For questions related to the solution and analysis of the heat equation.

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10 views

Thomas algorithm coefficients for heat transfer problem

I make a numerical simulation for two-dimensional heat transfer problem with two different materials with boundary $x = x'$ and the heat source function. So, the problem formulae is $\rho c ...
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0answers
30 views

Solving Heat Equation with Laplace Transform

I am trying an alternative method to separation of variables to the following equation $$ \begin{cases} u_{xx} =4u_t , 0 < x < 2, t>0\\ u(0,t)=0, u(2,t)=0, t>0\\ u(x,0)=2\sin(\pi x), 0 ...
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0answers
34 views

Scaling Two Equations

I recently got set this problem and am having trouble scaling the resulting equations. Any help would be appreciated. An incompressible thermal conducting fluid is contained between two infinite ...
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16 views

Flow between two infinite horizontal plates

I recently got set this problem and I was wondering if anyone would be able to give me some hints/intuition on how to solve it. Thanks. An incompressible thermal conducting fluid is contained between ...
2
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0answers
20 views

Deriving a certain delta-sequence with respect to its index

At the end of some calculations I've reached $$\lim \limits _{t \to 0_+} \int \limits _{\Bbb R ^n} \frac {h(t,x,y)} t f(y) \Bbb d y$$ where $$h(t,x,y) = \frac {\Bbb e ^{\frac {\Bbb i |x-y|^2} ...
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1answer
20 views

Prove $\sup_x|u_x(x,t)|\le Ct^{-\frac{3}{4}}\|f\|_2$ for all $t>0$.

Let $u$ be a bounded solution to the heat equation $u_t-u_{xx}=0$ in $-\infty<x<\infty,t>0$ with $u(x,0)=f(x)$ with $f\in L^2(\Bbb R)$. Prove that there is a constant $C>0$, independent ...
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0answers
19 views

Diffusion equation in polar coordinates with non-zero boundary conditions (BC)

I'm trying to solve the diffusion equation in polar coordinates: $$c_t = \frac{D}{r^2}[2r\,c_r + r^2\,c_{rr}] = \frac{D}{r}[2\,c_r + r\,c_{rr}] \tag{1}$$ with the following BC: $$c(0,t)=0, \quad ...
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1answer
62 views

Show that $\lim_{t\to \infty}u(x,t)=\frac{A+B}{2}$, for each $x\in\Bbb R$.

Let $u(x,t)$ be a $C^2$ bounded solution of $$u_t(x,t)-u_{xx}(x,t)=0,x\in \Bbb R, u(x,0)=f(x)$$ where $f\in C(\Bbb R)$ satisfies: $\lim_{x\to+\infty}f(x)=A,\lim_{x\to-\infty}f(x)=B$. Show that ...
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30 views

integration by parts in 2 dimensions with heat equation pde

I'm working on some PDE problems and my biggest issue is vector calculus facts. Let $u \in C^2(\Omega)$, where $\Omega$ is some bounded subset of $R^2$ with smooth boundary such that $u_{t}-\Delta ...
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0answers
18 views

A stochastic variant of the heat equation modulo $2\pi$ has weird unstable particle-antiparticle solutions. Does this equation have a name?

I implemented a discretization of a weird 2D heat equation "mod $2\pi$", $$\dot{f}(\mathbf{x},t)=\Delta^*f(\mathbf{x},t)$$ where (WARNING: handwavy, I'm not sure I understand it) ...
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1answer
40 views

What theorem is this? (in PDE)

I'm confused because it's titled as "Gauss's theorem about heat flux" (not in English though, I'm translating), but instead of the heat equation there's Laplace's equation written above the theorem. ...
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0answers
15 views

Heat equation with $x\in [0,+\infty[$ and non-homogeneous initial and boundary condition

The IVBP that i need to solve is the follow: \begin{equation} \begin{cases} u_t=au_{xx} & x>0,t>0,a\in\mathbb{R}^+\\ u(x,0)=B_0e^{-kx}\cos(kx) & x\geq ...
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1answer
45 views

