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Questions tagged [heat-equation]

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

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27 views

partial differential equation heat equation

I received this homework problem from my professor, but I am unsure of how to get it going. It doesn't correlate exactly with anything in our book. Consider the heat equation for $u(x,t)$ $$u_t = ...
2
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1answer
29 views

Solving Heat Equation IBVP

I have a hard time understanding how to proceed with the third condition of the following IBVP: \begin{cases} u_t = u_{xx}, \ x\in (0, 2), \ t>0 \\[6pt] u(x, 0) = x, \ x \in [0, 2] &(i)\\[6pt] ...
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22 views

Solving heat equation variant [on hold]

How would one approach solving the following PDE: \begin{cases} u_t-u_{xx}=\sin u &\text{in}\:\:U\times (0, \infty), \newline u=0 &\text{in} \:\: \partial U\times (0, \infty), \newline u=g &...
3
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1answer
43 views

Heat Equation with boundary conditions

I have the following PDE: \begin{cases} u_t = u_{xx}, \ x\in (0, 2), \ t>0 \\[6pt] u(x, 0) = \sin\frac{\pi x}{2}, \ x \in [0, 2] &(i)\\[6pt] u(0, t) = 1, \ t>0 &(ii)\\[6pt] u(2, t) = ...
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1answer
48 views

Heat equation on a finite graph and computing a ratio

Let $\Delta$ denotes the Laplace operator with $-\Delta \phi = \lambda\phi$ on the compact manifold $(M,g)$. In a paper it is stated that the solution of the heat equation \begin{align} (\partial_t -...
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0answers
46 views

Showing that the Heat Equation Solution is Bounded

I have the following problem and I'm not sure how to get started, any help is appreciated! Let u $\in C^2(U \times (0, \infty)) \cap C (\overline{U}\times[0,\infty)) $ be a solution of \begin{...
1
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1answer
35 views

Solving the Heat/Diffusion Equation with Piecewise Continuous Initial Condition

I'm trying to solve the following Cauchy problem in ${\rm \Bbb R}$ without using the fundamental solution. $$ \begin{cases} u_t = u_{xx} &\text{ for }\;\,(x,t)\in\Bbb R\times\{ 0<t<\infty\}\\...
2
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1answer
30 views

Uniqueness of Non-Linear Heat Equation

The following problem comes from an old exam: Consider \begin{cases} u_t - \Delta u + |u_{x_1}| = 0 \text{ in } \mathbb{R}^{n} \times (0,\infty) \\ u(x,0)=g(x) \text{ in } \mathbb{R}^n \end{cases} ...
0
votes
1answer
86 views

Evaluating Fourier coefficients to complete a Laplace equation solution

While solving a PDE problem involving the Laplace equation in 3D, I arrive at the following summation relation when i substitute the only non-homogeneous boundary condition available $$ \sum_{m=1}^{\...
2
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1answer
121 views

Laplacian with Integral BC(s)

I want to solve the three-dimensional laplacian $$\nabla^{2} T = 0$$ where $\nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}$ defined on $...
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0answers
17 views

Partial Differential Equations: Find an explicit solution of IVP for diffusion equation

Find an explicit solution of IVP for the diffusion equation: $u_{t}=u_{xx}, x \in\ {R}, t>0$ $u(0,x)=x, x\in R$ So, I used the normalization integral, differentiated it in t and then ...
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0answers
30 views

Discrete gradient operator

I am working on implementing a heat diffusion paper and I'm a bit stuck. Would be glad if someone can share their thoughts! I have a pointcloud $P \subset \mathbb{R}^n$. Each point has a value $\...
4
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0answers
105 views

3D Homogenous Laplace equation with integral boundary conditions

I have the 3D heat equation (Laplace equation) $$\nabla^{(3)}T_s=0$$ where $\nabla^{(3)}=(\frac{\partial^{2}}{\partial x^2}+\frac{\partial^{2}}{\partial y^2}+\frac{\partial^{2}}{\partial z^2})$ ...
0
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1answer
29 views

