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

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1answer
36 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 ...
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1answer
22 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
44 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
61 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
8 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
28 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
23 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
28 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
78 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
27 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
11 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: $$ ...
0
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1answer
114 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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0answers
15 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
14 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
12 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
36 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
37 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
37 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
22 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
29 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
29 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 ...
3
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1answer
37 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
25 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
47 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
148 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
14 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 & ...
1
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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
21 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
103 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 ...
1
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1answer
25 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
43 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 ...
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0answers
35 views

Homogeneous Heat Equation with Infinity

I need to solve the heat equation: $d_tu$ = $\nabla^2u$ with 0 < x < $\infty$ and 0 < y < $\pi$. Boundary conditions: u(0,y,t) = u(x,0,t) = u(x,$\pi$,t) = 0 and |u(x,y,t)| -> 0 as x ...
1
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1answer
13 views

Interchanging integral and derivative operations in the context of Duhamel's formula

I'll give you the whole context: In solving the heat equation $u_t = ku_xx$ with bounds $u(x,0)=0, u(0,t)=0, u(l,t)=f(t)$, let $v(x,t)$ be the solution for the special case $f(t)=1$. Use the Laplace ...
3
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1answer
42 views

Absorbing point in 2d heat equation - Why can the BC not be fulfilled?

I am trying to solve the radially symmetric 2D heat equation with an absorbing point at the origin in radial coordinates, i.e. (assuming no dependence on the angular variable) $$u_t = k ...
1
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1answer
29 views

Laplace equation for a lens-shaped volume - any approximate analytical solutions?

I need to solve Laplace equation for a lens (drop on a surface) with constant boundary condition on the top surface and zero on the bottom surface (this is the simplest case, not the only one I ...
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0answers
35 views

Maximum Principle of the Diffusion Equation

Consider a solution of the diffusion equation $u_{t} = u_{xx}$ in {$0\leq x\leq l,0\leq t<\infty$} a) Let $ M(t)$ = the maximum of $u(x,t)$ in the closed rectangle {$0\leq x\leq l,0\leq ...
0
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1answer
53 views

Separable solution to a nonlinear parabolic PDE

I seek a separable solution to the nonlinear parabolic partial differential equation, $\frac{\partial u}{\partial t} = u \frac{\partial u}{\partial x^2} + u^2.$ The physics of the problem allow ...
2
votes
1answer
69 views

Best Approximate Solution of Heat Equation (Diffusive Logistic Equation)

If $u(x,t) \ge 0$ in the domain $ (0 \times 1) \times (0,\infty)$, find a function that caps the value of $u(x,t)$ in the region $(0 \times 1) \times (0,T)$. $u(x,t)$ is a solution of the following ...
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0answers
23 views

How to integrate to solve a PDE with mixed partials in the integrand

Problem Statement: Determine the equlibrium temperature distribution inside a circular annulus $r_1\leq r \leq r_2$. If the outer radius is at temperature $T_2$ and inner radius at temp $T_1$. So ...
3
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0answers
26 views

Heat equation with mixed boundary conditions

I am trying to solve the following problem $$\left\{\begin{matrix}\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2},& 0<x<1,t>0,\\ u(0,t)=\frac{\partial u}{\partial ...
1
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1answer
22 views

uniqueness of heat equations and the squared integrable assumption

I am looking at the classical proof of uniqueness for the heat equation in Evans. Clearly, we differentiate under the integral sign of the square of $w$. A very basic question is, why are the ...
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0answers
45 views

long time behavior of heat equation

Given the heat equation \begin{align} {{u}_{t}}-{{u}_{xx}}&=0,\quad x\in \mathbb R,\,t>0 \\ u\left( x,0 \right)&=f\left( x \right),\quad x\in \mathbb R. \end{align} If ...
1
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1answer
24 views

checking that an initial condition holds for the heat equation

I'm trying to follow a video lecture on solving the heat equation. $I) \space u_t = ku_{xx}, x \in \mathbb{R}, t > 0$ $II) \space u(x,0)=\phi (x), $ $k$ is const, $\phi (x) $ is a ...
11
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0answers
194 views

Polynomial in the components of the curvature tensor

Consider a closed Riemannian manifold $(M,g)$ of dimension n and let $K(t,x,y)$ be its heat kernel. Then it is known that the heat kernel has an asymptotic expansion as $t\downarrow 0$: ...
1
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1answer
28 views

PDE Von Neumann Problem- Physical Interpretation

The Von Neumann Problem is as such: $\Delta u = f(x,y,z)$ in $\ D$ $\frac {\partial u} {\partial n} = 0$ on bdy $\ D$. Using this you can prove that for there to be a solution to this Von Neumann ...