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

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2
votes
1answer
23 views

Heat Equation, possible solutions

NOTE: This is a homework problem. Please do not solve. I was given a problem that asked me to find a function of the form $u_n(x,t)=\chi_n(x) \cdot T_n(t) $ that solves the heat equation with the ...
2
votes
1answer
48 views

Partial Differential Equation $\frac{\partial}{\partial t} p(x,t) = \frac{\partial^2}{\partial x^2} \left[ x^2 p(x,t) \right]$

In my research I have come across the partial differential equation \begin{equation} \frac{\partial}{\partial t} p(x,t) = \frac{\partial^2}{\partial x^2} \left[ x^2 p(x,t) \right]. \end{equation} ...
0
votes
0answers
32 views

How to compute an integral involving the error function

Solving a problem with one dimensional diffusion the following identity naturally arises $$\int _{-\infty }^{\infty }\frac{q}{2}{{\rm e}^{\eta\,t{\alpha}^{2}-\alpha \,x}} \left( {{\rm ...
0
votes
0answers
15 views

Fractional derivative of sine function

I try to reproduce the results of a paper. The authors are dealing with the fractional diffusion equation \begin{align} \partial_t^\alpha u - \partial_x^2 u = f \end{align} on the domain ...
0
votes
0answers
64 views

How could I solve this PDE?

Can anybody please let me know an idea about solving the following PDE for a given initial condition $T_0=0$ and boundary conditions $T(0,w)=u_0$? \begin{equation*} \nabla_w T_t(t,w) + T(t,w) . ...
0
votes
0answers
51 views

How to solve Energy Balance equation by numerical method

Good Day I am new to heat transfer technique please give me some suggestion on solving energy balance equation $$a \frac{\partial T_p}{\partial t}=\frac{\partial}{\partial x}\left(b\frac{\partial ...
1
vote
1answer
30 views

Unique solution for nonlinear heat equation

I have the following initial value problem: $$ \dfrac{\partial}{\partial t} u(x,t) - \frac{1}{2} \dfrac{\partial^2}{\partial x^2}u(x,t) = (u(x,t))^2$$ $$ u(x,0) = u_0(x)\in C^2(S^1)$$ and want to show ...
1
vote
1answer
25 views

Heat flow in 1D bar fourier series problem

I am stuck on this problem: The temperature $T$ in a one-dimensional bar whose sides are perfectly insulated obeys the heat flow equation $$ \frac{\partial T}{\partial t} = \kappa ...
4
votes
4answers
120 views

Connections between the solution of simple ordinary equation, normal distribution and heat equation

The solution to the following simple first-order linear ordinary differential equation: $$x'=-tx, x(0)=\frac{1}{\sqrt{2\pi}}$$ is the Standard normal distribution! One solution to another famous ...
0
votes
0answers
17 views

similarity reduction for heat equation

Given the Cauchy problem $$ \frac{\partial G}{\partial t} = \frac{\partial^2 G}{\partial x^2},G(x,0)=\delta(x) $$ with similarity reduction $$ G(x,t)=\frac{\phi(\xi)}{\sqrt{t}}, ...
1
vote
0answers
66 views

Integration of Heat Equation

how do i show that, $$ u(x, t)=\frac{1}{\sqrt{4\pi t}}\int_0^\infty y^2e^{-(x-y)^2/4t} \, dy = ( x^2+2t )N\left(\frac{x}{\sqrt{2t}}\right) + \sqrt{\frac{t}{\pi}} xe^{-x^2/4t} $$ where $$ N(z)= ...
0
votes
2answers
47 views

Lack of homogeneous boundary conditions of a Sturm-Liouville problem.

In my exercise bundle about Sturm-Liouville problems and solving partial differential equations with the separation method there is an exercise that goes as follows: Calculate the temperature ...
3
votes
0answers
53 views

Deriving a formula for an initial boundary-value problem

Given $g : [0,\infty) \to \mathbb{R}$, with $g(0)=0$, derive the formula $$u(x,t)=\frac{x}{\sqrt{4\pi}}\int_0^t \frac 1{(t-s)^{3/2}}e^{-\frac{x^2}{4(t-s)}}g(s)\,ds$$ for a solution of the ...
1
vote
1answer
20 views

Stokes' Rule for an initial-value problem

Assume $u$ solves the initial-value problem $$\begin{cases}u_{tt}-\Delta u = 0 & \text{in } \mathbb{R}^n \times (0,\infty) \\ u = 0, u_t = h & \text{on }\mathbb{R}^n \times \{t=0\}. ...
2
votes
0answers
46 views

Derive $u(x,t)$ as a solution to the initial/boundary-value problem.

Given $g : [0,\infty) \to \mathbb{R}$, with $g(0)=0$, derive the formula $$u(x,t)=\frac{x}{\sqrt{4\pi}}\int_0^t \frac 1{(t-s)^{3/2}}e^{-\frac{x^2}{4(t-s)}}g(s)\,ds$$ for a solution of the ...
7
votes
1answer
102 views

Solutions of the heat equation of the form $u(x,t)=v(x/\sqrt{t})$

Assume $n=1$ and $u(x,t)=v(\frac{x}{\sqrt{t}})$. (a) Show that $u_t = u_{xx}$ if and only if $$v''+\frac z2 v' = 0. \tag{$*$}$$ Show that the general solution of $(*)$ is $$v(z)=c \int_0^z ...
1
vote
0answers
25 views

Rewriting the heat diffusion equation with temperature dependent diffusion coefficient to include joule heating.

