Questions tagged [heat-equation]
For questions related to the solution and analysis of the heat equation.
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12 views
Complex Analysis and Heat Potential
What does the imaginary part of the complex heat potential tell us ? I understand that the real part gives the temperature distribution and that, since we add the real and imaginary parts, the ...
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0answers
8 views
Boundary condition for heat distribution in an insulated ball
Consider a hot ball whose temperature $T$ evolves according to the equation
$$k\frac{d}{dr}\left(r^2 \frac{dT}{dr}\right)+r^2\dot{q}=0,$$
where $\dot{q}$ is a constant. Solving this equation, we find ...
1
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0answers
28 views
2D Heat equation with Initial Data
I have the following 2D heat equation:
$$u_t - \Delta u = e^t$$ where $(x_1, x_2) \in \mathbb{R}^2, t > 0, u(x_1, x_2, 0) = cos(x_1) sin(x_2)$
I am looking to find the general solution $u(x_1, ...
2
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1answer
46 views
Heat equation initial value problem (General Solution)
I have the following equation: $u_t - u_{xx} = 0$ with initial data $u(x, 0) = e^{kx}$ for some constant $k$ and $x \in \mathbb{R}, t > 0$ I'm looking for the general solution $u(x, t)$.
So far, I ...
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0answers
27 views
Help me Solve this Steady State 2-D Heat equation
So I tried to solve this PDE by separation of variables which was not applicable because of non-homogeneous nature of this PDE. Then I assumed ( after reading from a book ) :
$$ T(x,y)=v(x,y) + \phi(...
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0answers
28 views
Homogeneous heat equation with time dependent boundary condition in sphere (General case)
I have been going through a derivation and am looking for some guidance.
The problem is as follows:
Flow of heat in a sphere given by:
$$\frac{\partial v}{\partial t} = k\left(\frac{\partial ^2v}{...
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0answers
15 views
Estimates of Hessian of Heat Equation
I am studying the heat equation
\begin{align*}
u_t - \Delta u = f
\end{align*}
where $u \in C^\infty(\bar{\Omega} \times (0,1])$ has compact support on $\Omega$ for all $t > 0$. My objective is to ...
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2answers
45 views
Regarding steady state solution of $u_{t}= c^2 u_{xx}$ with $u_{x}(0,t) = c_{1}$ and $u(L,t) = c_{1}$?
Suppose we have the one dimensional diffusion equation $u_{t} = c^2 u_{xx}$ with the boundary condition $u(L,t) = c_{2} $ and $u_{x}(0,t) = c_{1}$. I donot recognize which type of condition is it? ...
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0answers
39 views
Parabolic Partial Diff Equation ( Heat Equation )
Im having a really hard time with this question, I honestly don't even know how to start this...can anyone help me?
We are in the Parabolic section of PDE, and we are asked to find the unique ...
1
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1answer
21 views
PDE for heat conduction with loss
Consider the following PDE:
$$\frac{\partial }{\partial t}u = \alpha^2 \frac{\partial^2 }{\partial x^2}u -bu, \ \ b>0, 0<x<L$$
The problem asks to "set the time derivative in the PDE to ...
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1answer
33 views
2-D Heat Equation IVP
Find the solution to the two-dimensional heat equation, $u_t = u_{xx}
+ u_{yy}$ in the $x$-$y$ plane (that is, $−∞ < x < ∞$, $−∞ < y < ∞$) with initial data $u(x,y,0) = xe^{-y}$
I'm only ...
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0answers
18 views
Inhomogeneous heat equation on finite interval
My question is, how to solve an inhomogeneous heat equation on finite interval. The problem is $u_{t} - ku_{xx} = f(x, t) $ on finite interval with homogeneous Dirichlet boundary condition $ u(0, t) =...
1
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1answer
39 views
Hitting time of a sphere by a Brownian particle
Consider a Brownian particle in $\mathbb{R}^n$, starting at the origin. Let us consider a sphere of radius $r$ in $\mathbb{R}^n$ centered at the origin. We know that the probability that the particle ...
