ON CERTAIN INVERSE PROBLEMS FOR PARABOLIC EQUATIONS WITH FINAL AND INTEGRAL OBSERVATION

Solution of the fourth and fifth boundary value problems for a one-dimensional second-order parabolic equation in a curvilinear region

A coupled nonlinear hyperbolic-parabolic system

Subject of research: problems of numerical determination of point sources in inverse heat and mass transfer problems. Purpose of research: description of theoretical results (theorem of existence and uniqueness of solutions of the inverse problem), creation of an algorithm for solving the problem of numerical determination of point sources, study of its properties, numerical implementation of the algorithm and its testing and verification of stability. Objects of research: problems of numerical determination of point sources (right-hand side of a special type) in inverse problems of heat and mass transfer. The sources are specified as a sum of Dirac delta functions with coefficients that depend on time and characterize the power of the corresponding source. They are unknown and are to be determined together with a solution of the equation. The values of the solution at a certain set of points lying inside the domain serve as the overdetermination data. Methods of research: the algorithm is based on the finite element method for spatial variables and the finite difference method for time. The unknown right-hand side is determined at each time layer using the overdetermination condition. Main results of research: description of the algorithm for solving the problem, its properties, results of numerical experiments. In particular, the conditions are described when the algebraic system to which the problem is reduced has a unique solution. The data obtained as a result of the calculations are compared with test examples. The calculations are also carried out with the addition of random noise of various levels to the measurement data. The results show that the solution is stable under random perturbations of the data.

In the present paper, by means of the successive approximations method, the local or global existence and uniqueness theorems for a stochastic functional differential equation of the Ito type are proved.

A class of stochastic differential equations with highly singular drift fields is considered. Using a purely probabilistic approach, we can show the unattainability of the nodal set. Moreover, a global existence and uniqueness theorem for diffusion processes with singular drift fields is established. The finite action condition of Carlen and Zheng can be modified. We relate our results to the diffusions which describe the time evolution of quantum systems in stochastic mechanics.

Coefficient inverse problems related to identifying the right-hand side of an equation with use of additional information is of interest among inverse problems for partial differential equations. When considering non-stationary problems, tasks of recovering the dependence of the right-hand side on time and spatial variables can be treated as independent. These tasks relate to a class of linear inverse problems, which sufficiently simplifies their study. This work is devoted to finding the dependence of right-hand side of multidimensional parabolic equation on spatial variables using additional observations of the solution at the final point of time – the final overdetermination. More general problems are associated with some integral observation of the solution in time – the integral overdetermination. The first method of numerical solution of inverse problems is based on iterative solution of boundary value problem for time derivative with non-local acceleration. The second method is based on the known approach with iterative refinement of desired dependence of the right-hand side on spatial variables. Capabilities of proposed methods are illustrated by numerical examples for a two-dimensional problem of identifying the right-hand side of a parabolic equation. The standard finite-element approximation in space is used, whilst the time discretization is based on fully implicit two-level schemes.

Part 1 Composite type equations as the original mathematical object: equations of the composite type in mathematics and mathematical simulation canonical types of third order boundary-value problems for third order equations of composite type hyperbolic boundary-value problem uniqueness of regular solutions existence of the regularized solutions of the hyperbolic boundary-value problem the elliptic boundary-value problem uniqueness of regular solutions existence of generalized solutions of the elliptic boundary-value problem existence of regular solutions of the elliptic boundary-value problem correlation of the hyperbolic and elliptic boundary-value problems remark on the third order equations of the variable direction nonlocal problems for equations of composite type. Part 2 Solvability of inverse problems and other applications: inverse problems for equations with partial derivatives linear inverse problems for elliptic and parabolic equations on solvability of nonlinear inverse problems solvability of nonclassical boundary-value problems for equations of second order and third order the mixed problem for second order equations not solved for the time derivative boundary-value problems for the second order equations of the mixed type the problem with oblique derivative for equations of second and third orders a problem of viscous elasticity and the third order equation in noncylindrical domains connected with this problem.

In this paper, the influence of modelling, a priori information, discretization and measurement error to the numerical solution of inverse problems is investigated. Given an a priori approximation of the unknown parameter function in a parabolic problem, we propose a strategy for the regularized determination of a skeleton solution to the inverse problem. This strategy is based on a discretization control of the forward problem in order to find a trade-off between accuracy and computational efficiency. Numerical results with regard to a nonlinear inverse heat conduction problem illustrate the study.

Inverse spectral problems are studied for the non-self-adjoint matrix Sturm–Liouville differential equation on a finite interval. We give formulations of the inverse problems, prove the corresponding uniqueness theorems, and provide a constructive procedure for the solution of the inverse problems by the method of spectral mappings. The obtained results are a natural generalization of the classical results in inverse problem theory for scalar Sturm–Liouville operators.

A mixed problem is considered for a system of partial differential equations modeling the process of adsorption dynamics. An existence and uniqueness theorem is proved for this problem, and the solution properties are investigated. The inverse problem is posed, involving the determination of the system coefficient given additional information about the solution. A uniqueness theorem is proved for the solution of the inverse problem.

<abstract><p>In this paper, an inverse problem of determining the space-dependent volatility from the observed market prices of options with different strikes is studied. Being different from other inverse volatility problem with classical parabolic equations, we apply the linearization method and introduce some variable substitutions to convert the original problem into an inverse source problem in a degenerate parabolic equation in a bounded area, from which an unknown volatility can be recovered and deficiencies caused by artificial truncation can be solved. Based on the optimal control framework, the problem is transformed into an optimization problem and the existence of the minimizer is established. After the necessary conditions are deduced, the uniqueness and stability of the minimizer are proved. Then, the Landweber iterative method is used to obtain a stable numerical solution of the inverse problem and some numerical experiments are also performed. The numerical results show that the algorithm which we proposed is robust and the unknown coefficient is recovered quite well.</p></abstract>

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.

The inverse problem which consists in recovery of a nonlinear source in the equation of hyperbolic type is considered. The solution of the inverse problem is based on the additional boundary condition. The theorem of existence and uniqueness of solution of the inverse problem is proved in small. The convergence of the iterative method for solving the inverse problem is established.

We introduce new supplementary data to the set of the eigenvalues, to determine uniquely potential matrix in the inverse problem for Dirac canonical operator. Besides, we obtain others uniqueness theorems in inverse problems, which are the analogues of well-known Borg, Marchenko and McLaughlin-Rundell theorems in inverse Sturm-Liouville problems.

We consider a variable order differential operator on a graph with a cycle. We study inverse spectral problems for this operator by the system of spectra. Uniqueness theorems are proved, and constructive algorithms are obtained for the solution of the inverse problems.