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

learn more… | top users | synonyms

1
vote
1answer
17 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. ...
3
votes
2answers
47 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
21 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
22 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 ...
0
votes
0answers
16 views

Heat Transfer FEM 2D PDE Matlab [on hold]

I'd love to know if this looks right in any way since I'm unfamiliar with Heattransfer. The Domain is correct. The heat transfer coefficient is 1. The Dirichlet BC is u(0,x) = 1 and u(y,1) = 0 ...
1
vote
0answers
13 views

Heat Content Isoperimetric Inequality

This is a problem I have been working on and haven't really made any progress. Let $R(l, m)$ be a rectangle in $\mathbb{R}^2$ such that $l + m = p$ for $p$ some positive real number. Let $u(x, t)$ ...
0
votes
0answers
21 views

Bounds for the solution of heat equation using convolutions [closed]

We know that the solution of the heat equation $$u_t=u_{xx}$$ with initial condition $ u(0,x)=u_0(x)$ is given by $$u(t,x)=(u_o * H_t)(x)$$ where the heat kernel is given by ...
1
vote
0answers
79 views
+100

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 ...
2
votes
1answer
75 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
31 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
76 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
79 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
49 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
26 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
72 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
35 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
24 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
17 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
20 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
16 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
16 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
20 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
19 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
30 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 ...
0
votes
0answers
25 views

Heat Equation: Remains finite?

I am doing a PDE question. It's about heat equation, spherical coordinates (the usual stuff). The boundary condition is $\frac {\partial T}{\partial r} (1,t) = 0 $ and it also said for $T$ to remain ...
2
votes
0answers
48 views

PDE question: heat equation (third order??)

I am familiar with the usual heat equation, however, my lecturer gave me this problem and it does not look like anything I have ever seen (in my whole entire life and I am not just being dramatic). ...
3
votes
0answers
31 views

General setting of Varadhan's result for distance functions and heat kernels

For a senior project of mine, I would like to know what the most general setting of Varadhan's formula for the geodesic distance in terms of the limiting behavior of heat kernels is. The result I'm ...
1
vote
1answer
107 views

PDE Heat Equation with Variable Coefficient {Second ODE Variable Coefficient}

Another PDE question: If I have a non constant coefficients in my heat equation (PDE), how do I solve it? For example we have: $\frac {\partial T}{\partial t} =\frac {\partial ^2 T}{\partial r^2} + ...
1
vote
2answers
23 views

Time dependent parameter for heat equation

:\begin{cases} u_{t}=ku_{xx} & (x, t) \in [0, \infty) \times (0, \infty) \\ u(x,0)=g(x) & IC \\ u(0,t)=0 & BC \end{cases} What is the solution for $k$ as a function of $t$. I suspect it ...
2
votes
2answers
364 views

Heat Equation in spherical coordinates

Consider the problem of a sphere of material that starts at a non-uniform temperature, $T = r^{2}$ and is covered with insulation on the outer surface so that no heat gets out. We take the coordinate ...
1
vote
1answer
31 views

Long time behavior heat equation on infinite line

We know that a solution to the Cauchy problem on $\mathbb{R}$ : $u_{xx}=u_t$ with condition $u|_{t=0}=\varphi(x)$ is of the form $$u(x,t)=\dfrac{1}{2\sqrt{\pi ...
0
votes
1answer
19 views

Help solving Modified 1 dimensional heat equation

$u_t(t,x)=u_{xx}(t,x)-u(t,x)$ and $x\in(0,1), t>0$ with boundary and initial conditions $u(t,0)=0, u(t,1)=e-e^{-1}, u(0,x)=f(x)$. I tried using auxiliary function $v(t,x)=u(t,x)-(e-e^{-1})x$ but ...
1
vote
0answers
29 views

A physical model for pde

Do you know any mathematical model of a physical process such that it satisfies the following equations ? $$\mathrm{u_t=a(t)u_{xx}},\,\,\,0<x<1,\,\,\,t>0$$ ...
1
vote
0answers
25 views

Solving $du/dt=a \Delta u + b (\Delta u)^2$

Consider the function $u(\boldsymbol{x},t)$, where $u:\mathbb{R}^n \times \mathbb{R}_+ \rightarrow \mathbb{R}$ ($\mathbb{R}_+$ denotes nonnegative reals). My question is related to the PDE ...
0
votes
0answers
34 views

How do we deduce the condition for the solution?

Suppose that we have the differential equation $$u_t(x,t)=k^2u_{xx}(x,t), x \in (0,l), t>0$$ $$u(x,t): \text{ heat of rod at the position } x \ (0 \leq x \leq l )$$ If we have Dirichlet ...
0
votes
1answer
17 views

Verifying inhomogeneous solution of heat equation.

