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

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1answer
25 views

Proving Continuity and equivalence

I have posted ths on the Quant Finance page as it is part of a QF problem but realised I may get a swifter response here! Iam working on a problem where I have successfully reduced a version of Black ...
1
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1answer
45 views

Complex Fourier Series and using the square norm

Find the complex Fourier series of $f(x)=e^{(-πx/2)}$ on $-π < x < π$ Discuss the significance of $|C_n|$ in the solution. I've tried so far Using the Complex Fourier Series: $$ %% ...
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0answers
27 views

Regularity of semilinear heat equation

I'm facing following regularity issue and i wonder if anyone of you guys is able to help me. I'd like to show that the solution of a semilinear heat equation is classical, i.e. $C^2$ in space and ...
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0answers
16 views

coupled heat transfer equation

I want to try to solve a strong coupling problem, I have a variable $\zeta as$ : \begin{equation} \zeta(x,y,T)=\frac{\frac{R(x,y,T)}{\sqrt{2}}-F(T)}{F(T)-E(T)} \end{equation} Where F(T) and E(T) are ...
7
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0answers
58 views

heat kernel on n-sphere

I'm interested in diffusion, a.k.a. the heat kernel driven by the Laplace-Beltrami operator, on the $n$-dimensional sphere. There are lots of bounds showing that, for small times, it behaves in a way ...
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0answers
43 views

Derivation of fundamental solution of heat equation by reduction to ODE - Question on integration factor

In the derivation of fundamental solution for heat equation ( as in PDE by L.Evans ), we come across the reduction to following ODE : $\alpha w + {1\over2}r w'+ w'' +{n-1\over{r}}w' = 0$ Set ...
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0answers
32 views

Solving partial differential equation with mathematica.

I am trying to solve the heat equation in cylindrical coordinate. The object of inspection is a thick ring with height $\{z_0,z_L\}$ and radii $\{r_1,r_2\}$ within a heat generating environment. I ...
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0answers
33 views

Solving PDE via Fourier Transform & Uniqueness

When a PDE is solved via Fourier transform, is there already a uniqueness assertion that comes for free? For example, if we Fourier the heat equation \begin{align} \partial_t u(x,t) &= \Delta ...
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0answers
73 views

Compact support

From PDE Evans, 2nd edition, page 204 Example 9 (Wave equation from the heat equation). Next we employ some Laplace transform ideas to provide a new derivation of the solution for the wave ...
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0answers
9 views

Branching Brownian Motion and KPP equation

I have troubles understanding the proof of the connection between BBM and KPP equation. I mean the proof of the next lemma from the lecture notes of Anton Bovier about BBM, link. This is almost whole ...
3
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2answers
38 views

Heat Equation derivative in terms of Laplace

If the heat equation is $ \frac{\partial u}{\partial t} - \alpha \nabla^2 u=0$ Is the second derivative of u w.r.t t is the laplacian of the lapacian?
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35 views

Solve this heat equation problem.

$$ u_t-f(t)u_{xx}=0 \textrm{, over } \mathbb{R}^N\times]0,\infty[ \\ u(x,0)=u_0(x)\textrm{, over } x\in \mathbb{R}^N, u_0\in S(\mathbb{R}^N) $$ where $u\in C^2(\mathbb{R}^N\times]0,\infty[)\cap ...
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0answers
12 views

Heat equation on a circular plate

I'm in trouble with the following problem: assuming a circular plate of radius $R$, the heat equation on it, is: $$\partial_t ...
2
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3answers
57 views

2D Heat Equation with special initial condition

I want to solve the 2 dimensional heat equation on a square $\Omega = \{ (x,y) : 0 < x < \pi, 0 < y < 2\pi \}$ with the Fourier Method \begin{align*} \partial_t u - \Delta u & = 0 ...
2
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1answer
31 views