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0answers
32 views
dimension reduction for heat equation (from 2D to 1D)
I have two metal objects of different thickness and conductivity:
1) thicker, but poorly conductive. It's properties: $k_1, H$
2) thin and much better conductive. It's properties: $k_2, h$
...
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0answers
8 views
How to use IMEX scheme for the heat equation?
What I understood from reading online that the implicit-explicit (IMEX) method for solving pd for example taking the convection-diffusion equation $u_t+au_x=bu_{xx}$ it applies an implicit method to ...
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1answer
19 views
Pointwise bound of the gradient of solutions of heat equations in the half-space.
I want to investigate the decay of $L(x)$:
$$L(x) := \int_{\mathbb{R}^3_+} \nabla_x \Phi(x-y,1/2)(\eta(y)g(y))dy,$$
where $g:\mathbb{R}^3_+ \rightarrow \mathbb{R}^3$ is infinitely smooth away from ...
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0answers
17 views
Convolution of bounded Lipschitz function with heat kernel solves the heat equation
Let $\phi_t(x)=\Phi(x,t)=(4\pi t)^{-1/2}\exp(\frac{-x^2}{4t})$ be the heat kernel and
let $f$ be a bounded Lipschitz function.
Don't ask about the two different $\phi,\Phi$ notations for the heat ...
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0answers
9 views
Exponential decay of convolution with gradient of heat kernel in the half-space
I am wrestling with the decay of this integral:
$$I := \int_{\mathbb{R}^3_+} \nabla_x^k \Gamma(x-y,\frac{1}{2})(\xi(y)f(y))dy,$$
where $f:\mathbb{R}^3_+ \rightarrow \mathbb{R}^3$ is a smooth vector ...
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0answers
21 views
Separation of Variables method for 2-D Heat Equation
I have the heat equation below. and I am trying to solve it merely using separation of variables method.
$$E^2 w_{yy}+w_{xx}=E^2 w$$
$$w_x+αw=0 , x=0$$
$$w_y+βw=0 , y=1$$
$$w_y=0 , y=0$$
$$w(1,y)=G(y)...
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1answer
20 views
Use of L'Hosptial Rule in generalized setting related to heat equation
Let $X$ be a space of continuous functions with compact support in a bounded domain $\Omega \subset \mathbb{R}^{N}$ with Lipschitz continuous boundary, $F : X \to X$ be a Lipschitz continuous function,...
2
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1answer
62 views
Eigenfunction expansion to solve non homogenous heat equation
I've been really struggling to figure out how to solve this problem using Eigenfunction expansion, I can solve it using seperation of variables.
So this the problem is:
$$
\begin{cases}
u_t(x,t)=u_{...
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0answers
19 views
Coercivity of mapping A , sequence bounded
in Hess' article I don't understand why the sequence $u_{n\epsilon}$ is bounded (and $\frac{\partial u_{u\epsilon}}{\partial t}$ too)
$Q=\Omega \times (0,T)$
(A2) There exist constants $q \quad(1<...
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0answers
66 views
Backward Heat Equation with Transversality Condition
I have a backward parabolic equation of the form:
\begin{equation}
W_{\eta} + aW_{xx} - bW = 0
\end{equation}
s.t.
\begin{equation}
\lim_{\eta \rightarrow \infty}(x,\eta) = g(x)
\end{equation}
...
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0answers
12 views
(reference request) Heat kernel asymptotics of the Laplace-de Rham operator on forms
Let $(M,g)$ be a smooth closed manifold of $n$-dimension with a Riemannian metric $g$.
It is known that the heat kernel of the Laplace-Beltrami operator $\Delta_{LB}$ has the asymptotic expansion as ...
2
votes
0answers
60 views
How to solve the following (heat?) equation?
Let $u(x,t)$ be defined on $\Bbb R\times [0, T]$ such that:
$$u_t + (a - bx) u_x + \frac12 c^2 u_{xx} = xu,\quad u(x,T)\equiv 1$$
in which $a,b,c>0$ are constants. How to solve this PDE? Is it ...
