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Questions tagged [heat-equation]

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

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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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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 ...
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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, ...
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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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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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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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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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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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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 ...
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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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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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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) =...
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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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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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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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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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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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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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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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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,...
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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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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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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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(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 ...
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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 ...
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$W_{0}=x$ 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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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(...
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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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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 ...
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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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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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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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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 ...
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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 ...
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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 ...
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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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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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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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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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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|...
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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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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)...
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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(\...
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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, ...
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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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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 ...
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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 $$ \...
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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 ...
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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 $D>0$ ...
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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 ...