Dirichlet–Neumann inverse spectral problem for a star graph of Stieltjes strings

We solve two inverse spectral problems for star graphs of Stieltjes strings with Dirichlet and Neumann boundary conditions, respectively, at a selected vertex called root. The root is either the central vertex or, in the more challenging problem, a pendant vertex of the star graph. At all other pendant vertices Dirichlet conditions are imposed; at the central vertex, at which a mass may be placed, continuity and Kirchhoff conditions are assumed. We derive conditions on two sets of real numbers to be the spectra of the above Dirichlet and Neumann problems. Our solution for the inverse problems is constructive: we establish algorithms to recover the mass distribution on the star graph (i.e. the point masses and lengths of subintervals between them) from these two spectra and from the lengths of the separate strings. If the root is a pendant vertex, the two spectra uniquely determine the parameters on the main string (i.e. the string incident to the root) if the length of the main string is known. The mass distribution on the other edges need not be unique; the reason for this is the non-uniqueness caused by the non-strict interlacing of the given data in the case when the root is the central vertex. Finally, we relate of our results to tree-patterned matrix inverse problems.     

Reproducing-kernel-based splines for the regularization of the inverse ellipsoidal gravimetric problem

To solve the inverse gravimetric problem, i.e. to reconstruct the Earth's mass density distribution by using the gravitational potential, we introduce a spline interpolation method for the ellipsoidal Earth model, where the ellipsoid has a rotational symmetry. This problem is ill-posed in the sense of Hadamard as the solution may not exist, it is not unique and it is not stable. Since the anharmonic part (orthogonal complement) of the density function produces a zero potential, we restrict our attention only to reconstruct the harmonic part of the density function by using the gravitational potential. This spline interpolation method gives the existence and uniqueness of the unknown solution. Moreover, this method represents a regularization, i.e. every spline continuously depends on the given gravitational potential. These splines are also combined with a multiresolution concept, i.e. we get closer and closer to the unknown solution by increasing the scale and adding more and more data at each step.     

Study of motions of a vibrating string subject to solid friction

In this paper we consider a string moving in its plane and subject to solid friction (Coulomb's law). It is known that when the time increases indefinitely the string reaches an equilibrium position and by analogy with the case of a mass point we ask if the equilibrium is reached after a finite time. We prove that this is the case when the string is initially at rest and 1. when the initial shape possesses a second derivative bounded by certain limits, or 2. when the initial shape is formed by two straight line segments. In the last section we obtain some partial results when the string is initially at rest in the shape of a polygonal line. The case of an arbitrary initial position is still an open problem.     

Multiple Order Pick Sequencing in a Carousel System: A Solvable Case of the Rural Postman Problem

We consider the problem of sequencing picks in a set of orders on a single carousel. First we consider the situation in which the sequence of the orders is given. For this problem we present an efficient dynamic programming algorithm. Second, we consider the problem without a given order sequence. We simplify this problem to a Rural Postman Problem on a circle and solve this problem to optimality. Finally, we show that the solution of the Rural Postman Problem requires at most 1.5 revolutions more than a lower bound of an optimum solution to the original problem.     

Time dependent linear transport I. Existence,uniqueness, and continuous dependence

This paper concerns the time dependent linear transport equation posed in a multidimensional rectangular parallelepiped with partially reflecting walls. We consider the continuous transport equation and the discrete ordinate equations simultaneously. Our boundary condition, partial specular reflection, includes both vacuum and reflecting boundaries as special cases. We define strong and weak solutions of the problem, strong solutions being solutions in the ordinary sense and weak solutions being distributions, and show that a weak solution is a strong solution if it has space and time derivatives almost everywhere. For weak solutions we establish existence, uniqueness, and continuous dependence upon the initial data and the other functions which define the problem.     

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.     

Interpolation in Bernstein and Paley-Wiener spaces

We consider interpolation of discrete functions by continuous ones with restriction on the size of spectra. We discuss a sharp contrast between the cases of compact and unbounded spectra. In particular we construct ‘universal’ spectra of small measure which deliver positive solution of the interpolation problem in Bernstein spaces for every discrete sequence of knots.     

