The influence of two moving heat sources on blow-up in a reactive-diffusive medium

Two concentrated heat sources move through a one-dimensional reactive-diffusive medium. The sources, moving at constant speed in the same direction, are separated by a fixed distance. The leading source serves to preheat the medium ahead of the trailing source. Sufficient conditions, involving the speed and the separation distance, are found for blow-up at the site of the trailing source. Also, the asymptotic behavior of the solution near blow-up is examined.     

An inverse problem of Bilayer textile thickness determination in dynamic heat and moisture transfer

An inverse problem of bilayer textile thickness determination in dynamic heat and moisture transfer is presented satisfying the heat–moisture comfort level of human body. Heat and mass transfer law in bilayer textiles is displayed by proving the existence and uniqueness of solution to the coupled partial differential equations with initial-boundary value conditions. The finite difference method is employed to derive the numerical solution to partial differential equations. The regularized solution of the inverse problem is reformulated into solving function minimum problem through the Tikhonov regularization method. The golden section method is applied to solve the direct search problem and achieve the optimal solution to the inverse problem. Numerical algorithm and its numerical results provide theoretical explanation for textile materials research and development.     

Linearization method in two-dimensional inverse problem of magnetotelluric sounding

An algorithm is described for the two-dimensional inverse problem of magnetotelluric sounding with E-polarization, based on minimization of the Tikhonov smoothing functional by the Newton—Kantorovich method (linearization method). The method of integral equations is used in the algorithm to solve the direct problem of magnetotelluric sounding. An iterative method is constructed for the solution of the integral equation in the direct problem and a rate of convergence bound is derived for this method.Translated from Metody Matematicheskogo Modelirovaniya, Avtomatizatsiya Obrabotki Nablyudenii i Ikh Primeneniya, pp. 199–215, 1986.     

On the inverse problem of simultaneous determination of thermal conductivity and specific heat capacity

Translated from Sibirskii Matematicheskii, Vol. 35, No. 3, pp. 612–621, May–June, 1994.     

One nonlocal problem of determination of the temperature and density of heat sources

We consider one family of problems simulating the determination of the temperature and density of heat sources from given values of the initial and final temperature. The mathematical statement of these problems leads to the inverse problem for the heat equation, where it is required to find not only a solution of the problem, but also its right-hand side that depends only on a spatial variable. A specific feature of the considered problems is that the system of eigenfunctions of the multiple differentiation operator subject to boundary conditions of the initial problem does not have the basis property.We prove the unique existence of a generalized solution to thementioned problem.     

A two-dimensional inverse problem for the viscoelasticity equation

For the integrodifferential equation that corresponds to the two-dimensional viscoelasticity problem, we study the problem of determining the density, the elasticity coefficient, and the spaceintegral term in the equation. We assume that the sought functions differ from the given constants only inside the unit disk D = {x ∈ ?² | |x| < 1}. As information for solving this inverse problem, we consider the one-parameter family of solutions to the integrodifferential equation corresponding to impulse sources localized on straight lines and, on the boundary of D, there are defined the traces of the solutions for some finite time interval. It is shown that the use of a comparatively small part of the given information about the kinematics and the elements of dynamics of the propagating waves makes it possible to reduce the problem under consideration to three consecutively and uniquely solvable inverse problems that together give a solution to the initial inverse problem.     

Boundary determination for an inverse problem in elastodynamics

We consider the unique determination of material properties of an inhomogeneous, isotropic, elastic object from measurements made at the surface. The behavior of the 3-dimensional object is modeled by solutions of the hear. hyperbolic system of equations for elastodynamics. The material properties of the object (its density and elastic properties) correspond to the (leading) coefficients of these diffrential equations. Modeling surface measurements by the Dirichlet-to-Neumann map on a finite time interval, we show here that the Dirichlet-to-Neumann map uniquely determines the density and elastic properties of the surface of the object. Solution of this problem leads to a proof in [R] of unique determination of certain properties(the wave speeds) of the interior of the object. Techniques used here include geometric optics. propagation of singularities for systems of real principal type, and Hamilton-Jacobi theory.     

