Uniqueness Theorem For Inverse Problem Research Articles

Uniqueness Theorems for Inverse Problems of Discontinuous Sturm–Liouville Operator

In this paper, inverse spectral problems of discontinuous Sturm–Liouville problems contained in the discontinuous coefficient and discontinuity conditions at an interior point of the finite interval are, respectively, studied according to

On Inverse Problems for Strongly Degenerate Parabolic Equations under the Integral Observation Condition

Existence and uniqueness theorems for inverse problems of determining the right-hand side and lowest coefficient in a degenerate parabolic equation with two independent variables are proved. It is assumed that the leading coefficient of the equation degenerates at the side boundary of the domain and the order of degeneracy with respect to the variable $$x$$ is not lower than 2. Thus, the Black–Scholes equation, well-known in financial mathematics, is admitted. These results are based on the study of the unique solvability of the corresponding direct problem, which is also of independent interest.

On the determination of differential pencils with nonlocal conditions

Abstract Inverse problems for differential pencils with nonlocal conditions are considered. Uniqueness theorems of inverse problems from the Weyl-type function and spectra are proved, which are generalizations of the well-known Weyl function and Borg’s inverse problem for the classical Sturm–Liouville operators.

Recovering Dirac operator with nonlocal boundary conditions

In this paper inverse problems for Dirac operator with nonlocal conditions are considered. Uniqueness theorems of inverse problems from the Weyl-type function and spectra are provided, which are generalizations of the well-known Weyl function and Borg's inverse problem for the classical Dirac operator.

Recovering differential operators with nonlocal boundary conditions

Inverse spectral problems for Sturm–Liouville operators with nonlocal boundary conditions are studied. As the main spectral characteristics we introduce the so-called Weyl-type function and two spectra, which are generalizations of the well-known Weyl function and Borg’s inverse problem for the classical Sturm–Liouville operator. Two uniqueness theorems of inverse problems from the Weyl-type function and two spectra are presented and proved, respectively.

Discontinuous Sturm-Liouville Problems with Eigenvalue Dependent Boundary Condition

In this study, an inverse problem for Sturm-Liouville differential operators with discontinuities is studied when an eigenparameter appears not only in the differential equation but it also appears in the boundary condition. Uniqueness theorems of inverse problems according to the Prufer angle, the Weyl function and two different eigenvalues sets are proved.

Inverse problems for impulsive Dirac operators with spectral parameters contained in the boundary and multitransfer conditions

An inverse spectral problem is considered for Dirac operators with parameter‐dependent transfer conditions inside the interval, and parameter appears also in one boundary condition. The approach that was used in the investigation of uniqueness theorems of inverse problems for Weyl function or two eigenvalue sets is employed. Copyright © 2014 John Wiley & Sons, Ltd.

INVERSE PROBLEM FOR A CLASS OF DIRAC OPERATOR

In this paper, we consider a problem for the first order canonical Dirac differential equations system with piecewise continuous coefficient and spectral parameter dependent in boundary condition. The asymptotic behavior of eigenvalues, eigenfunctions and normalizing numbers of this system is investigated. The completeness theorem is proved. The spectral expansion formula with respect to eigenvector functions or equivalently Parseval equality is obtained. Weyl solution and Weyl function for the problem are constructed. Uniqueness theorem for inverse problem by the Weyl function and by the spectral data are proved.

Direct and inverse problems for differential systems connected with Dirac systems and related factorization problems

Uniqueness theorems for inverse problems for canonical differential systems of the form y'(t,λ) = iλy(t,λ)H(t)J when H(t) = X(t)NX(t)* for appropriately restricted X(t) and N are established. These results are obtained by showing that the canonical differential systems under consideration can be imbedded into a general framework for which uniqueness theorems were obtained by the authors earlier. Subsequently, uniqueness theorems for a class of systems of the form u'(t, A) = iλu(t, A)NJ + u(t, λ)V(t) that are related to Dirac systems are deduced on the basis of a factorization theorem that is developed in this paper. Finally, some refinements in a number of statements that are based on integral representations of matrix valued functions of the Schur class and the Caratheodory class are discussed briefly. A number of the basic observations in this last part were first noted by M.G. Krein.

Uniqueness theorems for inverse problems of kinematics and integral geometry

Uniqueness theorems for inverse problems of kinematics and integral geometry

Uniqueness theorems for inverse problems of heat potentials

Uniqueness theorems for inverse problems of heat potentials