On an inverse boundary-value problem for the equation of motion of a homogeneous elastic beam with pinned ends

t. This paper surveys some of the work our group has done in electrical impedance tomography.

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In this paper, we develop a non‐iterative way to solve biharmonic boundary value problems by using a radial basis meshless method. This is an original application of meshless method to solving inverse problems without any iteration, since traditional numerical methods for inverse boundary value problems mainly are iterative and hence very time‐consuming. Numerical examples are presented for inverse biharmonic boundary value problems and corresponding direct problems, since solving direct problems is a preliminary step for inverse problems. All our examples of direct and inverse problems are solved within seconds in CPU time on a standard PC, which makes our proposed technique a great potential candidate for wide‐spread applications to other inverse problems. Copyright © 2004 John Wiley & Sons, Ltd.

Determination of an Unknown Heat Source from Overspecified Boundary Data

This paper investigates an inverse non-stationary problem of the restoration of the spatial law of a homogeneous isotropic Timoshenko beam of finite length. Hinge support conditions are used as boundary conditions. Initial conditions are assumed to be zero. It is assumed that of the beam’s ends is fitted with sensors which in the course of corresponding experiment register the amount of deflection of the beam at the sensor points. The method of the solution of a direct problem is based on the principle of superposition where the deflection of the beam is associated with the space load the beam is exposed to, by means of an integral operator by the spatial coordinate and time. The kernel of such operator is so called influence function. This function is a fundamental solution of a system of differential equations of motion of the study beam. The construction of such solution represents a separate problem. The influence function is found by means of Laplace time transformation and expansion into Fourier series in a system of the problem’s eigenfunctions. The solution of the inverse problem at the first stage reduces to a system of algebraic equations for vector operator whose components are time convolutions of the coefficients of expansion series for an influence function with the desired coefficients of expansion of the load in a Fourier series. At the same time, the components of the vector of the rights parts are time dependencies registered by the sensors. The resulting system is ill-conditioned [1]. The second stage serves to resolve independent Volterra integral equations of the first kind for the desired coefficients of Fourier serials for the load.

In the current paper, in the domain $D=\{(t,x): t\in(0,T), x\in(0,L)\}$ we investigate the boundary value problem for the equation of motion of a homogeneous elastic beam $$ u_{tt}(t,x)+a^{2}u_{xxxx}(t,x)+b u_{xx}(t,x)+c u(t,x)=0, $$ where $a,b,c \in \mathbb{R}$, $b^2<4a^2c$, with nonlocal two-point conditions $$u(0,x)-u(T, x)=\varphi(x), \quad u_{t}(0, x)-u_{t}(T, x)=\psi(x)$$ and local boundary conditions $$u(t, 0)=u(t, L)=u_{xx}(t, 0)=u_{xx}(t, L)=0.$$ Solvability of this problem is connected with the problem of small denominators, whose estimation from below is based on the application of the metric approach. For almost all (with respect to Lebesgue measure) parameters of the problem, we establish conditions for the solvability of the problem in the Sobolev space. In particular, if $\varphi\in\mathbf{H}_{q+\rho+2}$ and $\psi \in\mathbf{H}_{q+\rho}$, where $\rho>2$, then for almost all (with respect to Lebesgue measure in $\mathbb{R}$) numbers $a$ exists a unique solution $u\in\mathbf{C}^{\,2}([0,T];\mathbf{H}_{q})$ of the problem considered.

The exact solution of the governing partial differential equations describing the motion of an anisotropic porous beam with axial-flexural coupling is presented. The motion of the beam is described by the classical Euler–Bernoulli theory. The pore-fluid pressure is governed by the generalized Darcy’s law with relaxation and retardation time parameters to account for the inertia and viscosity of the fluid. Solutions are sought in the frequency domain where the governing equations are converted into a polynomial eigenvalue structure and solved exactly. The wavenumbers and group speeds of propagating waves in the beam are studied in detail. It is found that the presence of fluid-filled porous micro-structure introduces three additional propagating modes, other than the axial and bending modes predicted by the classical beam theory. The effect of diffusion boundary conditions on the transverse motion of a porous beam is investigated in detail. It is also found that the material parameters have considerable influence on the magnitude of the transverse velocity, the group speed of propagation and the behavior of the pressure resultants.

In this exposition we give a unified presentation of the conformal mapping technique that was developed over the last decade by Akduman et al. [I. Akduman and R. Kress, Electrostatic imaging via conformal mapping, Inverse Probl. 18 (2002), pp. 1659–1672; R. Kress, Inverse Dirichlet problem and conformal mapping, Math. Comput. Simul. 66 (2004), pp. 255–265; H. Haddar and R. Kress, Conformal mappings and inverse boundary value problems, Inverse Probl. 21 (2005), pp. 935–953; H. Haddar and R. Kress, Conformal mapping and an inverse impedance boundary value problem, J. Inverse Ill-Posed Probl. 14 (2006), pp. 785–804; H. Haddar and R. Kress, Conformal mapping and impedance tomography, Inverse Probl. 26 (2010), p. 074002] for the inverse problem to recover three different types of inclusions in a homogeneous conducting background medium from Cauchy data on the accessible exterior boundary. The main ingredient of this method is a nonlinear and nonlocal ordinary differential equation for boundary values of a holomorphic function in an annulus bounded by two concentric circles that maps this annulus conformally onto the unknown domain. Furthermore, in a concluding section we illustrate how this differential equation also can be applied to numerically construct conformal mappings for doubly connected domains including numerical examples.