A reconstruction method for a two-dimensional inverse eigenvalue problem

In this paper we describe a method for constructing approximate solutions of a two-dimensional inverse eigenvalue problem. Here we consider the problem of recovering a functionq(x, y) from the eigenvalues of — +q(x, y) on a rectangle with Dirichlet boundary conditions. The potentialq(x, y) is assumed to be symmetric with respect to the midlines of the rectangle. Our method is a generalization of an algorithm Hald presented for the construction of symmetric potentials in the one-dimensional inverse Sturm-Liouville problem. Using a projection method, the inverse spectral problem is reduced to an inverse eigenvalue problem for a matrix. We show that if the given eigenvalues are small perturbations of simple eigenvalues ofq=0, then the matrix problem has a solution. This solution is used to construct a functionq which has the same lowest eigenvalues as the unknownq, and several numerical examples are given to illustrate the methods.     

Quasi-one-dimensional method for the two-dimensional magnetotelluric sounding inverse problem

The two-dimensional magnetotelluric sounding inverse problem is solved by a quasi-one-dimensional method. A linearization method is proposed for one-dimensional inverse problems. The appearance of false conducting structures in the solution of the inverse problem is considered and some counter-measures are suggested. The quasi-one-dimensional method is applicable to many-dimensional magnetotelluric sounding inverse problems if the electrical conductivity is a priori known to vary slowly along the Earth’s surface. __________ Translated from Prikladnaya Matematika i Informatika, No. 22, pp. 5–11, 2005.     

The inverse coefficient problem of heat conduction for an isotropic body

We discuss a method for determining the complex of thermophysical characteristics of isotropic materials based on the solution of the two-dimensional nonstationary heat conduction problem for a layer subject to a narrow-band heating by a heat flow. We take account of the finite speed of thermal action of an annular heater.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 33, 1991, pp. 4–9.     

The singular sources method for an inverse problem with mixed boundary conditions

We use the singular sources method to detect the shape of the obstacle in a mixed boundary value problem. The basic idea of the method is based on the singular behavior of the scattered field of the incident point-sources on the boundary of the obstacle. Moreover we take advantage of the scattered field estimate by the backprojection operator. Also we give a uniqueness proof for the shape reconstruction.     

The truncation method for a two-dimensional nonhomogeneous backward heat problem

We consider the backward heat problem

     

A two-dimensional inverse problem for an integro-differential equation of electrodynamics

An integro-differential equation corresponding to a two-dimensional problem of electrodynamics with dispersion is considered. It is assumed that the electrodynamic properties of a nonconducting medium with a constant magnetic permeability and the external current are independent of the x ₃ coordinate. In this case, the third component of the electric field vector satisfies a second-order scalar integro-differential equation with a variable permittivity of the medium. For this equation, we study the problem of finding the spatial part of the kernel entering the integral term. This corresponds to finding the part of the permittivity that depends on the electromagnetic frequency. It is assumed that the permittivity support is contained in some compact domain Ω ? ?². To find this coefficient inside Ω, we use information on the solution of the corresponding direct problem on the boundary of Ω on a finite time interval. An estimate for the conditional stability of the solution of the inverse problem is established under the assumption that the time interval is sufficiently large.     

Quasi-one-dimensional method for the two-dimensional inverse problem of magnetotelluric sounding

The article presents a quasi-one-dimensional method for solving the inverse problem of electromagnetic sounding. The quasi-one-dimensional method is an iteration process that in each iteration solves a parametric one-dimensional inverse problem and a two-dimensional direct problem. The solution results of these problems are applied to update the input values for the parametric one-dimensional inverse problem in the next iteration. The method has been implemented for a two-dimensional inverse problem of magnetotelluric sounding in a quasi-layered medium.     

An inverse problem for a system with a moving boundary

The problem of estimating the right-hand side of a nonlinear parabolic equation is considered. A finite-step algorithm based on the model positional control method and the finite element method is proposed. The algorithm is robust to informational noise and computational errors. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 23–33.     

An inverse problem for infinitely divisible moving average random fields

Statistical Inference for Stochastic Processes - Given a low frequency sample of an infinitely divisible moving average random field $${int _{mathbb {R}^d} f(x-t)varLambda (dx); t in mathbb...     

On a radially symmetric inverse heat conduction problem

In this paper, we treat an inverse problem for a radially symmetric heat equation, which arises from non-destructive evaluation by thermal imaging. The problem can also be considered as an inverse heat conduction problem. Based on a weighted energy method, we give a conditional stability estimate. A feasible regularization method is provided for numerical simulation. The reconstruction experiment is done for verifying the efficiency of the regularization method.