scholarly journals A Unified Analytical Approach to Fixed and Moving Boundary Problems for the Heat Equation

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 749
Author(s):  
Marianito R. Rodrigo ◽  
Ngamta Thamwattana
Keyword(s):  
Heat Equation ◽  
Initial Value Problem ◽  
Moving Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Moving Boundary Problems ◽  
Initial Value ◽  
Boundary Problems ◽  
Fixed Boundary Problems ◽  
Initial Boundary Value

Fixed and moving boundary problems for the one-dimensional heat equation are considered. A unified approach to solving such problems is proposed by embedding a given initial-boundary value problem into an appropriate initial value problem on the real line with arbitrary but given functions, whose solution is known. These arbitrary functions are determined by imposing that the solution of the initial value problem satisfies the given boundary conditions. Exact analytical solutions of some moving boundary problems that have not been previously obtained are provided. Moreover, examples of fixed boundary problems over semi-infinite and bounded intervals are given, thus providing an alternative approach to the usual methods of solution.

The initial value problem for a nonlinear semi-infinite string

1978 ◽  
Vol 82 (1-2) ◽  
pp. 19-26 ◽  
Author(s):  
R. W. Dickey
Keyword(s):  
Boundary Value Problem ◽  
Classical Solution ◽  
Initial Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Value ◽  
Galerkin Procedure ◽  
Initial Boundary Value ◽  
Infinite String

SynopsisThe existence of a classical solution to the initial boundary value problem for a semi-infinite extensible string is proved. The result is obtained by using a Galerkin procedure on a semi-infinite interval.

scholarly journals AN INTRODUCTION TO WELL-POSEDNESS AND FREE-EVOLUTION

2013 ◽  
Vol 28 (22n23) ◽  
pp. 1340015 ◽  
Author(s):  
DAVID HILDITCH
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Initial Value Problem ◽  
Boundary Value ◽  
Fourier Method ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Well Posedness ◽  
Initial Value ◽  
Initial Boundary Value

These lecture notes accompany two classes given at the NRHEP2 school. In the first lecture I introduce the basic concepts used for analyzing well-posedness, that is the existence of a unique solution depending continuously on given data, of evolution partial differential equations. I show how strong hyperbolicity guarantees well-posedness of the initial value problem. Symmetric hyperbolic systems are shown to render the initial boundary value problem well-posed with maximally dissipative boundary conditions. I discuss the Laplace–Fourier method for analyzing the initial boundary value problem. Finally, I state how these notions extend to systems that are first-order in time and second-order in space. In the second lecture I discuss the effect that the gauge freedom of electromagnetism has on the PDE status of the initial value problem. I focus on gauge choices, strong-hyperbolicity and the construction of constraint preserving boundary conditions. I show that strongly hyperbolic pure gauges can be used to build strongly hyperbolic formulations. I examine which of these formulations is additionally symmetric hyperbolic and finally demonstrate that the system can be made boundary stable.

Theorem on the existence of solutions of quasi-static moving boundary problems

1995 ◽  
Vol 6 (5) ◽  
pp. 529-538 ◽  
Author(s):  
Bart Klein Obbink
Keyword(s):  
Initial Value Problem ◽  
Moving Boundary ◽  
Existence Of Solutions ◽  
Functional Relation ◽  
Local Solution ◽  
Two Dimensional ◽  
Moving Boundary Problems ◽  
Initial Value ◽  
Boundary Problems ◽  
Iteration Technique

Using the theory of conformal mappings, we show that two-dimensional quasi-static moving boundary problems can be described by a non-linear Löwner-Kufarev equation and a functional relation ℱ between the shape of the boundary and the velocity at the boundary. Together with the initial data, this leads to an initial value problem. Assuming that ℱ satisfies certain conditions, we prove a theorem stating that this initial value problem has a local solution in time. The proof is based on some straightforward estimates on solutions of Löwner-Kufarev equations and an iteration technique.

scholarly journals Rapidly decreasing behaviour of solutions in nonlinear 3-D-thermoelasticity

1991 ◽  
Vol 43 (1) ◽  
pp. 89-99 ◽  
Author(s):  
Song Jiang
Keyword(s):  
Boundary Value Problem ◽  
Sobolev Spaces ◽  
Initial Value Problem ◽  
Weighted Sobolev Spaces ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Exterior Domains ◽  
Initial Value ◽  
Initial Boundary Value

In this paper we study the asymptotic behaviour, as |x| → ∞, of solutions to the initial value problem in nonlinear three-dimensional thermoelasticity in some weighted Sobolev spaces. We show that under some conditions, solutions decrease fast for each t as x tends to infinity. We also consider the possible extension of the method presented in this paper to the initial boundary value problem in exterior domains.

