Separation of Variables for a Lattice Integrable System and the Inverse Problem

The inverse problem for the heat equation plays an important area of study and application. Up to now, the backward heat problem (BHP) in Cartesian coordinates has been arisen in many articles, but the BHP in different domains such as polar coordinates, cylindrical one or spherical one is rarely considered. This paper’s purpose is to investigate the BHP on a disk, especially, the problem is associated with the perturbed diffusivity and the space-dependent heat source. In order to solve the problem, the authors apply the separation of variables method, associated with the Bessel’s equation and Bessel’s expansion. Based on the exact solution, the regularized solution is constructed by using the modified quasi-boundary value method. As a result, a Holder type of convergence rate has been obtained. In addition, a numerical experiment is given to illustrate the flexibility and effectiveness of the used method.

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Retrieval of the piecewise-constant mass density in the Bergman wave equation

10.22541/au.163975810.07656742/v1 ◽

2021 ◽

Author(s):

armand wirgin

Keyword(s):

Inverse Problem ◽

Wave Equation ◽

Mass Density ◽

Perturbation Technique ◽

Separation Of Variables ◽

Scattering Problem ◽

Acoustic Scattering ◽

Density Contrast ◽

Homogeneous Fluid ◽

Domain Integral

This investigation is concerned with the 2D acoustic scattering problem of a plane wave propagating in a non-lossy, isotropic, homogeneous ﬂuid host and soliciting a linear, isotropic, macroscopically-homogeneous, generally-lossy, ﬂat-plane layer in which the mass density and wavespeed are diﬀerent from those of the host. The focus is on the inverse problem of the retrieval of the layer mass density. The data is the transmitted pressure ﬁeld, obtained by simulation (resolution of the forward problem) in exact, explicit form via the domain integral form of the Bergman wave equation. This solution is exact and more explicit in terms of the mass-density contrast (between the host and layer) than the classical solution obtained by separation of variables. A perturbation technique enables the solution (in its form obtained by the domain integral method) to be cast as a series of powers of the mass density contrast, the ﬁrst three terms of which are employed as the trial models in the treatment of the inverse problem. The aptitude of these models to retrieve the mass density contrast is demonstrated both theoretically and numerically.

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On Bäcklund transformations and boundary conditions associated with the quantum inverse problem for a discrete nonlinear integrable system and its connection to Baxter’s $Q$-operator

Advances in Theoretical and Mathematical Physics ◽

10.4310/atmp.2001.v5.n4.a2 ◽

2001 ◽

Vol 5 (4) ◽

pp. 679-696 ◽

Cited By ~ 1

Author(s):

A. Ghose Choudhury ◽

A. Roy Chowdhury

Keyword(s):

Inverse Problem ◽

Boundary Conditions ◽

Integrable System ◽

Bäcklund Transformations ◽

Quantum Inverse ◽

Backlund Transformations ◽

Nonlinear Integrable System

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On an Inverse Problem of Reconstructing a Heat Conduction Process from Nonlocal Data

Advances in Mathematical Physics ◽

10.1155/2018/8301656 ◽

2018 ◽

Vol 2018 ◽

pp. 1-8 ◽

Cited By ~ 4

Author(s):

Makhmud A. Sadybekov ◽

Gulnar Dildabek ◽

Marina B. Ivanova

Keyword(s):

Inverse Problem ◽

Space Variable ◽

Separation Of Variables ◽

Heat Propagation ◽

Final States ◽

One Dimensional ◽

Existence And Uniqueness Results ◽

Uniqueness Results ◽

The Given ◽

Heat Conduction Process

We consider an inverse problem for a one-dimensional heat equation with involution and with periodic boundary conditions with respect to a space variable. This problem simulates the process of heat propagation in a thin closed wire wrapped around a weakly permeable insulation. The inverse problem consists in the restoration (simultaneously with the solution) of an unknown right-hand side of the equation, which depends only on the spatial variable. The conditions for redefinition are initial and final states. Existence and uniqueness results for the given problem are obtained via the method of separation of variables.

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