Integral equation methods for scattering by infinite rough surfaces

In this paper, we consider the Dirichlet and impedance boundary value problems for the Helmholtz equation in a non‐locally perturbed half‐plane. These boundary value problems arise in a study of time‐harmonic acoustic scattering of an incident field by a sound‐soft, infinite rough surface where the total field vanishes (the Dirichlet problem) or by an infinite, impedance rough surface where the total field satisfies a homogeneous impedance condition (the impedance problem). We propose a new boundary integral equation formulation for the Dirichlet problem, utilizing a combined double‐ and single‐layer potential and a Dirichlet half‐plane Green's function. For the impedance problem we propose two boundary integral equation formulations, both using a half‐plane impedance Green's function, the first derived from Green's representation theorem, and the second arising from seeking the solution as a single‐layer potential. We show that all the integral equations proposed are uniquely solvable in the space of bounded and continuous functions for all wavenumbers. As an important corollary we prove that, for a variety of incident fields including an incident plane wave, the impedance boundary value problem for the scattered field has a unique solution under certain constraints on the boundary impedance. Copyright © 2003 John Wiley & Sons, Ltd.     

The impedance boundary-value problem of diffraction by a strip

We consider the impedance boundary-value problem for the Helmholtz equation originated by the problem of wave diffraction by an infinite strip with imperfect conductivity. The two possible different situations of real and complex wave numbers are considered. Bessel potential spaces are used to deal with the problem, and the identification of corresponding operators of single and double layer potentials allow a reformulation of the problem into a system of integral equations. The well-posedness of the problem is obtained for a set of impedance parameters (and wave numbers), after the incorporation of some compatibility conditions on the data. At the end, an improvement of the regularity of the solution is derived for the same set of parameters previously considered.     

The method of integral equation for scattering problem with a mixed crack

We consider the scattering of an electromagnetic time‐harmonic plane wave by an infinite cylinder having a mixed open crack (or arc) in R² as the cross section. The crack is made up of two parts, and one of the two parts is (possibly) coated by a material with surface impedance λ. We transform the scattering problem into a system of boundary integral equations by adopting a potential approach, and establish the existence and uniqueness of a weak solution to the system by the Fredholm theory. Copyright © 2011 John Wiley & Sons, Ltd.     

Initial conditions for the cylindrical Korteweg‐de Vries equation

In this paper, we find suitable initial conditions for the cylindrical Korteweg‐de Vries equation by first solving exactly the initial‐value problem for localized solutions of the underlying axisymmetric linear long‐wave equation. The far‐field limit of the solution of this linear problem then provides, through matching, an initial condition for the cylindrical Korteweg‐de Vries equation. This initial condition is associated only with the leading wave front of the far‐field limit of the linear solution. The main motivation is to resolve the discrepancy between the exact mass conservation law, and the “mass” conservation law for the cylindrical Korteweg‐de Vries equation. The outcome is that in the linear initial‐value problem all the mass is carried behind the wave front, and then the “mass” in the initial condition for the cylindrical Korteweg‐de Vries equation is zero. Hence, the evolving solution in the cylindrical Korteweg‐de Vries equation has zero “mass.” This situation arises because, unlike the well‐known unidirectional Korteweg‐de Vries equation, the solution of the initial‐value problem for the axisymmetric linear long‐wave problem contains both outgoing and ingoing waves, but in the cylindrical geometry, the latter are reflected at the origin into outgoing waves, and eventually the total outgoing solution is a combination of these and those initially generated.     

On the linearization of operators related to the full waveform inversion in seismology

In this work, we analyze the parameter‐to‐solution map of the acoustic wave equation with respect to its parameters wave speed and mass density. This map is a mathematical model for the seismic inverse problem where one wants to recover the parameters from measurements of the acoustic potential. We show its complete continuity and Fréchet differentiability. To this end, we provide necessary existence, stability, and regularity results. Moreover, we discuss various implications of our findings on the inverse problem and comment on the Born series. Copyright © 2013 John Wiley & Sons, Ltd.     

具有多个非线性源项的波动方程

利用位势井方法研究在有界区域上具有多个非线性源项的波动方程初边值问题．给出了位势井的结构和位势井深度函数的性质．通过引进位势井族得到了在这些问题的流之下的一些集合不变性以及解的真空隔离,揭示了只要问题的初值属于位势井内或位势井外,则问题在今后所有时间内的解都存在于位势井内或井外,同时存在一个没有解的空间区域．进而给出了解的整体存在和不存在的门槛结果．最后,利用相同的方法讨论了具有临界初始条件的问题．     

Inverse spectral problem for the operators with non‐local potential

The main object under consideration in the paper is the second derivative operator on a finite interval with zero boundary conditions perturbed by a self‐adjoint integral operator with the degenerate kernel (non‐local potential). The inverse problem, i.e., the reconstruction of the perturbation from the spectral data, is solved by means of the step‐by‐step procedure based on the n‐interlacing property of the spectrum.     

