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Questions tagged [heat-equation]

Boundedness of solution operator

when dealing with the linear heat equation \begin{align*} \partial_t u &= \partial_{xx} u, \quad 0<x<l, 0<t<T\\ \partial_x u(0,t) &= \partial_x u(l,t) = 0, \quad 0<t<T\\ u(x,...

• 57

asked Oct 20 at 8:44

0 votes

0 answers

18 views

Heat kernel existence on non-compact riemannian manifold: a reference request

The existence of the heat kernel of the Laplace-Beltrami operator on a compact Riemannian manifold is well known (e.g. through the construction via a short time expansion coupled with a Volterra ...

• 33

asked Oct 18 at 16:05

1 vote

0 answers

14 views

Solving the following initial and boundary value problem

I've been doing some work on the heat equation recently and I'm having problems with the following equation: $$u_t-\frac{1}{1-t}u_{xx}=0, 0<x<π, t>0$$ with boundary equations: $$u(x,0)=g(x), ...

• 45

asked Oct 18 at 13:23

1 vote

0 answers

31 views

How to show the following effective boundary condition on a coated body with heat equation?

Let a body $\Omega_{1}$ (space shuttle or turbine engine) be thermally insulated by a thin coating $\Omega_{2}$ of thickness $\delta$. Assume the outer boundary of the coating is subject to a high ...

• 11

asked Oct 17 at 6:24

-1 votes

0 answers

8 views

Entropy increase after random jump

Suppose we have a discrete distribution X with Shannon entropy E. Let Y be a mean-zero random variable. What is a good lower bound of (entropy of X+Y)-(entropy of X)? Note: when Y is Gaussian, we can ...

• 9

asked Oct 17 at 2:16

1 vote

0 answers

14 views

Properties of heat flow semigroup

On a Riemannian manifold $(M^n,g)$, let $H_{B,t}^D$ be the heat flow semigroup associated with $\Delta$ and the Dirichlet boundary condition on a ball $B$, and let $h_{B,t}^D$ be its kernel. Let $h_t$ ...

• 41

asked Oct 16 at 2:38

0 votes

0 answers

37 views

Heat Kernel of a Square

I'm studying Marc Kac's 1966 famous article "Can One Hear the Shape of a Drum". In order to prove that the asymptotic behaviour of the heat kernel in the interior of a planar figure $\Omega\...

• 73

asked Oct 15 at 15:46

1 vote

0 answers

18 views

Odd initial condition makes heat equation odd

Is my proof correct for saying that if the diffusion equation has an odd initial condition then the diffusion equation must be odd. I have this: $$\partial_{t} u - \partial_{xx} u = 0\\ u(x,0) = f(x)\\...

• 33

asked Oct 13 at 1:11

1 vote

0 answers

24 views

Show that the solution to PDE converges to the initial condition in $L^2$

I have a solution, $u(x,t)$, to the heat equation $u_t-u_{xx} = 0$ with initial condition $u(x,0) = f(x)$. I would like to show that as $t \rightarrow 0$, the solution $u(x,t)$ tends to the initial ...

• 113

asked Oct 8 at 14:15

4 votes

0 answers

85 views

Self-similar solutions for a particular parabolic system

Consider the parabolic system \begin{align} \begin{cases} u_t - \Delta\Big((a_1 + a_{11} u + a_{12} v) u\Big) = 0, & t >0, \ x \in \mathbb R^n \\ v_t - \Delta\Big((a_2 + a_{22} v + a_{...

• 7,754

asked Oct 7 at 15:58

0 votes

0 answers

16 views

Solving a PDE for a short time after $t=0$

I have a heat-type equation below from some calculations (it is heat-type because $\rho$ depends on $t$): $\frac{\partial}{\partial t} \log h = \frac{\partial^2}{\partial \rho^2} \log h + \frac{\...

• 1,112

asked Oct 2 at 19:47

Simple doubt about partial differential equation?

I'm reading Farlow's Partial Differential Equations for Scientists and Engineers and am trying to solve the following problem: The answer is: I don't understand how he comes up to this answer. We ...

• 22.8k

asked Sep 29 at 15:09

0 votes

0 answers

15 views

Diffusion equation on Half Line for 2D

I am trying to solve the diffusion equation on the half line, I am very competent at solving this in 1D but cannot quite understand when i have it in 2D. I have: $u_t = \Delta u$ and initial ...

• 33

asked Sep 25 at 10:49

Is there a Theta-Method Numerical Scheme for General Semilinear Parabolic Partial Differential Equation?

Is there an analogue of the $\theta$-method for a general semilinear parabolic partial differential equation $$ \frac{\partial u}{\partial t} = a(x, t) \frac{\partial^{2} u}{\partial x^{2}} + b(x, t) \...

• 13

asked Sep 21 at 17:25

0 votes

0 answers

36 views

Numerical treatment of a 3-dim. partial differential equation in spherical coordinates - question re laplacian

I want to solve the 3-dim. anisotropic and inhomogeneous heat equation numerically. Starting point is $$ q = -A \cdot \nabla u $$ $$ \dot{q} = - \nabla q + f $$ with $x \in \mathbb{R}^3$, energy ...

• 205

asked Sep 21 at 7:00

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