Variation of the Diffusion Equation

Let $\mathcal{L}=D\dfrac{\partial^{2}}{\partial x^{2}}-v\dfrac{\partial}{\partial x}+\beta$ be a differential operator describing diffusion ($D$) with drift ($v$) and a source ($\beta$). As part of a ...
2
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0answers
69 views

Laplace [Heat] type equation with source terms

$$\lambda_h \frac{\partial^2 \theta_w}{\partial x^2} + \lambda_c V \frac{\partial^2 \theta_w}{\partial y^2} =\beta_h e^{-\beta_h x} \int e^{\beta_h x} \theta_w(x,y) \, \mathrm{d}x + \beta_c e^{-\...
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0answers
34 views

Laplace transform of heat conduction PDE in cylindrical coordinates.

I'm trying apply the Laplace transformation to solve the non-dimensional heat conduction PDE for a hollow cylinder with convection boundary conditions and a non-homogenous initial condition. $$\frac{...
0
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1answer
28 views

inhomogeneous heat equation

Find the solution u to the initial value problem $$u_t − u_{xx} =sin(x), u(x,0)=0$$ I know that the particular solution is $u_p = -sin(x)$ So $u(x,t) = h(x,t) + u_p$ $h(x,t) = u(x,t) - u_p$ $h(...
0
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1answer
50 views

Heat Equation: initial condition $\exp(-x^2)$

$u_{tt}=u_{xx}$ given $u_0(x)=e^{-x^2}$ I know that I just need to follow the formula $$u(x,t)=\frac{1}{\sqrt{4\pi t}}\int_{-\infty}^{\infty}e^{\frac{-(x-y)^2}{4t}}u_0(y)dy $$ but I get to a point ...
3
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1answer
65 views

Least Energy Path, Contour Following, around Hills toward Goal

I have a matrix of elevation values which could be said represents $h(x,y)$. I can obtain contours using this function that are like sides of hills, and I have a starting point and an end point. How ...
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0answers
21 views

Find all second-order polynomial solutions to the Heat Equation

The following problem is a homework and I am not asking for a solution but simply for a hint on how to approach it. The task is given as follows: Find all second-order polynomial solutions to the ...
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0answers
11 views

Large time behavior for the Neumann problem for the heat equation

Let $\Omega$ be a bounded open subset of $\mathbb{R}^n$ of class $C^{1,\alpha}$. Let $g \in C^{\frac{\alpha}{2},\alpha}(]0,+\infty[ \times \partial\Omega)$. Let $$ \Omega^- \equiv \mathbb{R}^n \...
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0answers
9 views

Literature request: modified heat equation (Fourier terms)

I am interested in a modified version of the heat equation: $u_t = |\nabla u| $. More precisely, I am in search of literature regarding the Fourier transform of its solution. I have found many ...
2
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1answer
28 views

How to solve the heat equation with initial condition of zero?

The solution of heat equation $$u_t = \kappa u_{xx}$$ with separation of variables is $$u(x,t) = \sum_{n=1}^{\infty}b_n e^{-\kappa n^2 \pi^2 t}\sin n \pi x$$ I spare the details as are well known. ...
0
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0answers
5 views

Proof of Proposition 5.2.3 An introduction to semilinear evolution equations / Thierry Cazenave and Alain Haraux

I am currently reading the book stated in the title and there is a part I do not understand in the proof. Before I state the proposition, I would like to clarify the general assumptions here. $\...
2
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0answers
26 views

Initial value problem related to heat equation.