I am modelling heat flow in a solid round copper conductor with a set area. I plan to discretize and solve numerically in Python. However, I only have a curve fit for thermal conductivity and specific ...
0
votes
1answer
18 views

Equation of heat conduction for spherical solid

What will be the correct Equation of heat conduction for a homogeneous spherical solid with constant thermal diffusivity K and no heat source?
6
votes
2answers
139 views

What physical information does the mean value property of heat equation convey?

I'm reading through the Evans' book on PDE, the chapter on heat equation. The definitions are the same as here. I see that mean value property of heat equation is useful for proving maximum principle ...
1
vote
0answers
26 views

Duhamel's principle in constructing heat kernel

I want to construct the heat kernel $k_t(x,y)$, which is the fundamental solution to the heat equation $(\partial_t + \Delta_x)u(t,x) = 0$. There is something that I don't understand about using ...
1
vote
1answer
36 views

Equilibrium Temperature in insulated rod from BVP

this problem is giving me some trouble. It is a review problem given to us to study for our Final Exam this week. I would love some help in understanding how to solve it so that I can study it and ...
2
votes
1answer
45 views

How do I accurately find a point within a grid of 4 based on varying values?

I am specifically trying to find a source of heat, with heat sensors giving readings from 4 different points eg below: The diagram above represents a 4 x 4 grid where the numbers are heat sensors ...
1
vote
0answers
22 views

Comparison principle for heat equation with smooth nonlinearity

Let $f : \mathbb{R} \to \mathbb{R}$ be $C^\infty$ and satisfy $f(0)=0$. Suppose $u_1,u_2 : \mathbb{R}^d \to \mathbb{R}$ are $C^2$ and satisfy $$\frac{\partial u}{\partial t} - \Delta u = f(u)$$ for ...
0
votes
0answers
34 views

How to solve one differential equation with two independent variables in heat transfer.

$$A\frac{ \partial T_a}{\partial t}=B(T_p-T_a)+C(D-T_a)-E\frac{\partial T_a}{\partial x}$$ Where $A, B, C, D, E$ are constants, $t$ is time and $x$ is $x$-axis of the box in which heat transfer is ...
0
votes
1answer
26 views

Matlab solution for non-homogenous heat equation using finite differences

Given the following PDE (non-homogenous heat equation): $$ u_t(x,t) = c^2u_{xx}(x,t) + f(x,t) $$ $$ u(0,t) = u(l,t) = 0 $$ $$ u(x,0) = g(x) $$ $$ 0 < x < l ; t > 0 ; c > 0 $$ I wrote the ...
1
vote
0answers
34 views

What is a reasonable manufactured solution to test finite difference method?

What is a reasonable manufactured solution to test the following equation against its finite difference approximation? I want it to look like a Cosine function about $0$, rotated about $Z$ axis ...
1
vote
1answer
34 views

Heat equation fundamental solution

The following is from a book of PDEs and I have cannot seem to figure out a particular step in it with regard to the derivation of the fundamental solution of the heat equation. I have highlighted it ...
2
votes
1answer
47 views

Is the on-diagonal heat kernel “local” with respect to the metric?

Question Let $X$ be a manifold, and $\mu_A$, $\mu_B$ two Riemannian metric on it which agree on an open subset $U\subset X$, i.e. $\mu_{A\,|U} = \mu_{B\,|U}$. Let $K_A(t;z,w)$ resp. $K_B(t;z,w)$ be ...
1
vote
0answers
40 views

Heat Equation on Manifold

Laplacian operator is defined well on Riemannian manifold, denoted by $\Delta$. Therefore people can study PDE $\Delta f=0$ on manifold. So is there any analogy to heat equation or wave equation on ...
0
votes
0answers
20 views

semi-infinite heat equation with Dirichlet BC via Laplace transforms

I am trying to solve the heat equation for a semi infinite rod with lateral surfaces insulated and $u(x,0)$ = $u_0$ for $x>0$, $u(0,t)=u_1$ for $t>0$, and the $\lim_{t\to\infty} u(x,t)=u_0$. I ...
1
vote
0answers
36 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 ...
0
votes
1answer
76 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 ...
1
vote
1answer
31 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 ...
1
vote
1answer
28 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 ...
0
votes
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 ...
0
votes
0answers
26 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.
1
vote
1answer
41 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 ...
0
votes
1answer
29 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 ...
1
vote
1answer
34 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. ...
0
votes
2answers
75 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 ${\rm u}\left(\, x,t\,\right)$ as temperature function,TAs ...
6
votes
2answers
128 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 ...
0
votes
0answers
21 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 ...
0
votes
0answers
17 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 ...
0
votes
1answer
22 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, \ ...
0
votes
4answers
28 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 ...
0
votes
0answers
17 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 ...
0
votes
1answer
26 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 ...
0
votes
1answer
48 views

Maximum principle for heat equation with Neumann boundary conditions

Consider the initial-boundary value problem $$ \frac{\partial u}{\partial t} = a\Delta u \;\mbox{ in } \;\Omega $$ $$ a\frac{\partial u}{\partial n} = g \;\mbox{ on }\;\Gamma = \partial \Omega $$ $$ ...
0
votes
0answers
30 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 ...
1
vote
2answers
43 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 ...