there is no guarantee that the calculated coefficient is indeed close to the correct one. In our terminology these are locally convergent numerical methods. In other words, their convergence to the correct solution can be guaranteed only if the starting point is located in a small neighborhood of this solution. Because of local minima, conventional numerical methods for multidimensional CIPs are locally convergent ones. However, in many important applications the first good guess is unavailable.

Multi-parameters identification problem for a degenerate parabolic equation

An effective technique for inverse coefficient problem solving of the shear modulus reconstruction for both inhomogeneous elastic and viscoelastic cylindrical waveguide is proposed. The identification is carried out using the displacement field data measured on the finite part of the outer boundary of the cylindrical waveguide in the torsional vibrations mode. An iterative scheme for solving the inverse problem is developed. At each iteration step, the correction to the shear modulus distribution function is calculated using the Fredholm integral equation of the first kind with a smooth kernel. Numerical results are obtained for both direct and inverse problems for different laws of the shear modulus variation.

In this paper, the weighted average-based differential quadrature method is presented for solving one-dimensional homogeneous first-order non-linear parabolic partial differential equation. First, the given solution domain is discretized by using uniform discretization grid point. Next, by using Taylor series expansion we obtain central finite difference discretization of the partial differential equation involving with temporal variable associated with weighted average of partial derivative concerning spatial variable. From this, we obtain the system of nonlinear ordinary differential equations and it is linearized by using the quasilinearization method. Then by using the polynomial-based differential quadrature method for approximating derivative involving with spatial variable at specified grid point, we obtain the system of linear equation. Then they obtained linear system equation is solved by using the LU matrix decomposition method. To validate the applicability of the proposed method, two model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supported the theoretical and mathematical statements and the accuracy of the solution is obtained. The accuracy of the present method has been shown in the sense of root mean square error norm and maximum absolute error norm and the local behavior of the solution is captured exactly. Numerical versus exact solutions and behavior of maximum absolute error between them have been presented in terms of graphs and the corresponding root mean square error norm and maximum absolute error norm presented in tables. The present method approximates the exact solution very well and it is quite efficient and practically well suited for solving the non-linear parabolic equation. The numerical result presented in tables and graphs indicates that the approximate solution is in good agreement with the exact solution.

We study the Cauchy problem and the initial boundary value problem (IBVP) for nonlinear parabolic equations in $\\mathcal{C}_b([0,T);L^p)$ and $L^q(0,T;L^p)$.\nWe give a unified method to construct local mild solutions of the Cauchy problem or IBVP for a class of nonlinear parabolic equations in $\\mathcal{C}_b([0,T);L^p)$ or $L^q(0,T;L^p)$ by introducing admissible triplet, generalized admissible triplet and establishing time space estimates for the solutions to the linear parabolic equations.\nMoreover, using our method, we also obtain the existence of global small solutions to the nonlinear parabolic equations.

The paper presents the coefficient inverse problem of heat and mass transfer. It is known that the process of moisture and heat transfer in soil is described by a system of partial differential equations. The inverse coefficient problem is solved for finding the thermophysical characteristics of the heat and mass transfer equations. The initial-boundary conditions for the heat and mass transfer system are set. To solve the problem, the measured values of temperature and moisture are additionally set at the accessible boundary of the considered area. To solve the inverse problem, the Dufort- Frankel scheme and Matlab Optimization Toolbox are used. The Jacobian is used to reduce the number of iterations. The results of numerical calculations with and without Jacobian are presented.