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

learn more… | top users | synonyms

0
votes
1answer
23 views

How can I use this initial condition for the heat equation

How can I use the following initial condition for a partial differential equation describing heat diffusion? $$f(x) = \begin{cases} 0, & 0<x<0.45 \\ 1, & 0.45<x<0.55 \\ 0, & 0....
0
votes
1answer
33 views

How to solve this nonhomogeneous heat equation

I don't know how to solve it $$ \left\lbrace \begin {array}{lcc} u_{t} \left( x,t \right) =u_{{\it xx}} \left( x,t \right) +2\,{{\rm e}^{-t}} \left( x-1+\sin \left( \pi\,x \right) \right) &0&...
6
votes
2answers
87 views

Why this abuse of notation correctly solves the heat equation

Here's a stupid method I observed to solve the heat equation in $\mathbb R^d$, \begin{align*} \partial_tu=\Delta u,\quad u|_{t=0}=u_0. \end{align*} Pretend that $\Delta$ is a constant so this just ...
0
votes
0answers
4 views

Heat Kernel in CD(k,N)/RCD(k,N) spaces

I'd like to know if some results are known concerning the heat kernel in CD(k,N) or RCD(k,N) spaces. I am mainly interested in two points: (gaussian) estimates asymptotic behavior when time goes to ...
0
votes
3answers
44 views

Solve the initial value problem for this inhomogeneous heat equation.

I'm trying to solve this IVP for heat equation, $$u_t-\frac{1}{4}u_{xx}=e^{-t}~~\text{ in }-\infty<x<\infty,~t>0,$$ $$u(x,0)=x^2.$$ By the superposition principle, the solution should equal ...
0
votes
0answers
36 views

Does anyone recognise this non-linear diffusion equation?

I'm doing some work on modelling cell migration, I've derived this particular form of a non-linear diffusion equation to describe the mean behaviour of a stochastic model I'm studying. I was wondering ...
1
vote
0answers
33 views

problem regarding exponential solution of heat equation?

Let the heat equation $$\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x_1^2}+ \frac{\partial^2 u}{\partial x_2^2}+ \frac{\partial^2 u}{\partial x_3^2}, \ t \geq 0 , \ x= (x_1, x_2, x_3)$...
1
vote
1answer
20 views

Analytical Solution of 3D Heat Equation - FDM

I'm writing a simple FDM algorithm for solving the well known 3D heat equation $$ \frac{\partial u}{\partial t} = \alpha \nabla^2 u + \frac{q}{c_p \rho} $$ where $q(x,y,z,t)$ represents the ...
0
votes
0answers
16 views

How do I plot this second order distribution in matlab?

http://imgur.com/ChrocF9 I'm trying to do a problem for heat transfer. I understand what's going on conceptually, but it wants me to do a 2D temperature plot in Matlab and I am stuck. The symmetry ...
0
votes
0answers
15 views

Solve $\frac{\partial u}{\partial t}-\Delta_x=f$

Let $f:\mathbb{R}^n\to\mathbb{R}$ be a smooth function with compact support. Put $y=(x,t)\in \mathbb{R}^n$, with $x\in \mathbb{R}^{n-1}$. Consider the following map: $$u(y)=\int\int \frac{f(z)e^{-i\...
2
votes
0answers
36 views

Maple: Symmetries of strange modified heat-equation

I thought about the following PDE (the $u(x=0,t)$ is not a boundary condition! It really is a part of the PDE): $$\dfrac{\partial u(x,t)}{\partial t}=\alpha \dfrac{\partial^2 u(x,t)}{\partial x^2}+u(...
0
votes
0answers
16 views

Intuition about heat equation with Neuman boundary data

On a bounded domain, consider the heat equation $u_t - \Delta u = 0$ with $\partial_\nu u = c$ (constant) and initial data $u_0$ which is non-negative. As usual $\nu$ is the outward normal vector. ...
0
votes
1answer
37 views

solving a partial differential equation (Damped heat equation)

I am trying to solve the below-mentioned PDE that represents a damped heat diffusion in one-dimensional space. I am using the separation of variables to solve it, however when I try to find the ...
3
votes
2answers
70 views

