Trusted frequency region of convergence for the enclosure method in thermal imaging

2017 ◽  
Vol 25 (1) ◽  
Author(s):  
Masaru Ikehata ◽  
Kiwoon Kwon
Keyword(s):  
Thermal Imaging ◽  
Space Dimension ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Frequency Region ◽  
Time Interval ◽  
Enclosure Method ◽  
Initial Boundary Value ◽  
Region Of Convergence ◽  
Large Frequency

AbstractThis study deals with the numerical implementation of a formula in the enclosure method as applied to a prototype inverse initial boundary value problem for thermal imaging in a one-space dimension. A precise error estimate of the formula is given and the effect on the discretization of the used integral of the measured data in the formula is studied. The formula requires a large frequency to converge; however, the number of time interval divisions grows exponentially as the frequency increases. Therefore, for a given number of divisions, we fixed the trusted frequency region of convergence with some given error bound. The trusted frequency region is computed theoretically using the theorems provided in this paper and is numerically implemented for various cases.

scholarly journals Global Existence and Uniqueness of Solutions for a Free Boundary Problem Modeling the Growth of Tumors with a Necrotic Core and a Time Delay in Process of Proliferation

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Shihe Xu ◽  
Minhai Huang
Keyword(s):  
Time Delays ◽  
Coupled System ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Time Interval ◽  
Unique Global Solution ◽  
Uniqueness Of Solutions ◽  
Global Existence And Uniqueness ◽  
A Free Boundary Problem ◽  
Initial Boundary Value

We study a mathematical model for the growth of necrotic tumors with time delays in proliferation. By transforming this problem into an initial-boundary value problem in fixed domain of a coupled system of a parabolic equation and one integrodifferential equation with time delays, in which all equations involve discontinuous terms, and using the approximation method combined with Schauder fixed point theorem, we prove that this problem has a unique global solution in any time interval[0,T].

Stability of discontinuous steady states in shearing motions of a non-Newtonian fluid

1990 ◽  
Vol 115 (1-2) ◽  
pp. 39-59 ◽  
Author(s):  
John A. Nohel ◽  
Robert L. Pego ◽  
Athanasios E. Tzavaras
Keyword(s):  
Shear Stress ◽  
Strain Rate ◽  
Newtonian Fluid ◽  
Space Dimension ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Steady States ◽  
Constitutive Law ◽  
Shear Strain Rate ◽  
Initial Boundary Value

SynopsisWe study the nonlinear stability of discontinuous steady states of a model initial-boundary value problem in one space dimension for incompressible, isothermal shear flow of a non-Newtonian fluid driven by a constant pressure gradient. The non-Newtonian contribution to the shear stress is assumed to satisfy a simple differential constitutive law. The key feature is a non-monotone relation between the total steady shear stress and shear strain-rate that results in steady states having, in general, discontinuities in the strain rate. We show that every solution tends to a steady state as t → ∞, and we identify steady states that are stable.

scholarly journals Nonexistence of Global Weak Solutions of Nonlinear Keldysh Type Equation with One Derivative Term

2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Kangqun Zhang
Keyword(s):  
Weak Solutions ◽  
Blow Up ◽  
Type Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Time Interval ◽  
Global Weak Solutions ◽  
Singular Initial Value Problem ◽  
Derivative Term ◽  
Initial Boundary Value

We focus on the nonexistence of global weak solutions of nonlinear Keldysh type equation with one derivative term. In terms of the analysis of the first Fourier coefficient, we show the solution of singular initial value problem and singular initial-boundary value problem of the nonlinear equation with positive initial data blow-up in some finite time interval.

The Rothe method for the wave equation in several space dimensions

1979 ◽  
Vol 84 (1-2) ◽  
pp. 1-18 ◽  
Author(s):  
Erich Martensen
Keyword(s):  
Wave Equation ◽  
Space Dimension ◽  
Continuous Solution ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
First Order ◽  
Rothe Method ◽  
Initial Boundary Value ◽  
The Given ◽  
Continuous Problem