Solving PDE with peridoc boundary conditions and a sine as initial condition

I need some help this exercise: I have to solve diffusion equation $u_t = Du_{xx}$ with periodic boundary conditions $u(t,-l) = u(t,l)$ and $u_x(t,-l) = u_x(t,l) $. The initial value should be ...
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0answers
19 views

heat equation with Interface Crank Nicolson

I am currently working on solving the heat equation with an interface numerically using Crank-Nicolson. There are jump discontinuities at the interface which are dealt with using fictitious values ...
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2answers
56 views

Heat equation with a disc

I'm faced with the problem of solving the Heat Equation on a two-dimensional disc: $$\frac{1}{\kappa} \frac{\partial T}{\partial t}=\Delta T$$ The boundary conditions in polar coordinates ...
1
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1answer
63 views

Heat transfer: boundary conditions with fluid velocity

The following equation is considered: $$ \frac{\partial u}{\partial t} - a\Delta u + \mathbf v \cdot \nabla u = f. $$ I have difficulties in formulating boundary conditions for this equation. If ...
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0answers
42 views

show than this PDE can be reduced to heat equation

How to reduce this PDE to heat equation $$x^2G_{xx}=G_t$$ ($G_{xx}$ is the 2nd order derivative on $x$, $G_t$ is the 1st derivative on $t$) We wish to obtain a form such that $G(x,t)=F(U(x,t))$, ...
0
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1answer
33 views

How to solve this heat equation with fourier method

Solve this via Fourier method: $$u_t-u_{xx}=0 \quad\quad 0< x<\pi, \quad t >0, $$ $$u(0,t)=u_x(\pi,t)=0, \quad\quad t \ge 0$$ $$u(x,0)=2\sin\left(\frac{3x}{2}\right) \, ...
4
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0answers
65 views

Convection-diffusion-reaction problem

I seek to solve to the system $$ \frac{\partial \phi_{a}}{\partial t} = D_{a} \frac{\partial^{2} \phi_{a}}{\partial x^{2}} - v_{a} \frac{\partial \phi_{a}}{\partial x} + \mathfrak{K}_{b}\phi_{b} ...
2
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1answer
52 views

How to show that $u(x,t)\le\pi^3-1+\sin(x)$ (heat equation)

Let $u(x,y)$ be a continuous solution of \begin{cases} u_t=u_{xx}+\sin(x) & 0<x<\pi,&t>0, \\[3ex] u(0,t)=u(\pi,t)=0 & t\ge0, \\[3ex] u(x,0)=4x(\pi-x)\quad&0\le x ...
0
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1answer
39 views

Fourier's Heat Law In Integral Form

I am having a little trouble with something. Here is a link (wikipedia article) to Fourier's Heat Law in integral form: http://en.wikipedia.org/wiki/Thermal_conduction#Integral_form What I am trying ...
3
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1answer
60 views

Behavior of solutions to the heat equation at infinity

I have read that for the solution $u$ of the heat equation $$u_t = u_{xx},$$ with $u(x,0)= a \exp(-bx^2)$ for some $a,b >0$, it holds $$\lim_{x \to \infty} u_x(x,t) = 0 = \lim_{x \to \infty} ...
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0answers
14 views

Maximum Principle - Proof

We want to show the maximum principle for a function $f = f(x,t)$ on a n-dimensional hypersurface $M,$ that is, (Corollary) Let $f = f(X,t)$ be a function on M, let $\vec{a}$ be a vector field on ...
0
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1answer
49 views

2d laplace equation with neumman boundary condition

$$\Delta u(x,y)=0$$ $$x,y\in(0,1),$$ $$\frac{\partial u(0,y)}{\partial x}=0,\quad \frac{\partial u(1,y)}{\partial x}=0,\quad\frac{\partial u(x,0)}{\partial y}=0,\quad\frac{\partial u(x,1)}{\partial ...
1
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1answer
49 views

Show this inequality in the “heat equation” problem.

