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

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2
votes
0answers
15 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) ...
1
vote
1answer
36 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. ...
0
votes
0answers
14 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 ...
3
votes
1answer
42 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 $c \le 0$ ...
1
vote
1answer
58 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 ...
1
vote
2answers
67 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 ...
0
votes
1answer
38 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 ...
-2
votes
0answers
11 views

1D Heat equation with “general” BC.

I'm trying to solve the equation $ku_{xx}=u_t$ with boundary conditions: $u(0,t)= \alpha(t)$ $u_x(L,t)=\beta(t)$ General Initial condition $u(x,0)=f(x)$. I really dont know how to start. I guess ...
1
vote
0answers
21 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 ...
0
votes
1answer
24 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 ...
0
votes
1answer
59 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 ...
0
votes
1answer
28 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 ...
1
vote
1answer
31 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 $$ ...
3
votes
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}\ ...
0
votes
1answer
37 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
votes
1answer
24 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 ...
0
votes
0answers
22 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?
7
votes
0answers
109 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: ...
1
vote
1answer
40 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 ...
1
vote
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.
0
votes
0answers
35 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) \\ ...
2
votes
1answer
31 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
votes
1answer
12 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 $u_t=u_{xx}+tx^2, t>0, x\in (-\infty,\infty)$ ...
0
votes
1answer
20 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 ...
0
votes
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 ...
0
votes
1answer
29 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 ...
-1
votes
1answer
65 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 ...
0
votes
0answers
20 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 ...
0
votes
0answers
56 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 $u(x,t)=x$ ...
1
vote
1answer
29 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. ...
4
votes
2answers
57 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 ...
1
vote
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 ...
1
vote
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 ...
2
votes
0answers
117 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
111 views

heat conduction problem

Find the solution of the heat conduction problem $U_{xx} =4U_t , 0 < x < 2, t>0;$ $U(0,t)=0, U(2,t)=0, t>0$; $U(x,0)=2\sin(\frac\pi2x)-\sin(\pi x) + 4\sin(3\pi x), 0 \le x \le 2 $ ok ...
0
votes
0answers
36 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
votes
1answer
82 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
votes
1answer
87 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
58 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
votes
1answer
32 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
82 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
48 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 ...
1
vote
0answers
33 views

Heat equation with heat source in form of delta function

Consider the problem \begin{equation} \left\{\begin{array}{cc}u_t-u_{xx}=\delta_0,&0<x<1,\ t>0\\ u_x(0,t)=u_x(1,t)=0,&t>0,\\ u(x,0)=0,& 0\leq x\leq 1.\end{array}\right. ...
0
votes
0answers
22 views

Heat equation boundedness of solutions

Consider $$u_t-u_{xx}=1,\ 0\leq x \leq 1,\ t>0,$$ with zero Dirichlet boundary conditions and vanishing initial conditions. The solution to the stationary problem, i.e. $$u^s_{xx}=1,\ 0\leq x \leq ...
0
votes
0answers
26 views

Question about heat equations?

I have to solve the heat equation with $u(0,t) = 0$, the end at $x=2$ insulated $\forall t \gt 0$ and initial condition $u(x,0)= 20\sin\frac{\pi x}{4}$. I interpreted the 2nd b.c to mean that ...
1
vote
0answers
17 views

Discrete maximum principle

\begin{align} \partial_t u(x,t) & = \kappa \partial_{xx}u(x,t), & -1 < x < 1, & \quad t>0 \quad \kappa > 0 \nonumber\\ u(-1,t) & = g_1(t) & t>0 & \\ u(1,t) ...
1
vote
0answers
19 views

Heat Equation and Composition of Functions

Let $u$ be a solution to the heat equation in a domain $U \times [0,T]$. Let $f$ be a $C^2$ function on the closure of $U \times [0,T]$. Assume that $$f = |\nabla f|=0 \text{ on } \partial U \times ...
1
vote
0answers
22 views

Heat on a wedge

I'm trying to solve the PDE given by $$\begin{array}\ u_t = \nabla^2 u, & 0 \lt r \lt 1, & 0 \lt \theta \lt \alpha \lt 2\pi\end{array} \\ u(r,\theta, 0) = f(\theta) \\ u(1,\theta,t) = u(r,0,t) ...
1
vote
0answers
22 views

A PDE question using variation of parameters

Variation of parameters: Consider IBVP \begin{align} u_t − u_{xx} = f(x, t) \qquad & \text{on } \Omega = (0, \pi) \times \Bbb R^+\\ u(x, 0) = \varphi(x) \qquad & \text{on } (0, \pi)\\ ...
1
vote
2answers
32 views

How fast does this function converge to zero?

Consider the function given by $$f:(0,\infty)\rightarrow \mathbb{R}, t\mapsto \int\limits_{-\delta}^{\delta}x^{2k}\frac{1}{\sqrt{4\pi t}}e^{-\frac{x^2}{4t}}dx,$$ where $k\in\mathbb{N}$ and $\delta ...