Application of Sinc-collocation method for solving an inverse problem

In this article, the identification of an unknown time-dependent source term in an inverse problem of parabolic type with nonlocal boundary conditions is considered. The main approach is to change the inverse problem to a system of Volterra integral equations. The resulting integral equations are convolution-type, which by using Sinc-collocation method, are replaced by a system of linear algebraic equations. The convergence analysis is included, and it is shown that the error in the approximate solution is bounded in the infinity norm by the norm of the inverse of the coefficient matrix multiplied by a factor that decays exponentially with the size of the system. To show the efficiency of the present method, an example is presented. The method is easy to implement and yields very accurate results.     

Inverse problem for a quasilinear system of partial differential equations with a nonlocal boundary condition

An initial boundary-value problem for a quasilinear system of partial differential equations with a nonlocal boundary condition involving a delayed argument is considered. The existence of a unique solution to this problem is proved by reducing it to a system of nonlinear integral-functional equations. The inverse problem of finding a solution-dependent coefficient of the system from additional information on a solution component specified at a fixed point of space as a function of time is formulated. The uniqueness of the solution of the inverse problem is proved. The proof is based on the derivation and analysis of an integral-functional equation for the difference between two solutions of the inverse problem.     

An Iterative Algorithm for the Coefficient Inverse Problem of Differential Equations

In this paper, an iterative algorithm for solving a coefficient inverse problem is submitted. The key of the method is to project an unknown coefficient function on a finite dimensional function space. Thus, the inverse problem can be changed into a nonlinear algebraic system of equations.     

The inverse problem for Lagrangian systems with certain non-conservative forces

We discuss two generalizations of the inverse problem of the calculus of variations, one in which a given mechanical system can be brought into the form of Lagrangian equations with non-conservative forces of a generalized Rayleigh dissipation type, the other leading to Lagrangian equations with so-called gyroscopic forces. Our approach focusses primarily on obtaining coordinate-free conditions for the existence of a suitable non-singular multiplier matrix, which will lead to an equivalent representation of a given system of second-order equations as one of these Lagrangian systems with non-conservative forces.     

Integral equations related to the study of an inverse coefficient problem for a system of partial differential equations

We consider an inverse coefficient problem for a linear system of partial differential equations. The values of one solution component on a given curve are used as additional information for determining the unknown coefficient. The proof of the uniqueness of the solution of the inverse problem is based on the analysis of the unique solvability of a homogeneous integral equation of the first kind. The existence of a solution of the inverse problem is proved by reduction to a system of nonlinear integral equations.     

An Imbedding Method for Computing the Generalized Inverses

This paper deals with a system of ordinary differential equations with known conditions associated with a given matrix. By using analytical and computational methods, the generalized inverses of the given matrix can be determined. Among these are the weighted Moore-Penrose inverse, the Moore-Penrose inverse, the Drazin inverse and the group inverse. In particular, a new insight is provided into the finite algorithms for computing the generalized inverse and the inverse.     

A geometric theory for preconditioned inverse iteration applied to a subspace

The aim of this paper is to provide a convergence analysis for a preconditioned subspace iteration, which is designated to determine a modest number of the smallest eigenvalues and its corresponding invariant subspace of eigenvectors of a large, symmetric positive definite matrix. The algorithm is built upon a subspace implementation of preconditioned inverse iteration, i.e., the well-known inverse iteration procedure, where the associated system of linear equations is solved approximately by using a preconditioner. This step is followed by a Rayleigh-Ritz projection so that preconditioned inverse iteration is always applied to the Ritz vectors of the actual subspace of approximate eigenvectors. The given theory provides sharp convergence estimates for the Ritz values and is mainly built on arguments exploiting the geometry underlying preconditioned inverse iteration.     

波动方程系数反问题的求解方法

讨论了一维波动方程系数反演的一种求解方法,将解进行一阶渐进展开,得到相应的反问题,将其转化为第二类Volttera型积分方程组,证明了反问题解的存在唯一性.     

微分连续正则化方法与一维声波方程系数反演问题求解

本文将求解非线性方程组的微分连续法与求解不问题的Ｔｉｋｈｏｎｏｖ正则化方法结合起来，形成了兼有大范围收性与稳定性的“微分连续正则化方法”，给出了方法的收敛性分析，文中最后给出的关于一维波动方程反问题的计算实例表明了方法的有效性。     

Solving the 2D and 3D nonlinear inverse source problems of elliptic type partial differential equations by a homogenization function method

In this article, the inverse source problems of 2D and 3D elliptic type nonlinear partial differential equations are resolved. For this purpose, a family of single-parameter homogenization functions that automatically meet the given boundary conditions are deduced and employed as the bases to expand the solution. We solve a linear algebraic equations system which satisfies the over-specified Neumann boundary condition to obtain the unspecified coefficients, and then the solution in the entire domain is permitted. Taking the solution into the governing equation, the unknown source function can be determined quickly. The present novel method is verified to be an accurate, effective, and robust scheme which is without solving nonlinear equations and iterations, and the additional data used are quite economical.     