Let $u(x, t)$ satisfy the IVP: $\frac{\partial u}{\partial t}= \frac{\partial^2 u}{\partial x^2}, x \in \mathbb{R}, t > 0$ and $$u(x, 0) = \begin{cases} 1, \ \ 0\leq x \leq 1\\ 0, \ \ \text{...
0
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0answers
32 views

Show the following result for a solution of the heat equation

Let $u \in C^{1}(0,T) \times C^{2}(0,1)$, $u \in C^{1}([0,T] \times [0,1])$ be a solution of the heat equation $\frac{\partial u}{\partial t} -\frac{\partial ^2 u}{\partial x^2}=0$. Let $0<t_0&...
2
votes
2answers
72 views

Heat equation inequality

Let $u \in C^{1}(0,\infty) \times C^{1}[0,1]$ be a solution to heat equation (1) with inital boundaries (2),(3) (1) $\partial_tu(t,x)-\partial_{xx}u(t,x)=0$, for all $(t,x) \in[0,\infty)\times[0,1]$ ...
4
votes
1answer
92 views

Newtons law of cooling applied to spherical region

I'm having trouble solving the following problem: Formulate a mathmatical model for a stationary (steady) temperature distribution inside the spherical volume $$ R^2\leq x^2+y^2+z^2\leq (2R)^2, $$ ...
5
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2answers
45 views

What are the differences between Heat equations and Poisson Equations?

Am fairly new into heat equations and wanted to have some clarifications. What are the distinguishing features between the heat equation and the Poisson equation?
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0answers
31 views

the heat equation with solution-dependent coefficients

I meet the problem $$u_t(t,x)=\frac{A}{B-u}u_{xx}(t,x)$$ with the boundary conditions: $$u(0,x)=B, x\in[0,H]$$ $$u(t,0)=u(t,H)=0, t>0$$ $$u(+\infty,x)=0$$ I have not any idea to start. Can you ...
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1answer
31 views

Heat Model Example for liquid and solid change in state

If i have a region $x>0$ that is initially liquid at constant temp $T_0$ above freezing temp $T_f$ (so $T_0 > T_f$). The surface at $x=0$ stays at $T_1$ below freezing temp (so $T_1<T_f<...
2
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0answers
19 views

Combustion Theory Book Recommendation

I am conducting research on highly volatile chemicals and have been recommended to read Mathematical Problems from CombustionTheory. I was wondering whether there is another book that could be ...
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0answers
14 views

A few questions on the finite difference approximation for the heat equation

I'm trying to learn for fun how to apply the heat equation to different scenarios. Suppose I have a 2-D hotplate (perhaps steel) and on top of it sits a cube of some other material, and I'm only ...
0
votes
1answer
33 views

Heat equation in cylindrical coordinates at origin

I'm trying to solve a heat equation in cylindrical coordinates $$\dfrac{\partial u}{\partial t} = a \left(\dfrac{\partial^2 u}{\partial r^2} + \dfrac{1}{r} \dfrac{\partial u}{\partial r} + \dfrac{1}{...
1
vote
1answer
41 views

Modified Heat Equation: k is not a constant

Given the heat equation, $$ u_{t} = ku_{xx}, $$ how do we modify the solution below (when $k$ is a constant) $$ u(x,t) = \frac{1}{\sqrt{4\pi kt}}\int\limits_{-\infty}^{\infty} g(y)e^{\frac{-(x-y)^{2}}{...
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0answers
20 views

Applying the method of lines to a partial differential equation and using Runge-kutta method

By method of lines I converted the PDE u_t=u_{xx}, with the initial and boundary conditions ...
9
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3answers
183 views

Heat Equation + Uniform Convergence in time -> Harmonic Limit

Assume we have $u \in C^3(\mathbb{R}^n \times (0,\infty))$ satisfying the heat equation $$ \Delta u(x,t) = \partial_t u(x,t)$$ and a function $u_0:\mathbb{R}^n \to \mathbb{R}$ with unknown regularity (...
0
votes
0answers
49 views

Analytical solution to unidimensional transient heat equation with mixed bounday conditions not zero

I need to solve the equation: ${\partial^2 T(x,t) \over \partial x^2 }= {1 \over \alpha} {\partial T(x,t) \over \partial t}$ With the boundary conditions $T(0,t)=T_0$ and ${\partial T(L,t) \over \...
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0answers
22 views

Mixed boundary condition for the heat equation

Would someone help me understand the way the solution obtained in this question: Heat Equation Mixed Boundaries Case: Fourier Coefficients I did not understand why in the final solution, he took $...
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1answer
40 views