Temperature/heat equation

I solved this problem $$\left\{\begin{array}{ll} u_{t}=ku_{xx}, & x\in(0,1), t>0 \\ u(0,t)=2, u(1,t)=3, & t>0 \\ u(x,0)=x^{2}+x+2, & x\in(0,1) \end{array}\right.$$ and I got this $$u(...
0
votes
1answer
32 views

Nonhomogeneous heat equation [closed]

I really don't know how to start to solve it: $$\left\{\begin{array}{ll} u_{t}=ku_{xx}-\lambda^{2}u, & x\in(0,\ell), t>0 \\ u(0,t)=u(\ell,t)=0, & t>0 \\ u(x,0)=h(x), & x\in(0,\ell) \...
1
vote
1answer
35 views

Leibniz rule; Partial Differential Equations

I'm stuck on a question :| So far I have, i) $$I'(t)=\int^L_0 2 \frac{\partial v(x,t)}{\partial x} [v(x,t)] dt$$ ii) $$I(0)=\int^L_0 [v(x,0)]^2 dx= 0$$
1
vote
0answers
44 views

Why is the maximum principle not valid here?

Why is the maximum principle not valid here ? $u:\mathbf R^n_{+}\times (0,1]$ and $u(x,t)=1-\frac{1}{\left(4\pi (t-1/2)\right)^{n/2}}e^{\frac{-|x+1|^2}{4t-2}}$ then $u$ is bounded by $1$ and ...
0
votes
1answer
23 views

Solving Neumann problem with only one boundary condition.

I have to find the solution of the problem $$ u_t(t,x)=u_{xx}(t,x)+2e^{t-x} \text{ in set } R_+\times R_+ $$ With boundary conditions: $$ \begin{cases} u(0,x)=7\cos x &\text{ for } x>0\\ u_x(...
1
vote
1answer
37 views

Understanding the notation $\nabla u \otimes \nabla u$

On a Riemannian manifold $(M,g)$ let $u = u(t,x)$ the solution to the heat equation $\partial_t u = \frac 12 \Delta u$. The Laplace-Beltrami operator etc. are taken with respect to the metric $g$. I'...
2
votes
0answers
28 views

A comparison principle for a nonlinear parabolic PDE

We know the following comparison principle holds for the diffusion equation: Suppose that $u(x,t)$ and $v(x,t)$ satisfy \begin{equation} \begin{cases} u_t\ge \Delta u+F(x,t,u), \ \ &x\in\Omega,\ \...
0
votes
0answers
24 views

Green functions

do you know some litterature about green functions for the heat equation ? in particular for the non-linear equation : $\frac{\partial u(x,y,t)}{\partial t}-\frac{\partial^2 \left[ f(u(x,y,t))u(x,y,t)...
3
votes
1answer
41 views

System control of an induction heated system

Previous Question i understand i asked my previous question regarding this topic the wrong way. thou before i had time to respond the question was closed. On the other side i couldn't rephrase my ...
0
votes
0answers
27 views

Heat Equation 1D in Cylindrical Region

I have heat equation 1D in cylindrical coordinates: $$u_{\rho\rho}+\frac{1}{\rho}u_{\rho}=u_{t},\;0<\rho<1,t>0 $$ with boundary conditions $u(1,t)=0,\;t>0$ and initial condition $u(\rho,0)=...
1
vote
1answer
32 views

Prove that this Cauchy problem has at most one solution.

