A spectral relation for Chebyshev-Laguerre polynomials and its application to dynamic problems of fracture mechanics

A new spectral relation for Chebyshev-Laguerre polynomials is derived and its use to construct an exact solution of the antiplane problem of the theory of elasticity on the diffraction of a shock SH-wave by a semi-infinite crack is described, when this wave is incident on the crack at an arbitrary angle. The problem is reduced to an integro-differential equation by the method of discontinuous solutions. An exact solution of this equation using the spectral relation obtained is given. A formula is obtained for the scattered wave and for the stress intensity factor.     

Numerical algorithm based on transmutation for solving inverse wave equation

This paper develops algorithms for solving an undetermined coefficient problem for a wave equation. The algorithms are based on an integral representation for the solution to the wave equation obtained by using transmutation. The convergence of the algorithm is studied and numerical experiments are performed.     

Tracking of solutions to parabolic equations with memory in a general class of controls

We consider a control problem for a parabolic equation with memory. It consists in constructing an algorithm for finding a feedback control which enables one to track a solution of the given equation with an unknown right-hand side. For this problem we propose two noise-resistant solution algorithms based on the method of extremal shift. The first algorithm is applicable in the case of continuous measurements of phase states, whereas the second one presumes discrete measurements.     

A Numerical Method for Computing Time‐Periodic Solutions in Dissipative Wave Systems

A numerical method is proposed for computing time‐periodic and relative time‐periodic solutions in dissipative wave systems. In such solutions, the temporal period, and possibly other additional internal parameters such as the propagation constant, are unknown priori and need to be determined along with the solution itself. The main idea of the method is to first express those unknown parameters in terms of the solution through quasi‐Rayleigh quotients, so that the resulting integrodifferential equation is for the time‐periodic solution only. Then this equation is computed in the combined spatiotemporal domain as a boundary value problem by Newton‐conjugate‐gradient iterations. The proposed method applies to both stable and unstable time‐periodic solutions; its numerical accuracy is spectral; it is fast‐converging; its memory use is minimal; and its coding is short and simple. As numerical examples, this method is applied to the Kuramoto–Sivashinsky equation and the cubic‐quintic Ginzburg–Landau equation, whose time‐periodic or relative time‐periodic solutions with spatially periodic or spatially localized profiles are computed. This method also applies to systems of ordinary differential equations, as is illustrated by its simple computation of periodic orbits in the Lorenz equations. MATLAB codes for all numerical examples are provided in the Appendices to illustrate the simple implementation of the proposed method.     

On efficient numerical methods for an initial-boundary value problem with nonlocal boundary conditions

Many physical phenomena are modeled by nonclassical hyperbolic boundary value problems with nonlocal boundary conditions. In this paper, the problem of solving the one-dimensional wave equation subject to given initial and non-local boundary conditions is considered. These non-local conditions arise mainly when the data on the boundary cannot be measured directly. Several finite difference methods with low order have been proposed in other papers for the numerical solution of this one dimensional non-classic boundary value problem. Here, we derive a new family of efficient three-level algorithms with higher order to solve the wave equation and also use a Simpson formula with higher order to approximate the integral conditions. Additionally, the fourth-order formula is also adapted to nonlinear equations, in particular to the well-known nonlinear Klein–Gordon equations which many physical problems can be modeled with. Numerical results are presented and are compared with some existing methods showing the efficiency of the new algorithms.     

Numerical solution method for the electric impedance tomography problem in the case of piecewise constant conductivity and several unknown boundaries

We study the electrical impedance tomography problem with piecewise constant electric conductivity coefficient, whose values are assumed to be known. The problem is to find the unknown boundaries of domains with distinct conductivities. The input information for the solution of this problem includes several pairs of Dirichlet and Neumann data on the known external boundary of the domain, i.e., several cases of specification of the potential and its normal derivative. We suggest a numerical solution method for this problem on the basis of the derivation of a nonlinear operator equation for the functions that define the unknown boundaries and an iterative solution method for this equation with the use of the Tikhonov regularization method. The results of numerical experiments are presented.     

Investigation of the behavior of weak waves in the solution of a vector variational wave equation

We consider nonsmooth solutions of the system of Euler-Lagrange equations corresponding to a variational problem with several unknown functions of several variables and with a quadratic functional. The propagation of weak discontinuities is described by the equations of the method of singular characteristics developed by Melikyan. The onset and interaction of weak discontinuities of the solution caused by nonsmooth initial conditions are studied by numerical-analytic methods. We develop two computer programs for shock-fitting and shock-capturing computations. The approach was earlier applied by the authors to the analysis of a variational wave equation, namely, to the solution of the Euler-Lagrange equation for a variational problem with a single unknown function.     

Algorithm for the continuous estimation of a disturbance in a stochastic differential equation

Based on the approach of the theory of dynamic inversion, the problem of continuous estimation of an unknown deterministic disturbance in an Ito stochastic differential equation is investigated with the use of inaccurate measurements of the current phase state. An auxiliary model equation with a control approximating the unknown input is derived. The suggested solution algorithm is constructive; an estimate for its convergence rate is written explicitly.     

Integral-Functional Equation Arising in the Study of an Inverse Problem for a Quasilinear Hyperbolic Equation

A problem with data on the characteristics is considered for a quasilinear hyperbolic equation. The inverse problem of determining two unknown coefficients of the equation from some additional information about the solution is posed. One of the unknown coefficients depends on the independent variable, and the other, on the solution of the equation. Uniqueness theorems are proved for the solution of the inverse problem. The proof is based on the derivation of the integro-functional equation and the analysis of the uniqueness of its solution.     