Let $(u,t)$ the $C^2$ solution of the equation $$ u_t=u_{xx}+u, \textrm{ over } [0,a]\times[0,T]\subset \mathbb{R}^2 $$ where $T>0$ Show that $$ \max\limits_{[0,a]\times[0,T]} |u| ...
0
votes
1answer
25 views

Diffusion Equation on the Half Line

Consider the Heat equation and take the Dirichlet boundary condition : $$v_t - kv_{xx} = =0 \ \ \ \ \ \ ( 0 < x < \infty, \ \ 0 < t < \infty) ,$$ $$ v(x,0) = \phi (x) \ \ \ \ \ \ ...
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0answers
23 views

Heat equation on the Whole Line

Consider the problem $$u_t = ku_{xx} \ \ \ \ \ \ (-\infty < x < \infty , 0 < t < \infty) \ \ \ \ \ \ \ \ (1)$$ $$u(x,0) = \phi(x) \ \ \ \ \ \ (2) $$ I have to Show that : ...
0
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1answer
60 views

Fundamental Solution for 1d heat equation

So this question says to take $u(x,t) = v(x^2/t)$ to solve the 1d heat equation. That is, $$ u_t = u_{xx} $$ and it gives the general solution in the form $$ v(z) = c\int_{0}^z e^{-s/4}s^{-1/2} ds + ...
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2answers
31 views

Diffusion equation by seperation of variables

A uniform rod of length $l$ has an initial (at time $t = 0$) temperature distribution given by $u(x, 0) = \sin(\frac{\pi x}{l})$, for $0 \leq x \leq l$. The temperature $u(x, t)$ satisfies the ...
0
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1answer
42 views

The solution of the heat equation is unique

I haven't really understood the following proof that the solution of the heat equation is unique. Could you explain it to me? Heat equation with Dirichlet boundary conditions: ...
1
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1answer
36 views

Multi dimensional heat equation

Can someone please help me with the following problem? The heat flow equation is $\nabla^2 u = \frac{1}{\alpha^2}\frac{\partial u}{\partial t}$, where $u(x,y,z,t)$ is the temperature, $\alpha^2$ ...
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0answers
12 views

Unbounded nonhomogeneous 1 dimensional heat equation

I have this following PDE to solve for $u(x,t): x\in(-\infty,\infty), t\in(0,T)$ $$u_t+\frac{1}{2}u_{xx} = f(x)u$$ Where $f(x) = \gamma x^2$. I purposed the Boundary conditions: $$u(\infty, t) = ...
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0answers
24 views

Question about meaning of evolution problem.

Consider the following "evolution problem" $f(t) - u_t(t) \in \partial \psi(u(t))$ $u(0) = u_0$ Where $f:[0,T] \rightarrow H$ $ u:[0,T] \rightarrow H$ $ \psi:H \rightarrow (-\infty,\infty]$ is ...
2
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0answers
30 views

Good family of kernels in $\mathbb{R}^n$

I'm trying to prove that, given the heat equation $u_t = \Delta u$ with boundary values $u(x,0) = f(x)$, the solution given by $$u(x,t) = f \star H_t^{(d)}(x)$$ is continuous up to the boundary ...
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0answers
46 views

In Maple or Mathematica, syntax : Finding the fundamental sol. of the Heat eq., with Dirac-Delta initial condition

I use this question as a case example for later solving more complex equations of this type : When trying to find the fundamental solution of the Heat equation using Maple (software) , I get the ...
0
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1answer
27 views

uniform limit of integral of heat kernel

For any $(x,t)\in \Bbb R^n\times(0,+\infty)$. Let $$K(x,t)=\frac 1{{(4\pi t)}^{\frac n2}}e^{-\frac {|x|^2}{4t}}$$ be the heat kernel and consider $$u(x,t)=\int_{\Bbb R^n}K(x-y,t)u_0(y)dy$$. Suppose ...
0
votes
1answer
48 views

