Methods of Moving Boundary Based on Artificial Boundary in Heat Conduction Direct Problem

2012 ◽  
Vol 490-495 ◽  
pp. 2282-2285
Author(s):  
Xue Yan Zhang ◽  
Qian Li ◽  
Da Quan Gu ◽  
Tai Ping Hou
Keyword(s):  
Neumann Boundary Condition ◽  
Direct Problem ◽  
Moving Boundary ◽  
Classical Problem ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Moving Boundary Problem ◽  
Artificial Boundary ◽  
Initial Boundary Value

The initial boundary value problem for parabolic equation with Neumann boundary condition is a kind of classical problem in partial differential equations. In this paper we use the artificial boundary to solve the moving boundary problem. Potential theory and difference method are discussed. Numerical results are given to support the proposed schemes and to give the compare of the two methods.

scholarly journals Global well-posedness in a chemotaxis system with oxygen consumption

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xujie Yang
Keyword(s):  
Smooth Boundary ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Liquid Environment ◽  
Bounded Convex Domain ◽  
Initial Boundary Value ◽  
Chemotaxis System ◽  
Bounded Convex ◽  
Uniformly Bounded

<p style='text-indent:20px;'>Motivated by the studies of the hydrodynamics of the tethered bacteria <i>Thiovulum majus</i> in a liquid environment, we consider the following chemotaxis system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{ \begin{split} &amp; n_t = \Delta n-\nabla\cdot\left(n\chi(c)\nabla{c}\right)+nc, &amp;x\in \Omega, t&gt;0, \ &amp; c_t = \Delta c-{\bf u}\cdot\nabla c-nc, &amp;x\in \Omega, t&gt;0\ \end{split} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>under homogeneous Neumann boundary conditions in a bounded convex domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb{R}^d(d\in\{2, 3\}) $\end{document}</tex-math></inline-formula> with smooth boundary. For any given fluid <inline-formula><tex-math id="M2">\begin{document}$ {\bf u} $\end{document}</tex-math></inline-formula>, it is proved that if <inline-formula><tex-math id="M3">\begin{document}$ d = 2 $\end{document}</tex-math></inline-formula>, the corresponding initial-boundary value problem admits a unique global classical solution which is uniformly bounded, while if <inline-formula><tex-math id="M4">\begin{document}$ d = 3 $\end{document}</tex-math></inline-formula>, such solution still exists under the additional condition that <inline-formula><tex-math id="M5">\begin{document}$ 0&lt;\chi\leq \frac{1}{16\|c(\cdot, 0)\|_{L^\infty(\Omega)}} $\end{document}</tex-math></inline-formula>.</p>

Stretching of viscous threads at low Reynolds numbers

2011 ◽  
Vol 683 ◽  
pp. 212-234 ◽  
Author(s):  
Jonathan J. Wylie ◽  
Huaxiong Huang ◽  
Robert M. Miura
Keyword(s):  
Asymptotic Expression ◽  
Classical Problem ◽  
Finite Extension ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Reynolds Numbers ◽  
Initial Surface ◽  
Initial Shape ◽  
Asymptotic Expressions ◽  
Initial Boundary Value

AbstractWe investigate the classical problem of the extension of an axisymmetric viscous thread by a fixed applied force with small initial inertia and small initial surface tension forces. We show that inertia is fundamental in controlling the dynamics of the stretching process. Under a long-wavelength approximation, we derive leading-order asymptotic expressions for the solution of the full initial-boundary value problem for arbitrary initial shape. If inertia is completely neglected, the total extension of the thread tends to infinity as the time of pinching is approached. On the other hand, the solution exhibits pinching with finite extension for any non-zero Reynolds number. The solution also has the property that inertia eventually must become important, and pinching must occur at the pulled end. In particular, pinching cannot occur in the interior as can happen when inertia is neglected. Moreover, we derive an asymptotic expression for the extension.

scholarly journals The critical Fujita number for a semilinear heat equation in exterior domains with homogeneous Neumann boundary values

2000 ◽  
Vol 130 (3) ◽  
pp. 591-602 ◽  
Author(s):  
H. A. Levine ◽  
Q. S. Zhang
Keyword(s):  
Boundary Value Problem ◽  
Global Solutions ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Boundary Values ◽  
Exterior Domains ◽  
Semilinear Heat Equation ◽  
Image Position ◽  
Initial Boundary Value

Let D be a domain in Rn with bounded complement and let n ≠ 2. For the initial-boundary value problem we prove that there are no non-trivial global (non-negative) solutions if 0 < n (p − 1) ≤ 2 and there exist both global non-trivial and non-global solutions if n (p − 1) > 2.