Here is my question. How (3.21) is induced? I tried several times but I could not get the answer.
0
votes
0answers
19 views

Conditions for always positive gradient of heat field in evolutionary thermo-elastic system

I am investigating stability and convergence of series of approximations for coupled thermoelasticity problem yielded by one-step recurrent time-integration scheme. I've managed to show that the ...
1
vote
0answers
63 views

solving PDE with state-dependent boundary conditions

I am interested in solving the following PDE (heat equation): $$\frac{\partial u}{\partial t} = \kappa \frac{\partial ^2 u}{\partial x^2}$$ In order to solve it, I discretize space uniformly into $N$ ...
0
votes
0answers
22 views

Two unknowns (heat transfer problem)

Hi I am attempting to work through a heat transfer problem and have reached the following formula, I am struggling to see how they have calculated J1 and J2 as seen below, any help would be ...
3
votes
1answer
61 views

Solution of $u_t=u_{xx}+xu_x$

I've been asked to solve the problem \begin{equation} \left\{\begin{array}{lc} u_t=u_{xx}+xu_x & \mbox{in }x\in\mathbb{R},t>0,\\ u(x,0)=g(x), & \mbox{on ...
0
votes
1answer
28 views

uniqueness heat equation

Consider the heat equation, $(1)$ $u_t=u_{xx}+f(x,t)$, $0<x<1$, $t>0$ $(2)$ $u(x,0)=\phi(x)$ $(3)$ $u(0,t)=g(t)$, $u(1,t)=h(t)$ When one wants to Show the uniqueness of solution of ...
0
votes
0answers
36 views

Finite Difference Method for Heat Equation with Neumann Boundary

I have read the book of Morton and Mayers. In chapter6, it said that the explicit finite difference scheme of a heat equation, $\frac{U^{n+1}_j-U^{n}_{j}}{\Delta ...
1
vote
0answers
20 views

Conservation of the $L^1$ norm in the Heat equation

I consider the homogeneous Heat equation in $\textbf{R}^d \times (0,T]$ for a fixed $T>0$ : \begin{cases} \partial_t f - \Delta f = 0 & \text{in } \textbf{R}^d \times (0,T] \\ f=f_0 & ...
0
votes
0answers
25 views

Parabolic Equation Solution Dependence on Coefficients

Given a heat equation of $u$ with spatial dependent diffusion coefficient $\alpha(x)\ge 0$ $$\frac{\partial u}{\partial t}=\Big(\frac{1}{2}\alpha_1^2\frac{\partial^2}{\partial ...
0
votes
1answer
62 views

Heat Equation Solution Dependence on Diffusion Coefficient

Given a heat equation of $u$ with spatial dependent diffusion coefficient $\alpha(x)\ge 0$ $$\frac{\partial u_\alpha}{\partial t}=\alpha(x) \frac{\partial^2 u_\alpha}{\partial x^2}$$ where $(t,x)\in ...
1
vote
0answers
29 views

Maximum Principle for the Derivatives of Parabolic PDE Solution

Is there a maximum principle for the spatial derivatives of the solution of a parabolic PDE with coefficients in front of the spatial partial derivatives depending on spatial variables only, similar ...
0
votes
1answer
35 views

Fourier series solution of the heat equation on $-2<x<2$

I have to solve the following boundary value problem: $u_t=u_{xx}$, $u(t,-2)=u(t,2)=0$ and $u(0,x)=f(x)$. I tried to solve the problem using the method of separation of variables. So assume ...
0
votes
1answer
48 views

Use energy method prove wave equation has a unique of solution.

Problem: Use energy method prove wave equation has a unique of solution. $\begin{align}U_{tt}+U_t-C^2U_{xx}=0 \\ U(a,t)=U(b,t)=0 \\ U(x,0)=f(x) \\ u_t(x,0)=g(x) \end{align} \quad \text{with} \quad ...
1
vote
0answers
20 views

uniform absolute convergence for function series

Consider the Theta function $\theta(x,t)=\sum\limits_{m=-\infty}^{\infty}K(x+2m,t)$, where $K(x,t)=\frac{e^{-\frac{x^2}{4t}}}{\sqrt{4 \pi t}}$. The $\theta$ function is seen in the solution of the ...
3
votes
1answer
40 views

Heat Equation on a disc

Question: The temperature distribution $u(r, θ, t)$ in a circular disc of unit radius $(0 ≤ r < 1, 0 ≤ θ < 2π)$ evolves according to the heat equation with unit thermal diffusivity $$u_t = ...