On tracking solutions of parabolic equations

We consider a control problem for a parabolic equation. It consists in constructing an algorithm for finding a feedback control such that a solution of a given equation should track a solution of another equation generated by an unknown right-hand side. We propose two noise-resistant solution algorithms for the indicated problem. They are based on the method of extremal shift well-known in the guaranteed control theory. The first algorithm is applicable in the case of “continuous” measurements of phase states, whereas the second one implies discrete measurements.     

Generalized differential equation arising in the solution of the inverse scattering problem in a layered medium

We consider a nonclassical ordinary differential equation containing not only an unknown function but also an unknown coefficient depending on the unknown function. We show that if the desired solution is assumed to have bounded variation and be a.e. constant on the interval where the equation is considered, then the problem of finding the solution and the unknown coefficient does not have a unique solution in terms of the classical derivative. We prove that if the derivative is understood as a distribution, than this problem has a unique solution. These results are used to show that the acoustic impedance and the damping factor in the inverse scattering problem in a layered dissipative medium can be determined simultaneously.     

一维波动方程联合反演的分步正则化法

本文只用一个纵波信息,对一维波动方程的速度和震源函数进行联合反演.并考虑到波动方程的反问题是一不适定问题,对震源函数和波速分别用正则化法分步迭代求解,大大减少了反问题的计算工作量,改善了该反问题的计算稳定性.为计算实际一维地震数据提供了一种方法.文中给出了只用一个反问题补充条件同时进行多参数反演的详细公式,并对相应的数值算例进行了分析和比较.     

Reconstruction of a time-dependent coefficient in nonlinear Klein–Gordon equation using Bernstein spectral method

In this work, we present a spectral method for recovering an unknown time-dependent lower-order coefficient and unknown wave displacement in a nonlinear Klein–Gordon equation with overdetermination at a boundary condition. We apply the initial and boundary conditions to construct the satisfier function and use this function in a transformation to convert the main problem to a nonclassical hyperbolic equation with homogeneous initial and boundary conditions. Then, we utilize the orthonormal Bernstein basis functions to approximate the solution of the reformulated problem and use a direct technique based on the operational matrices of integration and differentiation of these basis functions together with the collocation technique to reduce the problem to a system of nonlinear algebraic equations. Regarding the perturbed measurements, the method takes advantage of the mollification method in order to derive stable numerical derivatives. Numerical simulations for solving several test examples are presented to show the applicability of the proposed method for obtaining accurate and stable results.     

Exact solution of a model problem of subsurface sensing

An inverse electromagnetic wave radiation problem simulating subsurface radio sensing is considered. It is assumed that synchronous external currents with unknown spatial density emerge in the subsurface medium. It is shown that their distribution can be found from the pulsed radiation waveformes measured along the border of the examined half-space. In a model formulation, the problem is reduced to the reconstruction of a 2D function from its integrals over a set of semicircles. An explicit solution of that tomographic problem is found by means of the Darboux equation. Numerical examples are given. Bibliography: 13 titles.     

On the characteristic initial value problem for the wave equation in odd spatial dimensions with radial initial data

Summary The paper obtains an explicit solution of the characteristic initial value problem for the wave equation in odd spatial dimensions with radial initial data via solution of a characteristic boundary value problem involving a singular differential equation. The solution of the latter problem is obtained by a modified Riemann method. It is shown that on the time axis the solution of the original problem reduces to the solution that is obtainable by the use of Asgeirsson’s mean value theorem. Entrata in Redazione il 29 agosto 1971.     

半线性波方程的一个反问题

利用压缩映像原理讨论了一类半线性波方程确定未知系数的反问题,文中给出了该问题解的存在性、唯一性和稳定性。     

Generalized 1D Jörgens theorem

We study the Cauchy problem for a 1D nonlinear wave equation on R. The nonlinearity can depend on the unknown function and its first order spatial derivative. Using the fixed point theorem we prove the existence of a classical solution. Moreover, the existence of periodic and almost periodic solutions are shown.     

Interaction of a harmonic torsional wave with ring-shaped defects in an elastic body

We have solved the problem of determination of the stressed state in an isotropic elastic body near ring-shaped defects (a crack or a thin rigid inclusion) as a result of the action of a harmonic torsional wave. The method of solution is based on the use of discontinuous solutions of the equation of torsional vibrations and lies in the reduction of the initial boundary-value problems to integral equations for the unknown jumps of angular displacement or tangential stress.     

Dynamic reaction of a pseudo-ceramic half-space with a longitudinal shear crack to concentrated forces

The dynamic problem of electroelasticity for a piezoelectric half-space with tunnel cavities-cuts is examined. The problem is reduced to the solution of an integrodifferential singular equation in terms of the amplitude of the jump in displacements at the cut. Equations are obtained for the dynamic mechanial-stress intensity factor. The influence exerted by the curvature of the crack, its orientation, and the normalized wave number on the intensity factor is studied numerically.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 50–55, 1989.     

Identification of objects in an acoustic wave guide inversion II: Robin–Dirichlet conditions

In this paper we investigate the unknown body problem in a wave guide where one boundary has a pressure release condition and the other an impedance condition. The method used in the paper for solving the unknown body inverse problem is the intersection canonical body approximation (ICBA). The ICBA is based on the Rayleigh conjecture, which states that every point on an illuminated body radiates sound from that point as if the point lies on its tangent sphere. The ICBA method requires that an analytical solution be known exterior to a canonical body in the wave guide. We use the sphere of arbitrary centre and radius in the wave guide as our canonical body. We are lead then to analytically computing the exterior solution for a sphere between two parallel plates. We use the ICBA to construct solutions at points ranging over the suspected surface of the unknown object to reconstruct the unknown object using a least‐squares matching of computed, acoustic field against the measured, scattered field. Copyright © 2005 John Wiley & Sons, Ltd.