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

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36 views

Inverse non homogeneous heat problem unicity

Consider the inverse problem: Determine the pair $u = u(x, t)$ and $f = f(x)$ satisfying $$u_{t}(t, x) - u_{xx}(t, x) = f(x) \hspace{0.5cm} t > 0, x \in ]0, L[;$$ $$u(t, 0) = 0 \hspace{0.5cm} t ...
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26 views

Heat equation, boundary gradient singularity

Consider the Cauchy-Dirichlet problem for the heat equation in a non-cylindrical region $\Omega \subset \mathbf{R}^+ \times \mathbf{R}$: $\Omega = \{ (t,x): \; 0 \leq t \leq 1, \; x \leq ...
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1answer
36 views

Derivative of the fundamental solution of the heat equation

Let $\Gamma$ be the fundamental solution of the heat equation in $(0,\infty)\times\mathbb{R}^n$, that is \begin{equation} \Gamma(t,x)=\frac{1}{(4\pi t)^{\frac{n}{2}}}e^{\frac{-|x|^2}{4t}} \mbox{ per ...
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1answer
22 views

Intermediate Forms Between Parabolic and Hyperbolic PDE (numerically)

Greetings MSE community, I have recently conducted some rudimentary experiments in matlab coding of PDE's. I have explicit and implicit numerical solutions to both the heat and the wave equation, for ...
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1answer
27 views

What happens to other eigenvalue? - steady state heat equation

A circular plate is bounded by circles of radii $r=2$ and $r=4$, its surface is insulated and temperatures along boundaries are given by $u(2,\theta)=10\cos\theta + 6\sin\theta$ and ...
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1answer
22 views

What are $f(x)$ and $f(y)$ here in this heat equation problem?

Question: What is $f(x)$ and what is $f(y)$? You are given (no need to check) that the function $G(x-y,t)$ defined by $$G(x-y,t)=\frac{1}{\sqrt{4\pi c^2 t}}e^{-(x-y)^2/4c^2t}$$ satisfies the ...
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21 views

$2\operatorname{D}$ heat equation with non-zero initial condition

I cant find anything about $2\operatorname{D}$ heat equation with non-zero initial condition, need to find the temperature of the rectangle.
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1answer
35 views

Interesting counterexample of Strong Maximum Principle

Suppose $U=\Omega \cup (0,T)$ where $\Omega$ is a bounded domain. Let $u \in C_1^2 (U) \cap C(\overline U)$ satisfy $$u_t \le \Delta u+cu$$ in $U$ where $c \le 0$ is a constant. If $u \ge 0$, then ...
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1answer
20 views

How to transform parabolic equation into heat equation?

Consider the parabolic equation: $$u_t-k(\Delta u+\sum\limits_{i=1}^n a_i\frac{\partial u}{\partial x_i}+bu)=0$$ where $a_i,b,k$ are constants and $k>0$. How this equation can be transformed to the ...
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1answer
33 views

Energy dissipation

I've been asked to prove the following, but I don't find the way.. Let $\Omega\subset\left\{0<x_n<a\right\}$ be a subset of $\mathbb{R}^n$ such that it is bounded in the $n^{th}$ coordinate. ...
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2answers
50 views

heat equation with perfectly insulated end

In one of my tutorial question about $1$-dim heat equation,a question about heat equation with pefectly insulated end at $x=0 $ and $ x=l$ with $u(x,t)$ as temperature function,TAs used as perfectly ...
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2answers
115 views

Regularity of the heat kernel

Let $(M,g)$ be a compact Riemannian manifold. Let $H:M\times M\times\mathbb{R}_{>0}\to\mathbb{R}$ be the heat kernel. i.e. $H\in C^0(M\times M\times\mathbb{R}_{>0})$ is the unique continuous ...
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0answers
19 views

Inhomogeneous heat equation

I'm trying to find inhomogeneous heat equation with neumann boundary condition.We use subtraction method but i can't use it in my problem.So where can i find more examples? Can you suggest some books ...
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0answers
14 views

Product of $n$ $1$-dimensional solutions of heat equation

Suppose $u_{1},\ldots,u_{n}$ are solutions of the one-dimensional heat equation $\partial_{t}u=\partial_{y}^{2}u$. It is easy to verify that $$ v(x,t):=\prod_{j=1}^{n}u_{j}(x_{j},t) $$ solves the ...
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1answer
19 views