1
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1answer
17 views
$W_{0}=x$ for a Brownian motion - what is meant by $E_{x}(.)$?
$W_{0}=x$ for a Brownian motion - what is meant by $E_{x}(.)$?
That is, it it such that the expectation is taken with respect to a degenerate distribution, $W_{0}\text{~}DEGEN(x)$?
This comes up, for ...
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0answers
84 views
finding maximum value of BVP
Find the maximum value of $u$ in the following BVP
\begin{align*}
u_t &= u_{xx}, \; \; t,x \in (0,1) \\
u(0,t) &= 2t^2-t \\
u(1,t) &= \sin \pi t \\
u(x,0) &= x(...
2
votes
1answer
56 views
Three dimensional Laplace equation with constant Temp. on one face. [Solution not satisfying BC]
The governing differential equation is
$$\nabla^2 T=0 \tag A$$
The boundary conditions for this problem are as foll0ws:
$$T(0,y,z)=T_{hi} \tag {1A}$$
$$T(L,y,z) = T(x,0,z) = T(x,l,z) = T(x,y,0)= ...
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0answers
10 views
Prove finite difference operator on 1D heat is bounded
Could someone please check my answer and tell me if it is correct or what may be wrong?
Question:
Prove the operators $R_k$ for the 1D heat equation with forward Euler time stepping is a uniformly ...
3
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0answers
28 views
symmetry of partial differential equations (Heat equation)
Morning everyone, I am doing some problem sheets for my class in Partial differential equations where we dont have an actual textbook. we are given a pack on notes. I am having an issue discerning ...
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1answer
32 views
Heat semigroup representation
It is known that the Laplacian operator with Dirichlet boundary condition in $L^2(\Omega),$ (with $\Omega$ being a open subset of $R^n$) generates a $C_0-$semigroup in $L^2(\Omega)$). Moreover, in ...
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0answers
22 views
Heat equation - stationary solution
What are the conditions for the solution of the heat equation to become [sufficiently] stationary, i.e. to not change any more over time?
Here, I am talking about the 3D heat equation with Neumann ...
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votes
0answers
13 views
What is the defining properties of the heat kernel?
According to https://en.wikipedia.org/wiki/Fundamental_solution, the fundamental solution $u$ is defined as the solution of $Pu = \delta$. In the meantime, we know that the fundamental solution of the ...
1
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0answers
84 views
Laplace equation in 3D with numerous Non-Homogeneous BC(s) [Strategy Check]
I need to solve the three-dimensional Laplace equation ($\nabla^2T = 0$) where $\nabla^2=\frac{\partial^2}{\partial x^2} +\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}$ in the domain ...
1
vote
1answer
27 views
Heat Equation: Separation of Variables - Can't find solution
I am trying to solve the following problem on $[0,\pi]$ through separation of variables:
$u_t=u_{xx}$
$u(x,0)=x^3\space (1)$
$u(0,t)=0\space (2)$
$u(\pi,t)=\pi^3\space (3)$
So far I have come to the ...
0
votes
1answer
28 views
Separating variables in a PDE with multiple constants
My question is:
How do you use separation of variables on a PDE that has more than one constant in it?
All the examples I can find in my book/online only have one constant in it, like $$ \frac{\...
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votes
0answers
20 views
Directional heat equation
I know that the mappings $$ u(t,x) := G_{t} \ast u_0 (x) $$ are solutions of the heat equation $$\frac{\partial u}{\partial t}=\Delta u ,$$ where
$$u_0 \in C(\mathbb{R^n}), \lim_{x->\infty}u_0(x)...
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2answers
29 views
Transforming advection-diffusion equation into heat equation
I am working on this question:
Show that advection-diffusion equation
$$
u_t=Du_{xx}+Au_x+Bu,\quad x\in\Bbb R, t>0
$$
where $A, B, D > 0$ are constants, can be transformed into heat equation ...
