scholarly journals On the influence of the initial data in a combustion problem

1980 ◽  
Vol 22 (2) ◽  
pp. 193-209 ◽  
Author(s):  
K. K. Tam
Keyword(s):  
Partial Differential Equations ◽  
Initial Data ◽  
Differential Equations ◽  
Temporal Evolution ◽  
Initial Conditions ◽  
Coupled System ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential ◽  
Approximate Expressions ◽  
Combustion Problem

AbstractThe combustion of a material can be modelled by two coupled parabolic partial differential equations for the temperature and concentration of the material. This paper deals with properties of the solution of these equations inside a cylinder or a sphere and under given initial conditions. Bounds for the variation of the temperature with the initial conditions are first established by considering a decoupled form of the equations. Then the coupled system is used to obtain approximate expressions for the temporal evolution of temperature and concentration.

scholarly journals Three-Dimensional Fourth-Order Time-Fractional Parabolic Partial Differential Equations and Their Analytical Solution

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Yesuf Obsie Mussa ◽  
Ademe Kebede Gizaw ◽  
Ayana Deressa Negassa
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fourth Order ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Variable Coefficients ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential ◽  
Transform Method ◽  
Differential Transform

In this study, the fractional reduced differential transform method (FRDTM) is employed to solve three-dimensional fourth-order time-fractional parabolic partial differential equations with variable coefficients. The fractional derivative used in this study is in the Caputo sense. A few important lemmas which are essential to solve the problems using the proposed method are proved. The novelty of this method is that it uses appropriate initial conditions and finds the solution to the problems without any discretization, linearization, perturbation, or any restrictive assumptions. Two numerical examples are considered in order to validate the efficiency and reliability of the method. Furthermore, the FRDTM solution when α = 1 is compared with other analytical methods available in the existing literature. Computational results are shown in tables and graphs. The obtained results revealed that the method is capable and simple to solve fractional partial differential equations. The software used for the calculations in this study is Mathematica 7.

scholarly journals Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations

2021 ◽  
Author(s):  
Wayne Isaac Tan Uy ◽  
Benjamin Peherstorfer
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Initial Conditions ◽  
High Dimensional ◽  
Parabolic Partial Differential Equations ◽  
Error Estimator ◽  
Dimensional System ◽  
Partial Differential ◽  
Dimensional Solution ◽  
Reduced Models

This work derives a residual-based \emph{a posteriori} error estimator for reduced models learned with non-intrusive model reduction from data of high-dimensional systems governed by linear parabolic partial differential equations with control inputs. It is shown that quantities that are necessary for the error estimator can be either obtained exactly as the solutions of least-squares problems in a non-intrusive way from data such as initial conditions, control inputs, and high-dimensional solution trajectories or bounded in a probabilistic sense. The computational procedure follows an offline/online decomposition. In the offline (training) phase, the high-dimensional system is judiciously solved in a black-box fashion to generate data and to set up the error estimator. In the online phase, the estimator is used to bound the error of the reduced-model predictions for new initial conditions and new control inputs without recourse to the high-dimensional system. Numerical results demonstrate the workflow of the proposed approach from data to reduced models to certified predictions.

scholarly journals Aspects of Hadamard well-posedness for classes of non-Lipschitz semilinear parabolic partial differential equations

2016 ◽  
Vol 146 (4) ◽  
pp. 777-832 ◽  
Author(s):  
J. C. Meyer ◽  
D. J. Needham
Keyword(s):  
Partial Differential Equations ◽  
Initial Data ◽  
Differential Equations ◽  
Continuous Dependence ◽  
Parabolic Partial Differential Equations ◽  
Well Posedness ◽  
Partial Differential ◽  
Hölder Continuous ◽  
Lipschitz Continuous ◽  
Semilinear Parabolic

We study classical solutions of the Cauchy problem for a class of non-Lipschitz semilinear parabolic partial differential equations in one spatial dimension with sufficiently smooth initial data. When the nonlinearity is Lipschitz continuous, results concerning existence, uniqueness and continuous dependence on initial data are well established (see, for example, the texts of Friedman and Smoller and, in the context of the present paper, see also Meyer), as are the associated results concerning Hadamard well-posedness. We consider the situations when the nonlinearity is Hölder continuous and when the nonlinearity is upper Lipschitz continuous. Finally, we consider the situation when the nonlinearity is both Hölder continuous and upper Lipschitz continuous. In each case we focus upon the question of existence, uniqueness and continuous dependence on initial data, and thus upon aspects of Hadamard well-posedness.

scholarly journals Ultrasonic Waves in Bubbly Liquids: An Analytic Approach

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1309
Author(s):  
P. R. Gordoa ◽  
A. Pickering
Keyword(s):  
Mathematical Model ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Acoustic Waves ◽  
Elliptic Functions ◽  
Coupled System ◽  
Ultrasonic Waves ◽  
Partial Differential ◽  
Special Cases ◽  
Spherical Bubbles

We consider the problem of the propagation of high-intensity acoustic waves in a bubble layer consisting of spherical bubbles of identical size with a uniform distribution. The mathematical model is a coupled system of partial differential equations for the acoustic pressure and the instantaneous radius of the bubbles consisting of the wave equation coupled with the Rayleigh–Plesset equation. We perform an analytic analysis based on the study of Lie symmetries for this system of equations, concentrating our attention on the traveling wave case. We then consider mappings of the resulting reductions onto equations defining elliptic functions, and special cases thereof, for example, solvable in terms of hyperbolic functions. In this way, we construct exact solutions of the system of partial differential equations under consideration. We believe this to be the first analytic study of this particular mathematical model.

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