Steady-state solution

Obtain the steady-state solution of the problem $$\frac{\partial^2u}{\partial x^2}+\gamma^2(u-T)=\frac{1}{k}\frac{\partial u}{\partial t}, \ 0<x<a, \ t>0,$$ $$u(0,t)=T, \ u(a,t)=T, \ ...
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4answers
27 views

Why is $ g'(t)-cg(t)=0 \iff g(t)=Ae^{ct} $

Why is $$ g'(t)-cg(t)=0 \iff g(t)=Ae^{ct} $$ and how do I know that? This is a part of the heat equation that I don't understand, I must have missed this part in some other course... What do you call ...
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0answers
15 views

Zeta function and heat kernel

It is easy to prove that zeta function $$\zeta_{\Lambda}(s)=\sum \frac{1}{\lambda_{n}^{s}}$$ and trace of heat kernel $$K_{\Lambda}(t)=\sum e^{-\lambda_{n}t}$$ satisfy the relashion ...
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1answer
23 views

How to find the coefficients in the Fourier series solution of a 1-D heat equation?

I am trying to use Fourier's method to solve a problem. $u(x,t) = \sum \limits_{n=1}^\infty B_ne^{-(n\pi C / L)^2 t}\sin\left(\frac{n\pi x}{L}\right), B_n=\frac2L\int_0^L \sin\left(\frac{n\pi ...
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1answer
31 views

Maximum principle for heat equation with Neumann boundary conditions

Consider the initial-boundary value problem $$ \frac{\partial u}{\partial t} = a\Delta u \;\mbox{ in } \;\Omega $$ $$ a\frac{\partial u}{\partial n} = g \;\mbox{ on }\;\Gamma = \partial \Omega $$ $$ ...
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0answers
24 views

Hints on solution to $u_t-\Delta u+cu=f$

Consider the problem (Evans, Ch 2, 14) $$ u_t-\Delta u+cu=f ,x \in \mathbb R^n\times (0,\infty)$$ $$ u=g , \mathbb R^n\times {t=0} $$ If $u$ solves $ u_t-\Delta u=f$, $u=0$ on and $v$ solves ...
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2answers
38 views

heat equation with fourier series

Original PDE $$T_t=\alpha T_{xx}$$ I need to solve this equation numerically and analytically and compared them. I've already done the numerical part. But I need to solve it analytically now. Given ...
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1answer
47 views

Finding the heat equation (PDE)

I have the terms: $$\left\{\begin{array}{l l} u_t=u_{xx}, 0<x<1, t>0\\u(0,t)=1, u(1,t)=3, t>0\\ u(x,0)=2x+1-\sin(2\pi x), 0<x<1\end{array}\right.\ $$ And I should determine the ...
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2answers
36 views

Separation of variables in heat equation with decay

I just want to see if I completed this problem right. Here is the problem: Consider $\frac{\partial T}{\partial t} = k \frac{\partial^{2} T}{\partial x^2} -\alpha T$ where $k,\alpha >0$ are ...
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1answer
16 views

Separation of Variables of a PDE

I just want to see if I completed this problem right. Here is the problem: Consider $\frac{\partial T}{\partial t} = k \frac{\partial^{2} T}{\partial x^2} -\alpha T$ where $k,\alpha >0$ are ...
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2answers
60 views

The backwards heat equation is not well posed

I have some sort of issue on this problem. Here it is: Show that the backwards heat equation, $\frac {\partial u}{\partial t} = -k \frac {\partial^2 ...
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0answers
13 views

Book on transient flow

Can anyone recommend some books on transient flow, the heat equation or fluid dynamics that would be easy for a math major to read? I'm interested in modeling ground water and I would like a nice ...
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1answer
37 views

Heat equation: Why are these ratios of functions constant

One can solve the heat equation $$ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} $$ by a separation of variables such that $u(x, t) = f(x)g(t)$. Substituting this into the ...
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80 views

Max Principle for Heat Equation with Neumann Boundary Condition

This is a homework problem. I am looking for an additional hint or reference to theorems/ideas that can help. Some of my thought process is presented below. Suppose \begin{align*} u_t - \Delta u ...
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1answer
23 views

Heat equation, initial-boundary value problem

Let $u (x, t)$ be a solution of the initial -boundary value problem $$\left\{\begin{array}{ll} U_t - U_xx = 0 & 0 < x < L, t > 0 \\ U (0, t) = U (L, t) = 0 & t > 0 \\ U (x, 0) ...
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1answer
67 views

Heat equation convection

I want to solve the heat equation with convection $F_t = F_{xx} - F_x$ with initial condition $F(x,0) = f(x)$ So far what I've got is that if $F(x,t) = G(x-t,t)$ and G satisfies the heat equation ...
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1answer
25 views

How to prove that the L2 norm is a non-increasing function of time for a 2nd-order PDE?

