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

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0answers
18 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 ...
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0answers
14 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 ...
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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 $$\sum_{n} C_n\sin\left(\frac{n\pi x}{L}\right)$$ ...
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0answers
30 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 ...
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1answer
24 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 ...
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0answers
7 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 ...
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1answer
37 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) = ...
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2answers
81 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 ...
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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 ...
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1answer
25 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 ...
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0answers
14 views

heat equation-calculate the temperature of an bar

let an bar of lenght 50 cm, and temperature on t=0 is 100 degree. The question is calculate the degree on the middle of the bar. So i try to write the heat equation: $\dfrac{\partial u}{\partial t}= ...
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0answers
32 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. ...
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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 ...
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1answer
40 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\\ ...
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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 ...
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0answers
32 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
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1answer
27 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 ...
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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 ...
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1answer
48 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 ...
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1answer
62 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} ...
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0answers
10 views

Strong maximum principle for heat equation

Let $M$ be a closed Riemannian manifold. If $u \in H^1(0,T;L^2) \cap L^2(0,T;H^1)$ is a weak solution of $$u_t - \Delta u = f$$ $$u(0) =u_0$$ where $f \in L^2(0,T;L^2)$ with $f(t,x) \geq 0$ a.e. and ...
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0answers
29 views

Approximating the Heat Equation

Let us assume that we want to approximate the solution of $\partial_t a = \partial_{xx} a$ which is subject to the Dirichlet boundary condition $a(-1,t) = a(1,t) = 0$, with $t \geq 0$, by considering ...
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1answer
26 views

Neumann Laplacian heat kernel or semigroup representation

I have the equation $$u_t - \Delta u = f\text{ on $\Omega$}$$ $$\partial_\nu u = g\text{ on $\partial\Omega$}$$ $$u(0) = u_0$$ for $f \in L^2(0,T;H^1)$, $g \in L^2(0,T;H^1(\partial\Omega))$ and $u_0 ...
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0answers
30 views

Question about the solution to the heat equation?

The question I am attempting to solve is to show that the solution to the heat equation of a rod of length $10$ with initial temperature distribution given by $u(x,0)=f(x)$ is $$\frac{a_0}{2} ...
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0answers
80 views

When does heat kernel exist?

I have a question about heat kernel. Definition Let $(X,\mu)$ be a $\sigma$-finite measure space and $L$ be a densely defined closed linear operator on real Hilber space $L^{2}(X)$ such ...
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1answer
28 views

$\eta_{\epsilon}*u$ satisfies heat equation

If $u(x,t)$ satisfies the heat equation then $\eta_{\epsilon}*u$ also satisfies it, with $\eta(x)=e^{\frac{1}{|x|^2-1}}$ for $|x|<1$ and $0$ else and ...
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0answers
12 views

Problem with the numerical PDE solving (possibly lattice-pinning)

I am solving quite complicated PDE's. The behavior was unexpected, and I started to simplify it. Finally I found out, that the problem is in the modified heat equation. The equation is: $$ ...
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1answer
129 views

Model for soil temperature underground using the Heat Equation

Assuming that the temperature in the ground is a function of time $t$ and depth $x$ only and assuming that at x=0, ground level, the approximate temperature at the surface is $$u(0,t) = ...
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3answers
126 views

Find $\lim_{t\to\infty}u(1,t)$, where $u(x,t)$ is a solution of $\frac{\partial u}{\partial t}-\frac{\partial^2u}{\partial x^2}=0$

Let $u(x,t)$ be a solution of $$\frac{\partial u}{\partial t}-\dfrac{\partial^{2}u}{\partial x^{2}}=0\text{ with}\\ u(x,0)=\frac{e^{2x}-1}{e^{2x}+1}.$$Then $\lim\limits_{t\to\infty} u(1,t)$ is equal ...
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0answers
17 views

Explicit solutions heat equation

Is there anyway to represent explicit the solution of the following fractional heat equation: $$\partial_{t}u(x,t) + f(x,t)(-\Delta)^{\frac{1}{2}} u(x,t)=0$$ with given initial data ...
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0answers
17 views

Heat and Wave equation - Green's function versus Fourier series?

I am learning how to solve the heat and wave equation in bounded domains in 1 and 2D as well as in $\mathbb{R}$ and $\mathbb{R}^2$. In the latter case I have learned the representation formulas i.e. ...
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0answers
16 views

Heat loss equation

I'm given this heat loss equation $\frac{\partial v}{\partial t} = \gamma \frac{\partial^2 v}{\partial x^2} - \alpha v$, where $\gamma, \alpha > 0$. I need to show that the general solution to the ...
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1answer
16 views

Well/Ill posedness on Generalized heat equation

Suppose we have the following one dimensional generalized heat equation: $$u_t(x,t)=g(x,t)\Delta u(x,t) \ \ \ x\in\mathbb{R},t\in(0,\infty)$$ I need to prove that this equation is ill posed, for ...
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1answer
40 views

Analytical Solution for Second Order Linear PDE

I'm trying to solve the following PDE (derived from convective heat transfer in a fluid flow between two parallel plates) analytically and I'm not sure what path to take. I don't think separation of ...
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0answers
38 views