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0answers
31 views
Evans proof of smoothness of solutions to the heat equation
This is Theorem 8 in Section 2.3 of Evans:
Theorem. Suppose $u \in C_{1}^{2}(U_{T})$ solves the heat equation in $U_{T}$. Then $u \in C^{\infty}(U_{T})$.
Here, $U_{T} := U \times (0,T]$. In his ...
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0answers
27 views
How can I find the following one dimensional heat conduction solution?
$$\frac{\partial T}{\partial t}=\alpha\frac{\partial^2 T}{\partial^2 t}$$
with an initial condition and boundary conditions
$$T(x,0)=T_0$$
$$T(L,t)=T_0$$
$$-k\left.\frac{\partial T}{\partial x}\right|...
0
votes
1answer
31 views
The Heat Kernel to Solve an Initial Value Problem
The Question I have is:
$$u_t(x, t) − ku_{xx}(x, t) = 0$$ $$∀x ∈ \mathbb{R}, t > 0$$
subject to-
$u(x, 0) = x^2 − 3x − 1$ $∀x ∈ \mathbb{R}.$
I started off with
$$u(x,t)=\int_{\mathbb{R}}e^\...
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0answers
36 views
Using Duhamel's Principle and the Heat Kernel to Solve an IVP
I need to solve this:
$u_t(x,t)-ku_{xx}(x,t)=xte^{-t^2}$
By using Duhamel's Principle and the Heat Kernel.
So far this is what I've done:
$u_t(x,t)-ku_{xx}(x,t)=xte^{-t^2}$
where u(x,0)=0
$u(x,t)...
1
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0answers
21 views
Global $L^p$ estimates for the heat equation by approximation
Consider the heat equation $\partial_t u = \Delta u + f $ on $\mathbb{R}^N$ with $u(0) = 0$. Let $u \in C_c^\infty(\mathbb{R}^{N+1})$ be a solution of the heat equation to some $f \in C_c^\infty(\...
3
votes
1answer
47 views
Heat equation with variable dissapation
I've been asked to solve the diffusion equation with variable dissipation, I have given the start of my answer but can't seem to proceed. Would appreciate a full solution as the work is due soon, ...
1
vote
1answer
21 views
Existence of solution to the heat equation given boundary condition
Working on an old exam question:
Show that for every $f \in C(\mathbb{T}), \epsilon > 0$, there is an initial condition $g \in C(\mathbb{T})$ for which there is a solution $u(x,t)$ to the heat ...
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0answers
21 views
FDM variable coefficients
Usually authors of the books demonstrate the usage of FDM on the following equation
$\frac{\partial f}{\partial t} - a\frac{\partial^2 f}{\partial x^2} = f(x,t)$
where $a$ is some constant.
Is it ...
1
vote
1answer
46 views
Stability of numerical scheme for heat equation
We want to use the Fourier discrete transform to analyze the stability of leapfrog scheme for 1D diffusion eqn $v_t = \nu v_{xx}$
Thoughts
The leapfrog numerical discretization is given by
$$ \...
0
votes
0answers
22 views
Two-dimensional Laplace equation with weird Robin BC
I need to solve the steady-state heat equation a.k.a. Laplace equation over a rectangle
For $\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0$ defined on $x \in [0,a]$ and $y ...
3
votes
1answer
92 views
Solving diffusion equation with variable dissapation
I've been asked to solve the diffusion equation with variable dissipation:
$ \frac{∂u}{∂t} - D \frac{∂^2u}{∂x^2} + e^{-pt}u = 0 , x∈(-∞,∞), t∈[0,∞] $
subject to
$u(x,0) = φ(x),$
where $D>0$ ...
2
votes
1answer
62 views
Is $f(x-at)$ a solution to the heat equation?? ($u_t=u_{xx}$)
Show that the heat equation $u_t = u_{xx}$ has solutions of the form $u(x, t) = f (x − at)$ where $a$ is a constant. Show that $a$ can have any value, real or complex, and describe the behavior of the ...