Fourier transformation of $e^{-ax^2}$

Find the Fourier transformation of $e^{-ax^2}$, for any constant $a>0$. In general, $$\int \frac{1}{\sqrt{(2\pi \sigma^2)}}e^{-\frac{(x-\mu)^2}{2\sigma^2}} e^{-iλx }dx = e^{-i\muλ-\frac{1}{2}\sigma^...
1
vote
2answers
38 views

Well-posedness for Heat Equation with Robin Boundary Condition

Can anyone help me prove the well-posedness of the following heat equation with Robin boundary condition? $u_t(x,t)=u_{xx}(x,t)$ $u(0,t)=0$ $u_x(1,t)=-au(1,t)$ where $a>0$. The existence of ...
0
votes
1answer
28 views

$ \int_{0}^{1} \sum_{m=1}^{\infty} a_m \sin(m\pi x)\sin(l \pi x)dx = \frac{a_l}{2}$

Why is this true? $$\int_{0}^{1}u^0(x)\sin(l\pi x)dx = \int_{0}^{1}u(0,x)\sin(l\pi x)dx$$ $$ = \int_{0}^{1} \sum_{m=1}^{\infty} a_m \sin(m\pi x)sin(l\pi x)dx$$ $$ = \frac{a_l}{2}$$ where ...
1
vote
1answer
30 views

Is the steady state solution of the Heat Equation with Dirichlet boundary conditions always 0?

A heat equation problem with Dirichlet boundary conditions on the domain $[x_1,x_2]$ $$\frac{\delta u}{\delta t} = k \frac{\delta^2 u}{\delta x^2}$$ $$u(x_1,t) = u(x_2,t) = 0$$ Would have ...
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vote
0answers
111 views

Solution for the heat equation that doesn't belong to the Gevrey Class

Show that there exists a solution for the heat operator (in one spatial variabe) that doesn't belong to the Gevrey class of order $s$ for all $s<2$ I already defined the Gevrey class here: $\...
2
votes
2answers
50 views

Well-posedness of heat-equation PDE with only one initial condition

Consider the PDE given by $u_t = \alpha u_{xx}$ with initial condition $u(x, 0) = f(x)$. Now suppose we discretize the problem in the time variable, so we approximate $u_t(x, t)$ by a finite ...
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0answers
63 views

Relation of vector process to Heat Equation?

The user Dinisaur conjectured that the following vector process could maybe be analyzed via a discrete version of the heat equation. I am personally not aware of the heat equation and maybe someone ...
0
votes
0answers
41 views

Mixed Cauchy and Dirichlet and unspecified boundary conditions for Laplace equation on $I^2$

I am looking for a reference where the following problem is discussed: $u \in C^{\infty}(I^2)$ so that $\Delta u = 0$ $u(0,y) = f(y)$, $u(1,y) = g(y)$, $u(x,0) = h(x)$ $\nabla u(x,0) \cdot \hat{n} (...
0
votes
1answer
86 views

Convergence of Vector Combinations

I have $n$ vectors $v_1(t), \dots, v_n(t)$. Time is divided into discrete rounds. Initially, all vectors have length $\leq 1$. The vectors at the next time step $t+1$ can be calculated as follows: \...
1
vote
0answers
33 views

Level sets under heat flow

Let $f\in C^\infty([0,2\pi])$ be a smooth periodic function with mean zero ($\int f = 0$). Let $f(t,x)$ be the heat flow of $f(x)$, so that $f(t,x)$ solves $$ \partial_t f = f''. $$ Given a small $\...
0
votes
1answer
24 views

$u(x,t)=\left(f*\mathcal{H}_{t}^{(n)}\right)(x)$ is solution to the n-dimensional heat equation

Consider the time-dependent heat equation in $\mathbb{R^{n}}$: $$\displaystyle\frac{\partial u}{\partial t}=\frac{\partial^{2}u}{\partial x_{1}^{2}}+\cdots+\frac{\partial^{2}u}{\partial x_{n}^{2}},\...