Solving a modified numerical heat equation

I'm having a bit of trouble finding a good numerical form for this modified version of the heat/diffusion equation and I was just wondering if I am tackling this question the correct way. Firstly, I ...
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0answers
38 views

A simple assumption when deriving the fundamental solution of the heat equation

When deriving the fundamental solution in his book PDE (section 2.3.1, page 46), Evans comes to the equation $$r^{n-1}w'+\frac{1}{2}r^n w=a,$$ where $w:=w(y)=w(|y|)=w(r)$. After that he assumes that ...
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0answers
25 views

Solve the heat equation with green functions

i'm trying to solve this equation : $\frac{\partial u(t,x)}{\partial t} - \frac{\partial^2 u(t,x)}{\partial x^2} = 0 $ such as $u(t,x_1) = \delta(t)$ and what i would like to do is to express the ...
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1answer
33 views

Parabolic PDE with non-zero boundary conditions

I'm trying to solve the partial differential equation $\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}$ on the square $[0,\pi] \times [0,b[$ with initial conditions: $u(0,t) = 0$ ...
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1answer
30 views

Does exist an analytic solution for this PDE

i'm trying to solve this PDE : $ \frac{1}{g(x,y)}\frac{\partial g(x,y)}{\partial y} = d(x)\frac{1}{h(x,z)}\frac{\partial^2h(x,z)}{\partial z^2} $ Actually it's almost the heat equation but how do you ...
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1answer
39 views

Analyticity of solutions to the heat equation

Let us look at solutions to the linear heat equation on $\mathbb{R}$: $$ u_t = u_{xx}.$$ Is it true that solutions to the equation with nice enough initial datum are analytic after a certain time $T ...
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0answers
29 views

Neumann condition and Dirichlet condition at the same point

I am studying heat equation on a 1-D bar. We now that Neumann conditions at both ends leads to a singular matrix (for finite element methods) in equilibrium. Adding an initial condition can lead to ...
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2answers
48 views

Heat flow equation via Fourier Series

I know how to solve heat equations and wave equations defined on $\mathbb{R}^n\times(0,\infty)$ using Fourier transform. But I am having trouble solving similar equations defined on finite intervals ...
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0answers
152 views

Smoothness of solutions of the curve shortening flow given bounded curvature

I've been looking at the Lemma 1.5 of The Heat Equation Shrinks Embedded Plane Curves to Round Points (here), where Matthew Grayson proved that Suppose that $\kappa$ is bounded for $t\in[0,t_0)$. ...
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0answers
15 views

Piecewise nonhomogeneous PDE

The problem is $$u_{t}=u_{xx}+f(x) \\ u(0,t)=50 \\ u(\pi , t)=0 \\ u(x,0)=g(x)$$ $$0<x<\pi \\ t>0$$ $$f(x)=\begin{cases} 50 & 0<x<\frac{\pi}{2} \\ 0 & ...
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1answer
33 views

Prove that if $f$ is a solution of the heat equation.

Let $M^{k}\subset\mathbb{R}^{n}$ be a compact, oriented manifold, and assume that $f:M^{k}\times[0,\infty)\to\mathbb{R}$ is smooth. The heat equation is $$\triangle_{x}f(x,t)=\dfrac{\partial ...
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0answers
22 views

Boundedness of the solution of the integral equation associated to the heat kernel

(Cross-posting http://mathoverflow.net/questions/232720/boundedness-of-the-solution-of-the-integral-equation-associated-to-the-heat-kern ) Let $\Omega$ be a bounded open set of $\mathbb R^n$ ($n\geq ...
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0answers
107 views

Applying Fourier cosine transfer method with this Neumann boundary condition $\frac{dh(0,t)}{dx}=\frac{1}{a}(g(0,t)-h(x,t)) $ to solve heat equation!

Consider heat equation $$\frac{\partial h}{\partial t}= \text{D}\frac{\partial^{2} h}{\partial x^{2}}$$ I want to solve heat\diffusion equation with this Neumann boundary condition ...
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1answer
31 views

$L^p-L^q$ estimates for heat equation - regularizing effect

Where can I find a proof of the following estimate $$\|S(t)v\|_{L^p(\Omega)}\leq C t^{-\frac{N}{2}\left(\frac{1}{q}-\frac{1}{p}\right)}\|v\|_{L^q(\Omega)}, $$ where $1\leq p<q<+\infty$, ...
3
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0answers
44 views

Ill-posedness and well-posedness

Why is the backwards heat equation an ill-posed problem? $$\frac{∂u}{∂t}=-k\frac{∂^2u}{∂x^2}$$. And what makes this heat conduction equation $$\frac{∂u}{∂t}=k\frac{∂^2u}{∂x^2}$$ well-posed?
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0answers
45 views

Solution of partial differential equation - modified heat equation

I want to solve the "modified" heat equation $$ \frac{\partial y}{\partial t}=a\frac{\partial^2 y}{\partial x^2} +b\frac{\partial y}{\partial x} +cy+d $$ I assumed that a, b, c and d are all constant ...