离散形式的声阻抗反演问题的稳定性分析

1．引言考虑一维声波方程组从脉冲响应州附ω（0.t）反演声阻抗ε（x）,是一个适定问题（[1],[2],[3]）．声阻抗ε（X）具有明确的物理意义,它大于民与之对应的脉冲响应自然也应具有某种特殊的性质,本文对此进行深入的分析．本文首先引入KennethP．Bube和RobertBurrdge关于离散形式的声阻抗反演问题的一些基本结果,分析其实物理系统的脉冲响应的特征,说明了真实物理系统的脉冲响应具有某种正定性,这种特征是本质性的,由真实物理系统的内部固有性质所决定．在实际计算中,作为观测数据给出的脉冲响应数据如果具有这种特征,则反…     

An operator approach for an oblique derivative boundary‐transmission problem

We consider a boundary‐transmission problem for the Helmholtz equation, in a Bessel potential space setting, which arises within the context of wave diffraction theory. The boundary under consideration consists of a strip, and certain conditions are assumed on it in the form of oblique derivatives. Operator theoretical methods are used to deal with the problem and, as a consequence, several convolution type operators are constructed and associated to the problem. At the end, the well‐posedness of the problem is shown for a range of non‐critical regularity orders of the Bessel potential spaces, which include the finite energy norm space. In addition, an operator normalization method is applied to the critical orders case. Copyright © 2004 John Wiley & Sons, Ltd.     

Non‐relativistic continuous states in arbitrary dimension for a ring‐shaped pseudo‐Coulomb and energy‐dependent potentials

In this research, the non‐relativistic particle scattering has been investigated for an alternative pseudo‐Coulomb potential plus ring‐shaped and an energy‐dependent potentials in D‐dimensional space. The normalized wave functions of continuous states on the k/2π scale are expressed in terms of the hyper‐geometric series, and formula of phase shifts is presented. Analytical properties of the scattering amplitude and thermodynamics properties are discussed. Some of the numerical results of energy levels have been calculated too. Copyright © 2015 John Wiley & Sons, Ltd.     

The Robin problem for the scalar Oseen equation

We study the Robin problem for the scalar Oseen equation in an open n‐dimensional set with compact Ljapunov boundary. We prescribe two types of Robin boundary conditions, and prove the unique solvability of these problems as well as a representation formula for the solution in form of a scalar Oseen single layer potential. Moreover, we prove the maximum principle for the solution to the Robin problem of the scalar Oseen equation. Copyright © 2013 John Wiley & Sons, Ltd.     

On the Cauchy problem for thermoelastic plates

The Cauchy problem for an infinite thermoelastic plate with a non‐homogeneous governing system and homogeneous initial conditions is solved by means of an area potential. This is the first step in the construction of a potential theory for time‐dependent problems for thermoelastic plates, enabling the reduction of various initial‐boundary value problems to their versions for the homogeneous system of equations with homogeneous initial conditions, which, in turn, may then be solved by means of dynamic potentials. Copyright © 2006 John Wiley & Sons, Ltd.     

Stability of the space identification problem for the elliptic‐telegraph differential equation

The present paper is devoted to study the space identification problem for the elliptic‐telegraph differential equation in Hilbert spaces with the self‐adjoint positive definite operator. The main theorem on the stability of the space identification problem for the elliptic‐telegraph differential equation is proved. In applications, theorems on the stability of three source identification problems for one dimensional with nonlocal conditions and multidimensional elliptic‐telegraph differential equations are established.     

Solvability,asymptotic analysis and regularity results for a mixed type interaction problem of acoustic waves and piezoelectric structures

In the paper, we investigate the mixed type transmission problem arising in the model of fluid–solid acoustic interaction when a piezoceramic elastic body (Ω⁺) is embedded in an unbounded fluid domain (Ω^?). The corresponding physical process is described by the boundary‐transmission problem for second‐order partial differential equations. In particular, in the bounded domain Ω⁺, we have a 4×4 dimensional matrix strongly elliptic second‐order partial differential equation, while in the unbounded complement domain Ω^?, we have a scalar Helmholtz equation describing acoustic wave propagation. The physical kinematic and dynamic relations mathematically are described by appropriate boundary and transmission conditions. With the help of the potential method and theory of pseudodifferential equations based on the Wiener–Hopf factorization method, the uniqueness and existence theorems are proved in Sobolev–Slobodetskii spaces. We derive asymptotic expansion of solutions, and on the basis of asymptotic analysis, we establish optimal Hölder smoothness results for solutions. Copyright © 2017 John Wiley & Sons, Ltd.     