Help with a proof about heat equation

The question is Suppose $U=\Omega \times (0,T)$ where $\Omega \subseteq \Bbb{R}^n$ is a bounded domain. Let $u\in C_1^2(U)\bigcap C(\bar U)$ satisfy $u_t \le\Delta u + cu$ in $U$ where $c \le 0$ ...
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1answer
67 views

Heat equation — Modelling a real-life situation

I have read through a lot of books and lecture notes that cover the heat equation and I am still not sure how I would model the easiest real world situations. For example, take a rod at constant ...
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2answers
70 views

Do they have a mistake in this heat equation?

I need to know if there is a mistake in these notes: In the second page we have a representation of a function $f(x)$ as a $\sin$ series. Dont we need to have $f(0)=0=f'(l)$ for such a ...
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1answer
39 views

Heat equation with sin initial condition

How do i find the analytical solution of the heat equation: $$U_t = U_{xx} + \sin{\pi x}$$ subject to $u(0,t) = u(1,t) = 0$ and $u(x,0) = \sin(\pi x).$ I appreciate its a pretty common/general ...
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0answers
22 views

Reference For a PDE text that treats non homogenuous boundary conditions Rigorously

I am interested in reading a text or paper where elliptic and parabolic PDE's are discussed on bounded domains with non-homogeneous boundary conditions. I haven't been able to find anything in the ...
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1answer
25 views

Control of $L^{\infty}$ Norm of 3d Heat Equation Solution for $L^{3}$ Initial Data

Let $w_{t}$ denote the 3-dimensional heat kernel $$w_{t}(x)=(4\pi t)^{-3/2}e^{-\left|y\right|^{2}/(4t)},\qquad y\in\mathbb{R}^{3}, \ t > 0$$ Suppose $f\in L^{3}(\mathbb{R}^{3})$, and let ...
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1answer
63 views

Analytical solution to complex Heat Equation with Neumann boundary conditions and lateral heat loss

I have solved a PDE in this from numerically on Mathematica, but does anyone know if there is a way to solve the following PDE analytically, an analytical solution would really help me. This is an ...
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1answer
30 views

Stepping backwards with Forward Euler?

Let us say I want to use Forward Euler scheme to solve the heat equation $$ \frac{\partial u}{\partial t} = -\frac{\partial^2 u}{\partial x^2} $$ in the domain $t \in (0, 1)$, $x \in (0,1)$ but ...
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1answer
41 views

Maximum principle for the heat equation with Dirichlet conditions

Let us consider the Laplacian operator in a domain $\Omega\subset \mathbb{R}^n$, with Dirichlet boundary conditions. For all $f\in L^2(\Omega),$ we denote by $S(t)f$ the solution of the equation $$ ...
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0answers
28 views

Interpretation of a certain transform

I'm having troubles with understanding the physical meaning of a certain transform. If $u$ is a solution to the wave equation $$\partial_t^2u-\Delta u=0\ \mathrm{in}\ ...
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1answer
40 views

Is it a solution of Heat Equation?

I find in a Centrale's school document this solution for "short" time of the heat equation, I have not MAPLE or other calculus softwares, and I just want to be sure if my hand verification is correct ...
2
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1answer
31 views

Derive diffusion coefficient for heat equation from random walk simulation

I want to simulate the underlying stochastic process of diffusion on a microscopic level and compare the result with the solution of the heat equation. However, I'm not able to match the solution of ...
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0answers
26 views

Is it possible to construct a green function of the Dirichlet problem from the green function of the Cauchy problem?