Problem: $$ \begin{cases} u_t = u_{xx} +5 u_{x} &\text{ in } R_+\times(0,1)\\ u(0,x)=g(x) &\text{ for } x\in(0,1)\\ u_x(t,0)=\alpha(t) &\text{ for } t>1\\ u_x(t,1)=\beta(t) &\text{ ...
0
votes
0answers
19 views

Recommended books heat equation

I am studying mathematics methods in physics. I want some calculation problems in heat equation just like this link https://www.math.ubc.ca/~peirce/HeatProblems.pdf Can you recommend me a problem ...
0
votes
0answers
7 views

Mixed convection - Matlab

So I am trying to solve the following mixed convection problem in Matlab: for $x=0: u=v=0, T=Th$ (heated wall); for $x=L: u=v=0, \frac{\partial T}{\partial x}=0$; for $y=0: u=v=0, \frac{partial T}{\...
1
vote
1answer
88 views

Is there a way to solve this heat equation IBVP

I have ran into the following Heat equation IBVP, but I am not quite sure how to solve it as it has these time dependent boundary conditions $$ v_t = kv_{xx} \ \ \ \ \ \ ( 0 \le x \le \infty, \ \ ...
0
votes
0answers
17 views

Explicit heat kernels

For quite general domains, the Dirichlet heat kernel has a representation via the eigenfunctions of the corresponding Dirichlet problem. This form is usually not easy to analyse so I was wondering - ...
1
vote
1answer
24 views

the heat equation for mappings between closed Riemannian manifolds

Let $M$ be a closed (smooth) Riemannian manifold. Then we have the following existence and uniqueness theorem for the heat equation on $M$, which is considered more or less a standard result: Let $0&...
1
vote
1answer
22 views

Heat Equation IBVP on the Quarter plane

I have come across the following Heat equation IBVP but I am not quite sure how to solve it: $$v_t = kv_{xx} \ \ \ \ \ \ ( 1 < x < \infty, \ \ 0 < t < \infty) ,$$ $$ v(x,0) = \delta (x ...
1
vote
0answers
22 views

Determine the equilibrium temprature [closed]

By solving the heat equation determine the equilibrium temperature distribution for the circular ring $\theta\in[0,2\pi]$ by both (a) directly setting $u_t=0$, and finding the equilibrium solution, ...
1
vote
0answers
19 views

computing the heat kernel for small times

The heat kernel on a two-dimensional manifold $M$ has the well-known expression $$H(p,q,t) = \sum_{i=1}^\infty e^{\lambda_i t}\phi_i(p)\phi_i(q)$$ where $\phi_i, \lambda_i$ are the eigenfunctions and ...
2
votes
0answers
22 views

PDE realated to heat equation with exponential additive term

I want to solve a PDE realated to heat equation with exponential additive term $${\partial u\over\partial t}={1\over x^2}{\partial\over\partial x}(x^2{\partial u\over\partial x})+e^u$$ I dervived ...
0
votes
1answer
15 views

Possible to expand a constant function as a series of sines without phase?

Is it possible to expand a function such as $f(x) = C_0$, $C_0$ being an arbitrary positive real number, between $x = 0$ and $x = L$ in the form $$\sum_{n} C_n\sin\left(\frac{n\pi x}{L}\right)$$ ...
0
votes
0answers
31 views

Seperation of variable Heat equation

Consider a copper bar of length $L = 100cm$ which is kept at the temperature $u = 0\space °C$ at one end, and is perfectly insulated at the other end. The bar is initially heated according to the ...
1
vote
1answer
31 views

Lie Algebra: Optimal system of one-dimensional sub-algebras of the heat equation

This is a follow up question to Invariants of a PDE by Lie Symmetries, as I tried to follow the reasoning from the book Applications of Lie Groups to Differential Equations (Peter J. Olver, Example 3....
1
vote
0answers
10 views

Green's function of laplascian in 3D.

It is about heat equation in 3D. How to prove, that $$w(M,M_0)=\frac{1}{\lambda r(M,M_0)}$$ is a solution of $$\lambda \nabla^2w(M,M_0)+4\pi \delta(M,M_0)=0$$ ? I understand that I should integrate ...
1
vote
1answer
39 views

Under the given conditions. Prove that $\lim_{t\to\infty} u(x, t) = 0 $, uniformly in x.

For any $(x, t)\in R^n × (0, +∞)$ let $ K(x, t) := (\frac{1}{4πt})^\frac{n}{2} e^-\frac{|x|^2}{4t} $ be fundamental solution of the heat equation (also called the heat kernel) and consider $u(x, t) = ...
3
votes
2answers
92 views

Error Estimates. L1 or L2 norm?