A noniterative reconstruction method for the inverse potential problem with partial boundary measurements

In this paper, a noniterative reconstruction method for solving the inverse potential problem is proposed. The forward problem is governed by a modified Helmholtz equation. The inverse problem consists in the reconstruction of a set of anomalies embedded into a geometrical domain from partial or total boundary measurements of the associated potential. Since the inverse problem is written in the form of an ill‐posed boundary value problem, the idea is to rewrite it as a topology optimization problem. In particular, a shape functional measuring the misfit between the solution obtained from the model and the data taken from the boundary measurements is minimized with respect to a set of ball‐shaped anomalies by using the concept of topological derivatives. It means that the shape functional is expanded asymptotically and then truncated up to the desired order term. The resulting truncated expansion is trivially minimized with respect to the parameters under consideration that leads to a noniterative second order reconstruction algorithm. As a result, the reconstruction process becomes very robust with respect to the noisy data and independent of any initial guess. Finally, some numerical experiments are presented showing the capability of the proposed method in reconstructing multiple anomalies of different sizes and shapes by taking into account complete or partial boundary measurements.     

Well‐posedness of nonlinear wave equation with combined power‐type nonlinearities

In this paper, we study the initial boundary‐value problem with combined power‐type nonlinearities by utilizing potential well method. We provide an algorithm to compute the depth of the potential well with the help of Mathematica, and derive the invariant subsets, global existence and blowup of solutions. Moreover, we obtain the invariant subsets, global existence and blowup of solutions for the critical case. Copyright © 2011 John Wiley & Sons, Ltd.     

Inverse problem with a time‐integral condition for a fractional diffusion equation

We find the conditions for the unique solvability of the inverse problem for a time‐fractional diffusion equation with Schwarz‐type distributions in the right‐hand sides. This problem is to find a generalized solution of the Cauchy problem and an unknown space‐dependent part of an equation's right‐hand side under a time‐integral overdetermination condition.     

The Cauchy problem for Kawahara equation in Sobolev spaces with low regularity

This paper is devoted to studying the initial‐value problem of the Kawahara equation. By establishing some crucial bilinear estimates related to the Bourgain spaces X_{s, b}(R²) introduced by Bourgain and homogeneous Bourgain spaces, which is defined in this paper and using I‐method as well as L² conservation law, we show that this fifth‐order shallow water wave equation is globally well‐posed for the initial data in the Sobolev spaces H^s(R) with $s{>}-\frac{63}{58}$. Copyright © 2010 John Wiley & Sons, Ltd.     

The Cauchy problem for the generalized hyperelastic rod wave equation

In this paper we consider the Cauchy problem for the generalized hyperelasticrod wave equation which includes the Camassa‐Holm equation and the hyperelastic rod wave equation. Firstly, by using the Kato's theory, we prove that the Cauchy problem for the generalized hyperelastic rod wave is locally well‐posed in Sobolev spaces with . Secondly, we give some conservation laws, some useful conclusions and the precise blow‐up scenario and show that the Cauchy problem for the generalized hyperelastic rod wave equation has smooth solutions which blows up in finite time. Thirdly, we give the blow‐up rate of the strong solutions to the Cauchy problem for the generalized hyperelastic rod wave equation. Finally, we give the lower bound of the maximal existence time of the solution and the lower semicontinuity of existence time of solutions to the generalized hyperelastic rod wave equation.     

Boundary value problem for a global‐in‐time parabolic equation

The aim of this paper is to draw attention to an interesting semilinear parabolic equation that arose when describing the chaotic dynamics of a polymer molecule in a liquid. This equation is nonlocal in time and contains a term, called the interaction potential, that depends on the time‐integral of the solution over the entire interval of solving the problem. In fact, one needs to know the “future” in order to determine the coefficient in this term, that is, the causality principle is violated. The existence of a weak solution of the initial boundary value problem is proven. The interaction potential satisfies fairly general conditions and can have arbitrary growth at infinity. The uniqueness of this solution is established with restrictions on the length of the considered time interval.