For the heat equation. Is there a method to obtain the green function of the Dirichlet problem in a rectangular 2D domain from the green function of the Cauchy problem (infinite domain) PDE's?
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116 views

Heat equation asymptotic behaviour 2

Let $D$ be the domain defined as $D := \{ (x,t): t \in [0,1) , \; x < (1-t)^\alpha \}$. Let $u(x,t)$ satisfy the heat equation $u_t = \frac{1}{2}u_{xx}$ in $D$, with initial condition: ...
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1answer
43 views

Issues with solving PDE

It's been a while since I've had to solve the heat equation, and so I am having a slight issue. The question is as follows: A long, hollow, rigid tube, of length $L$ and constant cross section is ...
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1answer
25 views

General solution of laplace equation

I want to know that: What is the general solution of the (2 and 3 dimentional)Laplace equation $f_{xx}+ f_{yy}=0$ and $f_{xx}+ f_{yy} +f_{zz}=0$? With many thanks for your help.
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39 views

Heat equation with Neumann boundary conditions solution maximum/minimum

I have to show that, for the PDE: $$ \begin{cases} u_{t}-u_{xx}=0 & 0<x<\pi, 0<t< \infty\\ u_{x}(0,t)=0 & t\geq0 & (1) \\ u_{x}(\pi,t)=0 & t\geq 0 & (2) \\ ...
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1answer
32 views

Why is the general solution of this form?

I found the following in my lecture notes: $$u_t=u_{xx}, x \in \mathbb{R}, t>0 \\ u(x,0)=f(x)$$ $$u(x,t)=X(x)T(t)$$ $$\Rightarrow \frac{T'(t)}{T(t)}=\frac{X''(x)}{X(x)}=-\lambda \in \mathbb{R}$$ ...
2
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1answer
14 views

Derive solution of a heat equation.

Derive the following formula $u(t,x)={1 \over 3}t^3+{1 \over 2}t^2x^2$ for the one-dimensional non-homogeneous heat equation $u_t=u_{xx}+tx^2, t>0, x\in (-\infty,\infty)$ ...
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1answer
28 views

heat equation with a laplacian operator squared and power of 2

I've seen 2 variations of the laplacian operator fora heat equation. Some of which has a square on the laplacian operator and some of which does not have any power on the operator. What is the ...
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1answer
13 views

Laplace-Operator of distribution-valued function (heat equation)

I'm having trouble making sense of an exercise involving this definition of the heat equation: $u'(t) = \Delta (u(t))$, $u(0) = \delta_0$ for $t > 0$ where $u : [0, \infty) \rightarrow ...
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1answer
30 views

The initial condition for a heat equation with stationary solution subtracted

I am presented with the following question for exam revision: Heat is supplied at a prescribed rate $Q(x) > 0$ (per unit volume) to an isotropic conducting rod that occupies the region ...
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1answer
68 views

The goal of solving a Heat/Wave Equation

This is an example of a heat equation: $u_t=u_{xx}$, where $0<x<1$, $t>0$ $u_x(0,t)=u_x(1,t)=0$ and $u(0,t)=g(t)$ . The heat equation is a parabolic partial differential equation that ...
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0answers
22 views

Constructing a theoretical solution to a non-homogeneous Dirichlet problem from known solutions

To begin, let $\Omega\subset\Bbb R^n$ be whatever kind of domain we like, and let $$\begin{align}f&:\Omega\times(0,+\infty)\to\Bbb R \\ d &:\partial\Omega\to\Bbb R \\ g&:\Omega\to\Bbb ...
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0answers
60 views

Heat equation $u_t - \Delta u=0$

If I want to solve the heat equation on $\mathbb{R} \times (0,\infty)$ with the initial condition $g(x)=x^2$ and $g(x)=x$. Now, I actually did this by guessing that $u(x,t)=x^2+2t$ and $u(x,t)=x$ ...
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1answer
35 views

Heat Equation on $[0,l]$ with Neumann boundary conditions

I was reading the following pdf about the heat equation on an interval $[0,l]$ with Neumann conditions, http://texas.math.ttu.edu/~gilliam/fall03/m4354_f03/heat_N_web/heat_ex_homo_neum.pdf i.e. ...
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2answers
60 views