结构动力学逆特征值问题的同伦解法

将结构动力学反问题视为拟乘法逆特征值问题,利用求解非线性方程组的同伦方法来解决结构动力学逆特征值问题,这种方法由于沿同伦路径求解,对初值的选取没有本质的要求,算例说明了这种方法是可行的．     

The impedance boundary value problem for the time harmonic Maxwell equations

The existence of a unique solution to Maxwell's equations defined in an exterior domain with impedance boundary condition is established for all frequencies. This is accomplished by reducing this problem to that of solving a system of singular integral equations and then regularizing this system such that the Riesz theory is applicable. We also consider the inverse problem in which it is desired to determine the impedance from a knowledge of the far field pattern. By restricting the impedance to lie a priori in a compact set results are obtained on the existence, uniqueness, and stability of the solution to this inverse scattering problem.     

An inverse problem of identifying the diffusion coefficient in a coupled parabolic‐elliptic system

This work studies a nonlinear inverse problem of reconstructing the diffusion coefficient in a parabolic‐elliptic system using the final measurement data, which has important application in a large field of applied science. Being different from other works, which are governed by single partial differential equations, the underlying mathematical model in this paper is a coupled parabolic‐elliptic system, which makes theoretical analysis rather difficult. On the basis of the optimal control framework, the identification problem is transformed into an optimization problem. Then the existence of the minimizer is proved, and the necessary condition that must be satisfied by the minimizer is also given. Since the optimal control problem is nonconvex, one may not expect a unique solution universally. However, the local uniqueness and stability of the minimizer are deduced in this paper.     

Reconstructing parameters of a medium from incomplete spectral information

The problem of the synthesis of a stratified medium with specified amplitude and phase properties is investigated. The wave propagation in the medium is described by a system of differential equations. The synthesis problem considered in the paper relates to inverse problems of spectral analysis with incomplete spectral information. Using the contour integral method we study properties of spectral characteristics and obtain algorithms for the solution of the synthesis problem for differential equations with singularities.     

Prediction-control problem for some systems of equations of fluid dynamics

By methods of the theory of degenerate semigroups of operators, we investigate inverse problems for some systems of equations of fluid dynamics containing the incompressibility equation ? · v = 0. Namely, we obtain well-posedness criteria for inverse problems for the system of Sobolev equations, which describes the dynamics of small internal motions of a stratified fluid in an equilibrium state under some additional assumptions, and for the system of Oskolkov equations, which, in the linear approximation, describes the dynamics of a viscoelastic incompressible Kelvin-Voigt fluid of order 1.     

Determination of Lagrangians from Equations of Motion and Commutator Brackets

The question as to whether a given set of equations, which govern the dynamical evolution of a system, determine a Lagrangian is considered. This problem, which has come to be known as the inverse problem of the calculus of variations, is reviewed and theorems which contain systems of partial differential equations which determine a type of self-adjointness are developed. It is shown that, given a reasonable form for the classical correspondence, the usual quantum commutator brackets can be expressed in terms of classical quantities which satisfy a particular form of these equations.     

Criteria for the functional controllability and invertibility of non-linear systems

In the development of investigations on inverse problems [1, 2], criteria for the functional controllability and invertibility of non-linear systems of equations with an output are obtained. The solution is based on the construction of an inverse system for which the input action of the initial system is the output. An identification problem is considered which corresponds to the problem of invertibility with an unknown initial state. The properties of λ-invertibility and λ-identifiability, which arise in cases when the output signal is known in a set of trajectories, are investigated.     

Inverse Problem for A System of Integro-Differential Equations for SH Waves in A Visco-Elastic Porous Medium: Global Solvability

We consider a system of hyperbolic integro-differential equations for SH waves in a visco-elastic porous medium. The inverse problem is to recover a kernel (memory) in the integral term of this system. We reduce this problem to solving a system of integral equations for the unknown functions. We apply the principle of contraction mappings to this system in the space of continuous functions with a weight norm. We prove the global unique solvability of the inverse problem and obtain a stability estimate of a solution of the inverse problem.     

Multidimensional Inverse Scattering for First-Order Systems

A method for solving the inverse problem for a class of multidimensional first-order systems is given. The analysis yields equations which the scattering data must satisfy; these equations are natural candidates for characterizing admissible scattering data. The results are used to solve the multidimensional N-wave resonant interaction equations.