I simulate random walk on a divide difference grid to solve heat equation 1D. I want to prove numerically that this method has $Ν^{-1/2}$ error accuracy. My problem is that I don't know which norm ...
0
votes
1answer
15 views

Solving solution given initial condition condition

Suppose we know that: $$u_t=ku_{xx},~~~~~~~~0<x<l,~~~t>0$$ and $$u(x,t)=\sum_{i=0}^\infty[C_n~cos(n\pi x/l) ~e^{-w_nkt}]$$ where $w_n=\frac{n\pi}{l} ~~~ for~~n=1,2,3,...$ What if the ...
1
vote
1answer
28 views

heat equation conservation of energy

Suppose $u$ is a solution of the heat equation with the property that $|\int\limits_{-\infty}^{\infty}u(x,0)dx| < \infty$, and $u_{x}(x,t) \rightarrow 0$ as $x \rightarrow \pm \infty$. Then ...
1
vote
0answers
34 views

Computing a Green's function - where did I go wrong?

This is from a homework problem that was recently returned to me in a numerical analysis course. The grader even noted that he didn't know where I went wrong but the solution was marked as incorrect. ...
1
vote
0answers
14 views

Heat Equation stationary convergence

Consider the heat equation: $$u_t -\Delta u=0 \quad \text{in} \quad Q_T=\Omega \times(0,T) $$ $$u(\sigma,t)=0 $$ $$u(x,0)=g(x) \quad \text{in} \quad \Omega $$ a weak formulation is: find $u \in H^1(...
2
votes
1answer
42 views

calculate solution of heat equation with method of separation of variables

let the following problem $$ \begin{cases} \dfrac{\partial u}{\partial t}= \dfrac{\partial^2 u}{\partial x^2}, 0<x<1, t>0\\ u(0,t)=0, u(1,t)+ \dfrac{\partial u}{\partial x}(1,t)=0\\ u(x,0)=f(...
0
votes
0answers
15 views

Regularity of $u$ and $v$ satisfying $u_t - \Delta u = \Delta v - v_t$

I have that there are two functions $u$ and $v$ in $H^1(0,T;L^2)\cap L^2(0,T;H^2)$ satisfying weakly the equation on a bounded domain $\Omega$ $$u_t - \Delta u = \Delta v - v_t$$$ given some ...
0
votes
0answers
33 views

Is the solution to the heat equation always $C^k$, no matter what the boundary condition is?

Let $\Omega$ be a bounded Lipschitz domain (can be smooth if necessary). Consider the heat equation $$u_t - \Delta u = 0$$ $$u(0) = u_0$$ $$\text{some Robin boundary condition (B) for $u$}$$ We know ...
0
votes
1answer
33 views

Heat equation, maximum principle

For a solution of $v(x,t)$ in $C^2([-2,2]\times[0,T])$ of the heat eqution $v_t-v_{xx}=0$, if there is a point $(x_0,t_0)$ with $v_x(x_0,t_0)=v_t(x_0,t_0)=0$ and $v_{xx}(x_0,t_0)<0$ then $(x_0,t_0)\...
0
votes
1answer
12 views

How to find the differential of a series? And use it as a Substitution to solve the heat equation?

I have a question that says to solve the heat equation by substituting in $$\phi(x,t) = \frac{a_0(t)}{2} + \sum_{n=1}^{\infty} a_n(t)\cos(\frac{n\pi}{L} x)$$ I presume I must take the partial ...
1
vote
1answer
59 views

Heat equation - Evans

I have the following question. In Evan's PDE book it is stated (p 345, section 6.61) that if we take the differnential operator: $$ Lu=-\Delta u +cu $$ then there exists a $\mu>0$ such that for all ...
1
vote
1answer
65 views

Diffusion equation with advection and decay

I'm trying to solve the following initial value problem. $\begin{cases} u_t + u_x - u_{xx} &= -u, \quad \text{on} \quad \mathbb R \times \mathbb R_+\\ u(x,0) &= \frac{1}{4\pi} e^\frac{-x^2}{4}...