PDE: Heat equation problem

I'm trying this PDE: $$u_t = u_{xx} + g(x);\quad x\in[0,\pi]$$ With boundary conditions: $$u_x (0,t)=u_x(\pi,t)=0$$ And initial condition: $$u(x,0)=f(x)$$ I think variable separation proposing a ...
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1answer
29 views

Solution to the 1-D heat equation

In solutions to the heat equation $u_t(x,t)=cu_{xx}(x,t)$ I've seen they've used the set of boundary conditions $$u(0,t)=u(L,t)=0$$ $$u(x,0)=u_0(x)$$ These set of boundary conditions is set to model ...
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1answer
23 views

Incorporating the initial condition

I have solved the heat equation and have gotten to the stage of getting a general solution $$u(x,t)=x+\sum^\infty_{n=1} c_n \sin(\pi n x)e^{-\pi^2 n^2 t}$$ And I have the initial condition ...
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0answers
119 views

Behavior of a Solution to Heat equation Compactly Supported in Time

We say that a function $g\in C^{\infty}(\mathbb{R}^{n})$ is of Gevrey class $G^{\sigma}(\mathbb{R}^{n})$ if for each compact $K\subset\mathbb{R}^{n}$, there exist $C,R>0$ such that $$\sup_{x\in ...
3
votes
1answer
153 views

heat conduction problem

Question: Find the solution of the heat conduction problem $$ \begin{cases} U_{xx} =4U_t , 0 < x < 2, t>0\\ U(0,t)=0, U(2,t)=0, t>0\\ U(x,0)=2\sin(\frac\pi2x)-\sin(\pi x) + 4\sin(3\pi x), ...
0
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0answers
42 views

Nonlinear Differential Equation with Pure Neumann Boundary

Four governing equations concerning the reaction occurred in the porous electrode are \begin{equation} \nabla \cdot i_1 + \nabla \cdot i_2=0 \end{equation} \begin{equation} i_2 = -\kappa \nabla ...
4
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1answer
84 views

Can I combine the wave and heat equations?

I have this equation $$\frac{\partial^2u}{\partial x^2} = 2\frac{\partial u}{\partial t} + \frac{ \partial^2u}{\partial t^2}$$ Is it possible for me to use both the wave and heat equations to solve ...
4
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1answer
97 views

Backwards Heat Equation $ u_{t} = -\lambda^2 u_{xx}$

Problem Consider the backwards heat equation of the form $$ \left\{ \begin{aligned} u_{t} & = \lambda^2 u_{xx}, & x\in[0,L], \quad t\in[0,T]\\ u(0,t) &= u(L,t) = 0 \\ u(x,T) &= ...
2
votes
1answer
60 views

Uniqueness, symmetry, and energy behavior for the diffusion equation on an interval

Consider the boundary value problem (BVP) $u_t$ = $u_{xx}$ for $t>0$ and $x \in (0,1)$ with initial condition $$u(x,0) = \sin^4(2\pi x),\quad x \in(0,1) $$ and boundary conditions $u(0,t) = ...
0
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1answer
35 views

Inhomogeneous heat equation with source term orthogonality

This is a question on the lecture notes. Basically we have the usual heat equation: $$\frac{\partial y}{\partial t}(x,t)=k^2\frac{\partial^2 y}{\partial^2 x}(x,t)+F(x,t)$$ We also have the usual ...
4
votes
0answers
84 views

How to prove that this solution of heat equation is not a tempered distribution?

A theorem of PDEs sais that the following Cauchy problem for the heat equation \begin{align*} & \partial_t u = \partial_{x}^2 u, \quad (t,x) \in \mathbb{R_+} \times \mathbb{R}, \\ & u|_{ t = ...
2
votes
2answers
52 views

Change of variables for heat equation

How to make a change of variables to turn the equation $$\frac{\partial{u}}{\partial{t}}=D\frac{\partial^2{u}}{\partial{x}^2}+cu$$ back to the heat equation? Where can I read about change of ...