This paper is devoted to obtaining a unique solution to an inverse problem for a mul\-ti\-di\-men\-sio\-nal time-fractional integro-differential equation. In the case of additional data, we consider an inverse problem. The unknown coefficient and kernel are uniquely determined by the additional data. By using the fixed point theorem in suitable Sobolev spaces, the global in time existence and uniqueness results of this inverse problem are obtained. The weak solubility of a nonlinear inverse boundary value problem for a $d$-dimensional fractional diffusion-wave equation with natural initial conditions was studied in the work. First, the existence and uniqueness of the direct problem were investigated. The considered problem was reduced to an auxiliary inverse boundary value problem in a certain sense and its equivalence to the original problem was shown. Then, the local existence and uniqueness theorem for the auxiliary problem is proved using the Fourier method and contraction mappings principle. Further, based on the equivalency of these problems, the global existence and uniqueness theorem for the weak solution of the original inverse coefficient problem was established for any value of time.

The paper studies the problem of determining the boundary condition in the heat conduction equation for a rod consisting of homogeneous parts with different thermophys- ical properties. We consider the Dirichlet condition at the left end of the rod (at x = 0) corresponding to the heating of this end and the homogeneous condition of the first kind at the right end (at x = 1) corresponding to cooling during interaction with the environment as boundary conditions. At the point of discontinuity of the thermophysical properties (at x = x0), the conditions for the continuity of temperature and heat flux are set. In the inverse problem, the boundary condition at the left end is assumed to be unknown. To find it, the value of the direct problem solution at the point x0, i.e., the point of separation of the rod into two homogeneous sections, is set. In this work, we carried out an analytical study of the direct problem, which allowed us to apply the time Fourier transform to the inverse boundary value problem. The projection-regularization method is used to solve the inverse boundary value problem for the heat equation and obtain error estimates of this solution correct to the order.

In this paper we study an inverse boundary value problem with an unknown timedependent coefficient for a third-order pseudo-hyperbolic equation with an additional integral condition. The definition of the classical solution of the problem is given. The essence of the problem is that it is required together with the solution to determine the unknown coefficient. The problem is considered in a rectangular area. When solving the original inverse boundary value problem, the transition from the original inverse problem to some auxiliary inverse problem is carried out. The existence and uniqueness of a solution to an auxiliary problem are proved with the help of contracted mappings. Then the transition to the original inverse problem is again made, as a result, a conclusion is made about the solvability of the original inverse problem.

We study the inverse coefficient problem for the equation of longitudinal wave propagation with non-self-adjoint boundary conditions. The main purpose of this paper is to prove the existence and uniqueness of the classical solutions of an inverse boundary-value problem. To investigate the solvability of the inverse problem, we carried out a transformation from the original problem to some equivalent auxiliary problem with trivial boundary conditions. Applying the Fourier method and contraction mappings principle, the solvability of the appropriate auxiliary inverse problem is proved. Furthermore, using the equivalency, the existence and uniqueness of the classical solution of the original problem are shown.

Estimation of tractions and displacements on inaccessible boundaries, such as contact areas of solids, can be regarded as an inverse boundary value problem. In this study finite-element based inverse analysis schemes with regularization were applied to the estimation of the distributions of tractions and displacements on contact areas. The finite element equation was rewritten in terms of unknown boundary values on the contact area using over-prescribed boundary values. This equation was solved for the boundary values on the contact area. Like many other inverse problems, this inverse problem was severely ill-conditioned and the estimated distributions were very sensitive to the over-prescribed boundary values used in the estimation. To overcome the ill-posedness of this boundary value inverse problem, the function expansion method and Tikhonov regularization were introduced in the finite element-based inverse analysis scheme. The number of terms in the function expansion and the smoothing parameter in Tikhonov regularization were regarded as regularization parameters in the inverse analysis. To determine the optimum value of these regularization parameters, the estimated error criterion and the AIC were introduced. The usefulness of the finite element-based inversion scheme was examined by numerical simulations. It was found that the distributions of tractions and displacements can be estimated reasonably even from noisy observations by using the finite-element based inverse analysis schemes with regularization. The optimum value of the regularization parameters can be estimated by the estimated error criterion or by the AIC.

We develop a theoretical analysis for special neural network architectures, termed operator recurrent neural networks , for approximating nonlinear functions whose inputs are linear operators. Such functions commonly arise in solution algorithms for inverse boundary value problems. Traditional neural networks treat input data as vectors, and thus they do not effectively capture the multiplicative structure associated with the linear operators that correspond to the data in such inverse problems. We therefore introduce a new family that resembles a standard neural network architecture, but where the input data acts multiplicatively on vectors. Motivated by compact operators appearing in boundary control and the analysis of inverse boundary value problems for the wave equation, we promote structure and sparsity in selected weight matrices in the network. After describing this architecture, we study its representation properties as well as its approximation properties.We furthermore show that an explicit regularization can be introduced that can be derived from the mathematical analysis of the mentioned inverse problems, and which leads to certain guarantees on the generalization properties. We observe that the sparsity of the weight matrices improves the generalization estimates. Lastly, we discuss how operator recurrent networks can be viewed as a deep learning analogue to deterministic algorithms such as boundary control for reconstructing the unknown wave speed in the acoustic wave equation from boundary measurements.

Direct and inverse boundary value problems for models of stationary reaction–convection–diffusion are investigated. The direct problem consists in finding a solution of the corresponding boundary value problem for given data on the boundary of the domain of the independent variable. The peculiarity of the direct problem consists in the inhomogeneity and irregularity of mixed boundary data. Solvability and stability conditions are specified for the direct problem. The inverse boundary value problem consists in finding some traces of the solution of the corresponding boundary value problem for given standard and additional data on a certain part of the boundary of the domain of the independent variable. The peculiarity of the inverse problem consists in its ill-posedness. Regularizing methods and solution algorithms are developed for the inverse problem.