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.

scholarly journals THE ANALYTIC SOLUTION OF INITIAL BOUNDARY VALUE PROBLEM INCLUDING TIME FRACTIONAL DIFFUSION EQUATION

2020 ◽  
Vol 35 (1) ◽  
pp. 243
Author(s):  
Süleyman Çetinkaya ◽  
Ali Demir ◽  
Hülya Kodal Sevindir
Keyword(s):  
Boundary Value Problem ◽  
Analytic Solution ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
One Dimension ◽  
Sturm Liouville ◽  
Fractional Differential ◽  
Initial Boundary Value

The motivation of this study is to determine the analytic solution of initial boundary value problem including time fractional differential equation with Neumann boundary conditions in one dimension. By making use of seperation of variables, the solution is constructed in the form of a Fourier series with respect to the eigenfunctions of a corresponding Sturm-Liouville eigenvalue problem.

Boundedness and large time behavior in a quasilinear chemotaxis model for tumor invasion

2018 ◽  
Vol 28 (07) ◽  
pp. 1413-1451 ◽  
Author(s):  
Dan Li ◽  
Chunlai Mu ◽  
Pan Zheng
Keyword(s):  
Tumor Invasion ◽  
Large Time ◽  
System Modeling ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Large Time Behavior ◽  
Time Behavior ◽  
The Matrix ◽  
Initial Boundary Value

This paper deals with the quasilinear chemotaxis system modeling tumor invasion [Formula: see text] under homogenous Neumann boundary conditions in a smoothly convex bounded domain [Formula: see text] [Formula: see text], where [Formula: see text] is a given function satisfying [Formula: see text] for all [Formula: see text] with [Formula: see text] and [Formula: see text]. Here the matrix-valued function [Formula: see text] fulfills [Formula: see text] for all [Formula: see text] with some [Formula: see text] and [Formula: see text]. It is shown that for all reasonably regular initial data, a corresponding initial-boundary value problem for this system possesses a globally defined weak solution under some assumptions. Based on this boundedness property, it can finally be proved that in the large time limit, any such solution approaches the spatially homogenous equilibrium [Formula: see text] in an appropriate sense, where [Formula: see text], [Formula: see text] and [Formula: see text] provided that merely [Formula: see text] on [Formula: see text]. To the best of our knowledge, there are the first results on boundedness and asymptotic behavior of the system.

scholarly journals Inverse Problem for Source Function in Parabolic Equation at Neumann Boundary Conditions

2001 ◽  
Vol 14 (4) ◽  
pp. 445-451
Author(s):  
Victor K. Andreev ◽  
Keyword(s):  
Inverse Problem ◽  
Parabolic Equation ◽  
Source Function ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Integral Overdetermination Condition ◽  
Integral Overdetermination ◽  
Initial Boundary Value ◽  
Problem Data

The second initial-boundary value problem for a parabolic equation is under study. The term in the source function, depending only on time, is to be unknown. It is shown that in contrast to the standard Neumann problem, for the inverse problem with integral overdetermination condition the convergence of it nonstationary solution to the corresponding stationary one is possible for natural restrictions on the input problem data

scholarly journals A Numerical Schemefor the Probability Density of the First Hitting Time for Some Random Processes

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1907
Author(s):  
Jorge E. Macías-Díaz
Keyword(s):  
Stochastic Model ◽  
Moving Boundary ◽  
Numerical Study ◽  
Random Variable ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Structural Features ◽  
Stability And Convergence ◽  
General Stochastic Model ◽  
Initial Boundary Value

Departing from a general stochastic model for a moving boundary problem, we consider the density function of probability for the first passing time. It is well known that the distribution of this random variable satisfies a problem ruled by an advection–diffusion system for which very few solutions are known in exact form. The model considers also a deterministic source, and the coefficients of this equation are functions with sufficient regularity. A numerical scheme is designed to estimate the solutions of the initial-boundary-value problem. We prove rigorously that the numerical model is capable of preserving the main characteristics of the solutions of the stochastic model, that is, positivity, boundedness and monotonicity. The scheme has spatial symmetry, and it is theoretically analyzed for consistency, stability and convergence. Some numerical simulations are carried out in this work to assess the capability of the discrete model to preserve the main structural features of the solutions of the model. Moreover, a numerical study confirms the efficiency of the scheme, in agreement with the mathematical results obtained in this work.

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