scholarly journals On the inverse problem for a heat-like equation

1987 ◽  
Vol 1 (2) ◽  
pp. 81-97 ◽  
Author(s):  
Igor Malyshev
Keyword(s):  
Inverse Problem ◽  
Boundary Data ◽  
Mixed Type ◽  
Parabolic Type ◽  
Time Dependent ◽  
Nonlinear Volterra Integral Equation ◽  
Parabolic Case ◽  
Special Cases ◽  
Uniqueness Of The Solution ◽  
Explicit Formulae

Using the integral representation of the solution of the boundary value problem for the equation with one time-dependent coefficient at the highest space-derivative three inverse problems are solved. Depending on the property of the coefficient we consider cases when the equation is of the parabolic type and two special cases of the degenerate/mixed type. In the parabolic case the corresponding inverse problem is reduced to the nonlinear Volterra integral equation for which the uniqueness of the solution is proved. For the special cases explicit formulae are derived. Both “minimal” and overspecified boundary data are considered.

An inverse problem of determining the time-dependent potential in a higher-order Boussinesq-Love equation from boundary data

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Mousa Huntul ◽  
Mohammad Tamsir ◽  
Abdullah Ahmadini
Keyword(s):  
Inverse Problem ◽  
Boundary Data ◽  
Numerical Solutions ◽  
Physical Property ◽  
Neumann Boundary ◽  
Higher Order ◽  
Time Dependent ◽  
Potential Coefficient ◽  
Content Type ◽  
Dependent Potential

PurposeThe paper aims to numerically solve the inverse problem of determining the time-dependent potential coefficient along with the temperature in a higher-order Boussinesq-Love equation (BLE) with initial and Neumann boundary conditions supplemented by boundary data, for the first time.Design/methodology/approachFrom the literature, the authors already know that this inverse problem has a unique solution. However, the problem is still ill-posed by being unstable to noise in the input data. For the numerical realization, the authors apply the generalized finite difference method (GFDM) for solving the BLE along with the Tikhonov regularization to find stable and accurate numerical solutions. The regularized nonlinear minimization is performed using the MATLAB subroutine lsqnonlin. The stability analysis of solution of the BLE is proved using the von Neumann method.FindingsThe present numerical results demonstrate that obtained solutions are stable and accurate.Practical implicationsSince noisy data are inverted, the study models real situations in which practical measurements are inherently contaminated with noise.Originality/valueThe knowledge of this physical property coefficient is very important in various areas of human activity such as seismology, mineral exploration, biology, medicine, quality control of industrial products, etc. The originality lies in the insight gained by performing the numerical simulations of inversion to find the potential co-efficient on time in the BLE from noisy measurement.

Asymptotic expansion of solutions to an inverse problem of parabolic type with non-homogeneous boundary conditions

2011 ◽  
Vol 141 (4) ◽  
pp. 777-817 ◽  
Author(s):  
Davide Guidetti
Keyword(s):  
Inverse Problem ◽  
Asymptotic Expansion ◽  
Boundary Conditions ◽  
General Result ◽  
Parabolic Problem ◽  
Source Function ◽  
Parabolic Type ◽  
Time Dependent ◽  
Asymptotic Expansion Of Solutions ◽  
Inverse Parabolic Problem

We consider an inverse parabolic problem of reconstruction of the source function, together with the traditional solution. In contrast with older literature, we consider non-homogeneous and time-dependent boundary conditions. We are able to prove a general result of convergence to a stationary state, and of asymptotic expansion as t → ∞.

scholarly journals UNIFORM BMO ESTIMATE OF PARABOLIC EQUATIONS AND GLOBAL WELL-POSEDNESS OF THE THERMISTOR PROBLEM

2015 ◽  
Vol 3 ◽  
Author(s):  
BUYANG LI ◽  
CHAOXIA YANG
Keyword(s):  
Parabolic Equations ◽  
Elliptic Equations ◽  
Boundary Data ◽  
Time Dependent ◽  
Well Posedness ◽  
Global Classical Solution ◽  
Thermistor Problem ◽  
Degenerate Elliptic ◽  
Uniqueness Of The Solution ◽  
Temperature Equation

We prove global well-posedness of the time-dependent degenerate thermistor problem by establishing a uniform-in-time bounded mean ocsillation (BMO) estimate of inhomogeneous parabolic equations. Applying this estimate to the temperature equation, we derive a BMO bound of the temperature uniform with respect to time, which implies that the electric conductivity is an $A_{2}$ weight. The Hölder continuity of the electric potential is then proved by applying the De Giorgi–Nash–Moser estimate for degenerate elliptic equations with an $A_{2}$ coefficient. The uniqueness of the solution is proved based on the established regularity of the weak solution. Our results also imply the existence of a global classical solution when the initial and boundary data are smooth.

On the Theory of Thermally Weighted Electron-Phonon Transition Probabilities

1980 ◽  
Vol 35 (9) ◽  
pp. 902-914
Author(s):  
J. Schupfner
Keyword(s):  
Transition Probabilities ◽  
Algebraic Method ◽  
Calculation Technique ◽  
Radiative Transitions ◽  
Special Cases ◽  
Explicit Formulae ◽  
The Common ◽  
Common Transition ◽  
Franck Condon ◽  
Refined Calculation

Abstract We present a refined calculation method for the phonon part (Franck-Condon Overlaps) of the transition probabilities of electron-phonon radiative and non-radiative transitions in crystals. The evaluation of the thermal averaged Franck-Condon integrals is a purely algebraic method and the transition probabilities we use are derived from first principles and completely atomistic. For the electronic transitions we take into account the frequency shift of the lattice and the change of the phonon normal coordinates. Explicit formulae of the phonon parts are derived and it is shown that the common transition probabilities used in literature are special cases of our functional calculation technique.

Lipschitz stability in an inverse problem for a hyperbolic equation with a finite set of boundary data

2008 ◽  
Vol 87 (10-11) ◽  
pp. 1105-1119 ◽  
Author(s):  
M. Bellassoued ◽  
D. Jellali ◽  
M. Yamamoto
Keyword(s):  
Inverse Problem ◽  
Hyperbolic Equation ◽  
Boundary Data ◽  
Lipschitz Stability ◽  
Finite Set

SENSITIVITY ANALYSIS AND IDENTIFIABILITY OF A MATHEMATICAL MODEL OF A CHEMICAL REACTION

2021 ◽  
pp. 14-18
Author(s):  
Л.Ф. Сафиуллина
Keyword(s):  
Mathematical Model ◽  
Inverse Problem ◽  
Sensitivity Analysis ◽  
Chemical Reaction ◽  
Reaction Model ◽  
Numerical Calculations ◽  
Specific Reaction ◽  
Uniqueness Of The Solution ◽  
The Mathematical Model ◽  
Kinetics Of

В статье рассмотрен вопрос идентифицируемости математической модели кинетики химической реакции. В процессе решения обратной задачи по оценке параметров модели, характеризующих процесс, нередко возникает вопрос неединственности решения. На примере конкретной реакции продемонстрирована необходимость проводить анализ идентифицируемости модели перед проведением численных расчетов по определению параметров модели химической реакции. The identifiability of the mathematical model of the kinetics of a chemical reaction is investigated in the article. In the process of solving the inverse problem of estimating the parameters of the model, the question arises of the non-uniqueness of the solution. On the example of a specific reaction, the need to analyze the identifiability of the model before carrying out numerical calculations to determine the parameters of the reaction model was demonstrated.

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