Direct computation for Heat Equation

Prove by direct computation $(1)$ $K(x,t)=t^{-\frac n2}$ $e^{-\frac {|x|^2}{4t}}$ the heat equation for $t\gt0.$ $(2)$ For any $\alpha \gt0$,$G(x,t)=(1-4\alpha t)^{-\frac n2} e^{\frac ...
0
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1answer
25 views

Linear Operator for Heat Equation

Show that if $u(x,t)$ satisfies $u_t = k \Delta u$ in a bounded region G, then for any $L>0$, $u(Lx, L^2t)$ solves the same equation for $x \in L^{-1}G$, where $L^{-1}G$ is the set of points ...
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2answers
67 views

Question about separation of variables

This is for the heat equation, where $$\frac{\partial U}{\partial t}-k \frac{\partial^2 U}{\partial x^2}=1$$ with the conditions $$U(0,t)=0, \; U(x,0)=0 \text{ and } \frac{\partial U}{\partial t} ...
0
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0answers
37 views

PDE Heat Equation Question: Finding T(x,t) with limited information.

Say our equation for temperature at position x and time t is shown by: $$ T(x)=T_0(1-x/a) $$ This equation holds for a rod of length a from x=0 to x=a. Initially T(0,t)=$T_0$ and T(a,t)=0. ...
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0answers
50 views

Finding an upper bound on the gradient of the solution to the heat equation

I have a function $u:\mathbb{R}^n\times [0,T]\to \mathbb{R}$ that solves the heat equation $u_t=\Delta u$, is bounded, and $u(x,0)=g(x)$. I need to show that $$\max|\nabla u(x,t)|\leq ...
1
vote
1answer
29 views

Chain or product rule for heat diffusion equation

A portion of the heat diffusion equation for a 1-D solid is given as: $$\frac{1}{r} \frac{\partial}{\partial r} \left(r \; k \frac{\partial T}{\partial r} \right)$$ Apparently this can be expanded ...
1
vote
1answer
77 views

Solution to Anisotropic Heat Equation

I am trying to find the solution to a 1-D anisotropic heat equation. The domain is a line segment of length L (i.e., it's a line segment extending from $x = 0$ to $x = L$). The form of the equation ...
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0answers
35 views

Converting this sum to integral (possible?). The goal is to get error function

The solution to the heat equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}$$ with boundary conditions and ini condition as followed: $$u(0\ or\ 1, t)=0\qquad u(x,0)=1$$ ...
0
votes
1answer
22 views

Existence of a subsequence and convergence to $0$ of function solving heat equation

Let $f^j(x)$ be a sequence of integrable functions on the circle such that $$\int_{-\pi}^{\pi}|f^j (x) |^2 dx = 1.$$ ALso, let $u^j(x,t)$ solve the heat equation on the circle with initial data ...
0
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1answer
37 views

How to get from heat equation with final condition to one with initial condition?

How do I get from the heat equation with end condition $$\frac{d}{dt}u(x,t) + \Delta u(x,t) = f(x,t)$$ $$u(x,T) = u_0(x)$$ where $t \in (0,T)$ and $x \in \Omega$, to a normal heat equation with ...
1
vote
1answer
33 views

Integral involving convolution with Poisson kernel.

Suppose that $f \in L^2(\mathbb{R}^n)$ and let $P_y(x)$ ($x \in \mathbb{R}^n$, $y > 0$) be the dilation of the Poisson kernel: $$P_y(x) = \frac{C_n y}{(y^2 + |x|^2)^\frac{n+1}{2}},$$ where $C_n$ ...
0
votes
1answer
99 views

Obtaining fundamental solution of the heat equation (1-d) through Laplace transform

A classic problem I'm having problems with (problem requires to use Laplace transform) $\frac{\partial ^2}{\partial x^2} u(x,t)=\frac{\partial}{\partial t} u(x,t) $ with conditions: ...