Analysis of a meshless method for the time fractional diffusion-wave equation

In this paper a numerical technique is proposed for solving the time fractional diffusion-wave equation. We obtain a time discrete scheme based on finite difference formula. Then, we prove that the time discrete scheme is unconditionally stable and convergent using the energy method and the convergence order of the time discrete scheme is \(\mathcal {O}(\tau ^{3-\alpha })\). Firstly, we change the main problem based on Dirichlet boundary condition to a new problem based on Robin boundary condition and then, we consider a semi-discrete scheme with Robin boundary condition and show when \(\beta \rightarrow +\infty \) solution of the main semi-discrete problem with Dirichlet boundary condition is convergent to the solution of the new semi-discrete problem with Robin boundary condition. We consider the new semi-discrete problem with Robin boundary condition and use the meshless Galerkin method to approximate the spatial derivatives. Finally, we obtain an error bound for the new problem. We prove that convergence order of the numerical scheme based on Galekin meshless is \(\mathcal {O}(h)\). In the considered method the appeared integrals are approximated using Gauss Legendre quadrature formula. The main aim of the current paper is to obtain an error estimate for the meshless Galerkin method based on the radial basis functions. Numerical examples confirm the efficiency and accuracy of the proposed scheme.     

Linear programming based approaches for the discrete time/cost trade-off problem in project networks

In project management, the activity durations can often be reduced by dedicating additional resources. The Time/Cost Trade-off Problem considers the compromise between the total cost and the project duration. The discrete version of the problem assumes a number of time/cost pairs, called modes, and selects a mode for each activity. In this paper, we consider the Discrete Time/Cost Trade-off Problem. We study the Deadline Problem, that is, the problem of minimizing total cost subject to a deadline on the project duration. To solve the Deadline Problem, we propose optimization and approximation algorithms that are based on the optimal Linear Programming Relaxation solutions. Our computational results from large-sized problem instances reveal the satisfactory behaviour of our algorithms.     

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.     

The modified Green function technique for the exterior Dirichlet problem in linear thermoelasticity

In this work, the modified Green function technique for the exterior Dirichlet problem in linear thermoelasticity is presented. Expressing the solution of the problem as a double‐layer potential of an unknown density, we form the associated boundary integral equation that describes the problem. Exploiting that the discrete spectrum of the irregular values of the associated integral equation is identified with the spectrum of eigenvalues of the corresponding interior homogeneous Neumann problem for the transverse part of the elastic displacement field, we introduce a modification of the fundamental solution of the elastic field. We establish the sufficient conditions that the coefficients of the modification must satisfy to overcome the problem of nonuniqueness for the thermoelastic problem.     

Discrete String Simulating the Musical Scale

A discrete string with fixed endpoints carrying a finite number of beads is determined by the masses of beads and the distances between them. The string possesses a set of simple eigenfrequencies corresponding to harmonic eigenmodes. In this paper, the following problem is studied: to find a discrete string carrying seven beads such that its eigenfrequencies coincide with the freqiencies of the notes of the first octave of the musical scale. The problem is solved in two steps. First, the spectral inverse problem is considered, i.e., we recover the string from its spectrum and a set of constants related to the normalized eigenmodes. A procedure of solving this problem is described. One of the main results of the paper is a necessary and sufficient condition for the solvability of the spectral inverse problem. The second step is numerical realization of the procedure. Bibliography: 3 titles.     

Full discretization of a nonlinear parabolic problem containing volterra operators and an unknown dirichlet boundary condition

The reconstruction of an unknown solely time‐dependent Dirichlet boundary condition in a nonlinear parabolic problem containing a linear and a nonlinear Volterra operator is considered. The inverse problem is converted into a variational problem in which the unknown Dirichlet condition is eliminated using a given integral overdetermination. A time‐discrete recurrent approximation scheme is designed, using Backward Euler's method. The convergence of the approximations towards a solution of the variational problem is proved under appropriate assumptions on the data and on the Volterra operators. The uniqueness of this solution is shown in the case that the nonlinear Volterra operator satisfies a particular inequality. Moreover, the Finite Element Method is used to discretize the time‐discrete approximation scheme in space. Finally, full‐discrete error estimates are derived for a particular choice of the finite elements. The corresponding convergence rates are supported by a numerical experiment. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1444–1460, 2015     