I am having a test in few days and I saw an interesting question while I was skimming through the book problems. The problem is concerned about initial-boundary value problem of 2nd order PDEs. To ...
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1answer
22 views

PDEs Diffusion eq dirichlet BC

I want to find the $2l$ periodic solution of the diffusion equation: $u_t = u_{xx}$, $∀x∈ (0, l), ∀t ∈ ℝ$ with initial condition $u(x,0) = x$ and the Dirichlet boundary condition $u(0,t) = 0$ ...
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0answers
21 views

Separation of variables and Fourier transformation

I know there's another question very similar to this argument. In the book "Probabilità e modelli aleatori" of Enzo Orsingher, at pag 134, it shows that the transiction function of an absorbing ...
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23 views

How do I show that $u(x, t) = e^{-\alpha^2k^2t}\sin(kx)$ is a solution for $u_t = \alpha^2u_{xx}$.

I want to show that $u(x, t) = e^{-\alpha^2k^2t}\sin(kx)$ is a solution for $u_t = \alpha^2u_{xx}$. I did the following: $$u_x = e^{-\alpha^2k^2t}\cos(kx)k$$ $$u_{xx} = ...
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0answers
29 views

The heat kernel as a fundamental solution

From my undergraduate studies I know that a fundamental solution to a partial differential operator $P$ is a distribution $u$ such that $Pu= \delta$ (no reference to any boundary or initial ...
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2answers
23 views

Heat Equation Existence of Fourier Series

I'm currently doing a bit of digging with the Heat Equation and the Fourier Series. It seems that the boundary condition $u(x,0)=f(x)$ can be arbitrary. At some point, we get something like (in a ...
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1answer
21 views

Heat Equation Two Conditions

I'm currently working on solving the Heat Equation in a one dimensional rod of length $L$. However, instead of the 'usual' singular condition $u(x,0)=f(x)$ for all $0\leq x\leq L$, I am given ...
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1answer
34 views

Smoothing effect for weak solutions of heat equation

Let $u_0 \in L^2$ and $f \in L^2(0,T;H^{-1})$ and consider the solution $u \in L^2(0,T;H^1)\cap H^1(0,T;H^{-1})$ of $$u_t - \Delta u = f$$ $$u(0) = u_0$$ with some BC (eg. zero Dirichlet). I am ...
0
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1answer
38 views

Solve this heat equation using separation of variables and Fourier Series

I'm working on a practice question and just a little confused at some parts, would greatly appreciate some help. Here is the question: $ \frac{\partial u}{\partial t} = K \frac{\partial^2 ...
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1answer
99 views

Using Fourier Transform to solve heat equation

the heat equation of $U(x,t)$ on $-\infty<x<+\infty$ and $t>0$ is $$U_t=U_{xx}+\exp\left({\frac{-x^2}{2}}\right)$$ where $$U(x,t)\rightarrow 0 \quad as\quad x\rightarrow\pm\infty$$ and the ...
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1answer
74 views

Deriving an estimate in regularity theory of the heat equation

I have another question from PDE Evans 2nd edition, this time from pages 380-381. It's about a step in the formal derivation of estimates. Given the initial-value problem for the heat equation ...
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22 views

Newton heat transfer question with step functions

The temperature outside of the building was stable 0◦ Celsius. At time t = 0 inside the building it was 18◦. The time constant 1 for the building equals 4 hrs. In 1/3 hr a source of heat was turned ...
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1answer
49 views

Lp bounds of the Heat Kernel

These days, I am struggling with a problem which seems very straightforward (and I'm pretty sure it is straightforward) but it resists to my attempts to prove it. Here it is: Let $\mathcal H_t$ be ...
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1answer
31 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 ...
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1answer
57 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
46 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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29 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 ...
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93 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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74 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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40 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 ...