SynopsisThe interior initial-boundary value problem for the wave equation in m ≧ 1 space dimension is considered for vanishing boundary values. Certain regularity, dependent on m, is required for the solution and additionalboundary conditions, the number of which being also dependent on m, are imposed on the given right hand side. Emphazising the case m = 3, the Rothe method is applied after the problem has been rewritten as a hyperbolic first order evolution problem for m + 1 unknown functions. The sequence of discrete solutions obtained is shown to be discretely convergent to the continuous solution in the sense of uniform convergence if the solution of the continuous problem is assumed to exist. A priori estimates are derived both for the discrete solutions and the continuous solution.

scholarly journals A viscosity solution of the Hamilton–Jacobi equation with exponential dependence of Hamiltonian on the momentum

2021 ◽  
pp. 273-276
Author(s):  
Lyubov Shagalova
Keyword(s):  
Viscosity Solution ◽  
Sufficient Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Generalized Solution ◽  
The State ◽  
Exponential Dependence ◽  
Time Interval ◽  
Momentum Variable ◽  
Initial Boundary Value

The initial – boundary value problem is considered for the Hamilton-Jacobi of evolutionary type in the case when the state space is one-dimensional. The Hamiltonian depends on the state and momentum variables, and the dependence on the momentum variable is exponential. The problem is considered on fixed bounded time interval, and the state variable changes from a given fixed value to infinity. The initial and boundary functions are subdifferentiable. It is proved that such a problem has a continuous generalized viscosity) solution. The representative formula is given for this solution. Sufficient conditions are indicated under which the generalized solution is unique. Hamilton-Jacobi equations with an exponential dependence on the momentum variable are atypical for theory, but such equations arise in practical problems, for example, in molecular genetics.

THE INITIAL BOUNDARY VALUE PROBLEM FOR THE FOUR-VELOCITY PLANE BROADWELL MODEL

1991 ◽  
Vol 01 (03) ◽  
pp. 293-310 ◽  
Author(s):  
GIUSEPPE TOSCANI ◽  
WŁODZIMIERZ WALUŚ
Keyword(s):  
Boundary Value Problem ◽  
Specular Reflection ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Time Interval ◽  
Step Method ◽  
Fractional Step Method ◽  
Uniqueness And Existence ◽  
Initial Boundary Value

The initial boundary value problem for the four-velocity Broadwell equations on a square with specular reflection on the boundary is considered. Uniqueness and existence of solutions on finite time interval is established. The length of the time interval depends on the initial data. The proof of existence uses the procedure known in numerical analysis as the fractional step method. It shows that the method is convergent justifying its usage in computations.

Convergence of Finite Fifference Method for Parabolic Problem

2003 ◽  
Vol 3 (1) ◽  
pp. 45-58 ◽  
Author(s):  
Dejan Bojović
Keyword(s):  
Convergence Rate ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Parabolic Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Rate Estimate ◽  
Convergence Rate Estimate ◽  
Initial Boundary Value

Abstract In this paper we consider the first initial boundary-value problem for the heat equation with variable coefficients in a domain (0; 1)x(0; 1)x(0; T]. We assume that the solution of the problem and the coefficients of the equation belong to the corresponding anisotropic Sobolev spaces. Convergence rate estimate which is consistent with the smoothness of the data is obtained.

ON THE SOLVABILITY OF A MIXED PROBLEM WITH DEGREE LAPLACE OPERATORS WITH NONLOCAL BOUNDARY CONDITIONS

2020 ◽  
pp. 85-104
Author(s):  
Shakirbai G. Kasimov ◽  
◽  
Mahkambek M. Babaev ◽  
◽  
Keyword(s):  
Boundary Conditions ◽  
Spectral Problem ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Boundary Conditions ◽  
Laplace Operators ◽  
Initial Boundary Problem ◽  
Initial Boundary Value ◽  
Delayed Time

The paper studies a problem with initial functions and boundary conditions for partial differential partial equations of fractional order in partial derivatives with a delayed time argument, with degree Laplace operators with spatial variables and nonlocal boundary conditions in Sobolev classes. The solution of the initial boundary-value problem is constructed as the series’ sum in the eigenfunction system of the multidimensional spectral problem. The eigenvalues are found for the spectral problem and the corresponding system of eigenfunctions is constructed. It is shown that the system of eigenfunctions is complete and forms a Riesz basis in the Sobolev subspace. Based on the completeness of the eigenfunctions system the uniqueness theorem for solving the problem is proved. In the Sobolev subspaces the existence of a regular solution to the stated initial-boundary problem is proved.

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