Spectral methods for initial boundary value problems with nonsmooth data

Summary. In this paper we consider hyperbolic initial boundary value problems with nonsmooth data. We show that if we extend the time domain to minus infinity, replace the initial condition by a growth condition at minus infinity and then solve the problem using a filtered version of the data by the Galerkin-Collocation method using Laguerre polynomials in time and Legendre polynomials in space, then we can recover pointwise values with spectral accuracy, provided that the actual solution is piecewise smooth. For this we have to perform a local smoothing of the computed solution. Received August 1, 1995 / Revised version received August 19, 1997     

双对称非负定阵一类逆特征值问题的最小二乘解

１．引言 逆特征值问题在工程中有广泛的应用,其研究已有一些很好的结果［１－５］．最近,文［６］还研究了双对称矩阵逆特征值问题,即研究了如下两个问题： 问题Ａ．已知Ｘ∈Ｒｎｘｍ,Ａ＝ｄｉａｇ（λ１…,λｍ）,求Ａ∈ＢＳＲｎｘｎ使 ＡＸ＝ＸＡ,其中 Ｒｎｘｍ表示全体 ｎ ｘ ｍ实矩阵集合, ＢＳＲｎｘｎ表示全体 ｎ ｘ ｎ双对称阵集合． 问题Ｂ．已知Ａ＊ＥＲｎｘｎ,求Ａ∈ＳＥ使 ｜｜Ａ＊－Ａ｜｜＝ ｉｎｆ ｜｜Ａ＊－Ａ｜｜ ＡＦＳＥ其中 ＳＥ是问题 Ａ的解集合,｜｜． ｜｜表示 Ｆｒｏｂｅｎｉｕｓ范数． 在实际问题中, …     

Solution of a possibilistic multiobjective linear programming problem

The estimate of the parameters which define a conventional multiobjective decision making model is a difficult task. Normally they are either given by the Decision Maker who has imprecise information and/or expresses his considerations subjectively, or by statistical inference from the past data and their stability is doubtful. Therefore, it is reasonable to construct a model reflecting imprecise data or ambiguity in terms of fuzzy sets and several fuzzy approaches to multiobjective programming have been developed 1, 9, 10, 11. The fuzziness of the parameters gives rise to a problem whose solution will also be fuzzy, see 2, 3, and which is defined by its possibility distribution. Once the possibility distribution of the solution has been obtained, if the decision maker wants more precise information with respect to the decision vector, then we can pose and solve a new problem. In this case we try to find a decision vector, which approximates as much as possible the fuzzy objectives to the fuzzy solution previously obtained. In order to solve this problem we shall develop two different models from the initial solution and based on Goal Programming: an Interval Goal Programming Problem if we define the relation “as accurate as possible” based on the expected intervals of fuzzy numbers, as we showed in [4], and an ordinary Goal Programming based on the expected values of the fuzzy numbers that defined the goals. Finally, we construct algorithms that implement the above mentioned solution method. Our approach will be illustrated by means of a numerical example.     

Determining surface heat flux in the steady state for the Cauchy problem for the Laplace equation

In this paper, we consider the Cauchy problem for the Laplace equation, in a strip where the Cauchy data is given at x = 0 and the flux is sought in the interval 0<x?1. This problem is typical ill-posed: the solution (if it exists) does not depend continuously on the data. We study a modification of the equation, where a fourth-order mixed derivative term is added. Some error stability estimates for the flux are given, which show that the solution of the modified equation is approximate to the solution of the Cauchy problem for the Laplace equation. Furthermore, numerical examples show that the modified method works effectively.     

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.     

Discretized Tikhonov regularization by reproducing kernel Hilbert space for backward heat conduction problem

In this paper we propose a numerical reconstruction method for solving a backward heat conduction problem. Based on the idea of reproducing kernel approximation, we reconstruct the unknown initial heat distribution from a finite set of scattered measurement of transient temperature at a fixed final time. Standard Tikhonov regularization technique using the norm of reproducing kernel is adopt to provide a stable solution when the measurement data contain noises. Numerical results indicate that the proposed method is stable, efficient, and accurate.     

On Starting Control of Vibrations of a String

Within the framework of the theory of games, we consider the problem of starting control of oscillations of points of a string according to a given law. As control parameters for players, the initial position and the starting velocity of the string are taken. We determine the optimal control for players in both discrete case and continuous case.