一类抛物型偏微分方程反问题的稳定性

本讨论一类抛物型偏微分方程反问题，研究测量值在特定边界上给定时源项确定的稳定性，在合理的假设下证明了该反问题具有按Lipschitz型连续依赖于测量值的稳定性，推广了Yamamoto的结果。     

一类双曲型微分方程反问题

本文对地震勘探中的一类双曲型微分方程反问题，给出了解的整体唯一性定理。     

泛函微分方程解的指数渐近稳定性与有界性

本研究了具有有限时滞中立型泛函微分方程解的有界性问题，得到了方程解的指数渐近稳定性蕴涵有界解的存在性的新的结果。     

双曲型积分微分方程的一个反问题

该文研究了一类双曲型积分微分方程确定未知系数的反问题,利用Ｂａｎａｃｈ压缩映像原理证明了该问题的局部存在唯一性和稳定性。     

一类非线性无穷时滞中立型泛函数分方程系统的一致稳定性

本文讨论一类无穷时滞非线性中立型泛函微分方程解的渐近性态与零解的一致稳定性，得到若干简单的稳定性判据。     

双曲型积分方程的一个反问题

本文研究一类双曲型积分微分方程的一个反问题，即确定方程Ｄ２ｔｕ－Ｄ２ｘｕ＝∫ｔ０ｋ（Ｔ）ｕ（ｘ，ｔ－Ｔ）ｄＴ中ｕ（ｘ，ｔ）和积分核ｋ（ｔ），得到了解的存在性和唯一性．     

数值求解NDDEs系统的单支方法的非线性稳定性

该文探讨了单支方法关于一类中立型延迟微分方程（ＮＤＤＥｓ）系统的整体稳定性和渐近稳定性．在适当的条件下，获得了单支方法关于ＮＤＤＥｓ系统的一些新的非线性稳定性判据．     

马尔科夫切换型中立型随机泛函微分方程

尽管具有马尔科夫切换型随机微分方程的稳定性受到了人们的关注,但是关于具有马尔科夫切换型中立型随机泛函微分方程的稳定性的研究则很少.本文的主要目的是试图研究这一问题,我们证明了解的存在唯—性,并得到了p-阶指数稳定性和几乎处处指数稳定性的判据.     

一类具分布型滞量的中立型微分系统的稳定性

研究了一类具分布型滞量的线性中立型系统,得得到了系统零解的一致稳定性及渐近稳定性的充分条件.     

泛函微分方程安全全局渐近稳定性的一类判别准则

本利用分离变量型V函数，建立了泛函微分方程安全全局渐近稳定性的一类Razumikhin型定理，并对一类变时滞线性微分差分方程给出简明的安全全局渐近稳定性判别准则。     

Asymptotic properties of fractional delay differential equations

In this paper we study the asymptotic properties of d-dimensional linear fractional differential equations with time delay. We present necessary and sufficient conditions for asymptotic stability of equations of this type using the inverse Laplace transform method and prove polynomial decay of stable solutions. Two examples illustrate the obtained analytical results.     

Solvability and stability of the inverse Sturm–Liouville problem with analytical functions in the boundary condition

The paper deals with the Sturm–Liouville eigenvalue problem with the Dirichlet boundary condition at one end of the interval and with the boundary condition containing entire functions of the spectral parameter at the other end. We study the inverse problem, which consists in recovering the potential from a part of the spectrum. This inverse problem generalizes partial inverse problems on finite intervals and on graphs and also the inverse transmission eigenvalue problem. We obtain sufficient conditions for global solvability of the studied inverse problem, which prove its local solvability and stability. In addition, application of our main results to the partial inverse Sturm–Liouville problem on the star-shaped graph is provided.     

Fractional differential equations solved by using Mellin transform

In this paper, the solution of the multi-order differential equations, by using Mellin transform, is proposed. It is shown that the problem related to the shift of the real part of the argument of the transformed function, arising when the Mellin integral operates on the fractional derivatives, may be overcame. Then, the solution may be found for any fractional differential equation involving multi-order fractional derivatives (or integrals). The solution is found in the Mellin domain, by solving a linear set of algebraic equations, whose inverse transform gives the solution of the fractional differential equation at hands.     

Identifiability and Stability of an Inverse Problem Involving a Fredholm Equation

The authors study a linear inverse problem with a biological interpretation, which is modelled by a Fredholm integral equation of the first kind, where the kernel is represented by step functions. Based on different assumptions, identifiability, stability and reconstruction results are obtained.     

On convexity of the functional for inverse problems of hyperbolic equations

In this paper, a strategy to design a functional for inverse problems of hyperbolic equations is proposed. For an inverse source problem, it is shown that the designed functional is globally strictly convex. For an inverse coefficient problem, we can only prove that it is strictly convex near true solution. This strategy can be generalized to other inverse problems, as long as Lipschitz stability is given.     

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…     

Finite element theory on curved domains with applications to discontinuous Galerkin finite element methods

In this paper we provide key estimates used in the stability and error analysis of discontinuous Galerkin finite element methods (DGFEMs) on domains with curved boundaries. In particular, we review trace estimates, inverse estimates, discrete Poincaré–Friedrichs' inequalities, and optimal interpolation estimates in noninteger Hilbert–Sobolev norms, that are well known in the case of polytopal domains. We also prove curvature bounds for curved simplices, which does not seem to be present in the existing literature, even in the polytopal setting, since polytopal domains have piecewise zero curvature. We demonstrate the value of these estimates, by analyzing the IPDG method for the Poisson problem, introduced by Douglas and Dupont, and by analyzing a variant of the hp-DGFEM for the biharmonic problem introduced by Mozolevski and Süli. In both cases we prove stability estimates and optimal a priori error estimates. Numerical results are provided, validating the proven error estimates.     

Stability estimate for a multidimensional inverse spectral problem with partial spectral data

In Bellassoued, Choulli and Yamamoto (2009) [4] we proved a log-log type stability estimate for a multidimensional inverse spectral problem with partial spectral data for a Schrödinger operator, provided that the potential is known in a small neighbourhood of the boundary of the domain. In the present paper we discuss the same inverse problem. We show a log type stability estimate under an additional condition on potentials in terms of their X-ray transform. In proving our result, we follow the same method as in Alessandrini and Sylvester (1990) [1] and Bellassoued, Choulli and Yamamoto (2009) [4]. That is we relate the stability estimate for our inverse spectral problem to a stability estimate for an inverse problem consisting in the determination of the potential in a wave equation from a local Dirichlet to Neumann map (DN map in short).     

On differential equations with interval coefficients

In this study, a new approach is developed to solve the initial value problem for interval linear differential equations. In the considered problem, the coefficients and the initial values are constant intervals. In the developed approach, there is no need to define a derivative for interval-valued functions. All derivatives used in the approach are classical derivatives of real functions. The reason for this is that the solution of the problem is defined as a bunch of real functions. Such a solution concept is compatible also with the robust stability concept. Sufficient conditions are provided for the solution to be expressed analytically. In addition, on a numerical example, the solution obtained by the proposed approach is compared with the solution obtained by the generalized Hukuhara differentiability. It is shown that the proposed approach gives a new type of solution. The main advantage of the proposed approach is that the solution to the considered interval initial value problem exists and is unique, as in the real case.     

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.