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.

scholarly journals On global dynamics of 2D convective Cahn–Hilliard equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaopeng Zhao
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Dynamics ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Initial Value ◽  
Hilliard Equation ◽  
Long Time ◽  
Cahn Hilliard Equation ◽  
Initial Boundary Value

AbstractIn this paper, we study the long time behavior of solution for the initial-boundary value problem of convective Cahn–Hilliard equation in a 2D case. We show that the equation has a global attractor in $H^{4}(\Omega )$ H 4 ( Ω ) when the initial value belongs to $H^{1}(\Omega )$ H 1 ( Ω ) .

scholarly journals SPECIAL SPLINES OF HYPERBOLIC TYPE FOR THE SOLUTIONS OF HEAT AND MASS TRANSFER 3-D PROBLEMS IN POROUS MULTI-LAYERED AXIAL SYMMETRY DOMAIN

2017 ◽  
Vol 22 (4) ◽  
pp. 425-440
Author(s):  
Harijs Kalis ◽  
Andris Buikis ◽  
Aivars Aboltins ◽  
Ilmars Kangro
Keyword(s):  
Differential Equations ◽  
Transfer Problem ◽  
Circular Cross Section ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Rate Of Change ◽  
Wood Block ◽  
Initial Value ◽  
Special Integral ◽  
Initial Boundary Value

In this paper we study the problem of the diffusion of one substance through the pores of a porous multi layered material which may absorb and immobilize some of the diffusing substances with the evolution or absorption of heat. As an example we consider circular cross section wood-block with two layers in the radial direction. We consider the transfer of heat process. We derive the system of two partial differential equations (PDEs) - one expressing the rate of change of concentration of water vapour in the air spaces and the other - the rate of change of temperature in every layer. The approximation of corresponding initial boundary value problem of the system of PDEs is based on the conservative averaging method (CAM) with special integral splines. This procedure allows reduce the 3-D axis-symmetrical transfer problem in multi-layered domain described by a system of PDEs to initial value problem for a system of ordinary differential equations (ODEs) of the first order.

Initial-boundary value problem for a fractional type degenerate heat equation

2018 ◽  
Vol 28 (06) ◽  
pp. 1199-1231
Author(s):  
Gerardo Huaroto ◽  
Wladimir Neves
Keyword(s):  
Heat Equation ◽  
Fractional Laplacian ◽  
Dirichlet Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Diffusion ◽  
Laplacian Operator ◽  
Initial Boundary Value ◽  
Fractional Laplacian Operator ◽  
Fractional Type

In this paper, we study a fractional type degenerate heat equation posed in bounded domains. We show the existence of solutions for measurable and bounded non-negative initial data, and homogeneous Dirichlet boundary condition. The nonlocal diffusion effect relies on an inverse of the [Formula: see text]-fractional Laplacian operator, and the solvability is proved for any [Formula: see text].

scholarly journals Initial value problem for fractional Volterra integrodifferential pseudo-parabolic equations

2021 ◽  
Author(s):  
Nguyen Duc Phuong ◽  
Nguyen Anh Tuan ◽  
Devendra Kumar ◽  
Nguyen Huy Tuan
Keyword(s):  
Mild Solution ◽  
Parabolic Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Value ◽  
Nonlinear Memory ◽  
Main Equation ◽  
Memory Source ◽  
Whole Space ◽  
Initial Boundary Value

In this paper, we investigate the initial boundary value problem for the Caputo time-fractional pseudo-parabolic equations with fractional Laplace  of order $ 0<\nu\le1 $ and the nonlinear memory source term. For $ 0<\nu<1 $, the Problem will be considered on a bounded domain of $ \R^d $. By some Sobolev embeddings and the properties of Mittag-Lefler function, we will give some results on the existence and the uniqueness of mild solution for the Problem \eqref{Main-Equation} below. When $ \nu=1 $, we will introduce some $ L^p-L^q $ estimates, and base on them we derive the global existence of a mild solution in the whole space $ \R^d. $

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