A Stability Estimate for a Solution to a Three-Dimensional Inverse Problem for the Maxwell Equations

We consider the problem of determining dielectric permittivity and conductivity in the Maxwell equations. As additional information we prescribe the traces of the tangential components of the electromagnetic field on the lateral surface of a cylindric domain. We establish a stability estimate for a solution to the inverse problem and a uniqueness theorem.     

Uniqueness and local stability for the inverse scattering problem of determining the cavity

Considering a time-harmonic electromagnetic plane wave incident on an arbitrarily shaped open cavity embedded in infinite ground plane, the physical process is modelled by Maxwell's equations. We investigate the inverse problem of determining the shape of the open cavity from the information of the measured scattered field. Results on the uniqueness and the local stability of the inverse problem in the 2-dimensional TM (transverse magnetic) polarization are proved in this paper.     

WELL-POSEDNESS OF A PARABOLIC INVERSE PROBLEM

1.IntroductionItiswellknownthatinverseproblemsinpartialdifferentialequations,mostofwhichhavenotyetbeensolveduptonow,remainasachallengeinappliedmathematics.Therefore,manymathematiciansstudiedvariousinverseproblemsforparabolicequa-tions.FOrasimplesurveywereferto[1,2,8,9]foridentifyingcoefficients,[7]foridentifyingboundaryvalues,[4,10]foridentifyingsourcetermsofparabolicequations.Wehavenotincludedalotofpapersconcerningthecomputationalmethodsusedforsolvinginverseparabolicproblems.Inthispaperthein…     

On the impulsive implicit Ψ-Hilfer fractional differential equations with delay

In this paper, we investigate the existence and uniqueness of solutions and derive the Ulam-Hyers-Mittag-Leffler stability results for impulsive implicit Ψ-Hilfer fractional differential equations with time delay. It is demonstrated that the Ulam-Hyers and generalized Ulam-Hyers stability are the specific cases of Ulam-Hyers-Mittag-Leffler stability. Extended version of the Gronwall inequality, abstract Gronwall lemma, and Picard operator theory are the primary devices in our investigation. We provide an example to illustrate the obtained results.     

一类对偶积分方程组正则化为Cauchy奇异积分方程组解法

本文基于Mellin变换法求解复杂更一般形式的对偶积分方程组．通过积分变换，由实数域化成复数域上的方程组，引入未知函数的积分变换，移动积分路径，应用Cauchy积分定理，实现退耦正则化为Cauchy奇异积分方程组，由此给出一般性解，并严格证明了对偶积分方程组退耦正则化为Cauchy奇异积分方程组与原对偶积分方程组等价性，以及对偶积分方程组解的存在性和唯一性．给出的解法和理论解，作为求解复杂对偶积分方程组一种有效解法，可供求解复杂的数学、物理、力学中的混合边值问题应用．     

Problem of determining the nonstationary potential in a hyperbolic-type equation

We solve the problem of determining the hyperbolic equation coefficient depending on two variables. Some additional information is given by the trace of the direct problem solution on the hyperplane x = 0. We estimate the stability of the solution of the inverse problem under study and prove the uniqueness theorem. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 2, pp. 220–225, August, 2008.     

A Stability Estimate for a Solution to the Problem of Determination of Two Coefficients of a Hyperbolic Equation

We consider the problem of determination of two coefficients (x) and q(x) in a hyperbolic equation. Here (x) is the coefficient of the first derivative with respect to t and q(x) is the coefficient of the solution itself. We suppose that these coefficients are small in some norm and supported in a disk D. Oscillations are excited by the impulse function (t)(x·) supported on the straight line t=0, x·=0. Here is a unit vector playing the role of a parameter of the problem. The acoustic field generated by this source lying outside D is measured at the points of the boundary of D together with the normal derivative on some time interval of a fixed length T for two different values of the parameter . We prove that, for a sufficiently large T, the given information determines the sought coefficients uniquely. We obtain a stability estimate for a solution to the problem.     

Regularized Solutions of a Nonlinear Convolution Equation on the Euclidean Group

In this paper we apply Fourier analysis on the two and three dimensional Euclidean motion groups to the solution of a nonlinear convolution equation. First, we review the theory of the irreducible unitary representations of the motion group and discuss the corresponding Fourier transform of functions on the motion group. The main reasons why exact solutions of this convolution equation do not exist in many cases are discussed. Techniques for regularization of the problem and numerical methods for finding approximate solutions are presented. Examples are considered and approximate solutions are found.     

Analysis of nonlinear integral equations with Erdélyi-Kober fractional operator

This paper initiates the investigation of nonlinear integral equations with Erdélyi-Kober fractional operator. Existence and uniqueness results of solutions in a closed ball are obtained by using a directly computational method and Schauder fixed point theorem via a weakly singular integral inequality due to Ma and Pec?ari? [20]. Meanwhile, three certain solutions sets Y_K,σ, Y_1,λ and Y_1,1, which tending to zero at an appropriate rate t^−ν, 0 < ν = σ (or λ or 1) as t → +∞, are constructed and local stability results of solutions are obtained based on these sets respectively under some suitable conditions. Two examples are given to illustrate the results.     

Unique solvability of a linear problem with perturbed periodic boundary values

We investigate the problem with perturbed periodic boundary values