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

learn more… | top users | synonyms

0
votes
1answer
11 views

Help solving Modified 1 dimensional heat equation

$u_t(t,x)=u_{xx}(t,x)-u(t,x)$ and $x\in(0,1), t>0$ with boundary and initial conditions $u(t,0)=0, u(t,1)=e-e^{-1}, u(0,x)=f(x)$. I tried using auxiliary function $v(t,x)=u(t,x)-(e-e^{-1})x$ but ...
0
votes
0answers
19 views

A physical model for pde

Do you know any mathematical model of a physical process such that it satisfies the following equations ? $$\mathrm{u_t=a(t)u_{xx}},\,\,\,0<x<1,\,\,\,t>0$$ ...
1
vote
0answers
21 views

Solving $du/dt=a \Delta u + b (\Delta u)^2$

Consider the function $u(\boldsymbol{x},t)$, where $u:\mathbb{R}^n \times \mathbb{R}_+ \rightarrow \mathbb{R}$ ($\mathbb{R}_+$ denotes nonnegative reals). My question is related to the PDE ...
0
votes
0answers
21 views

How do we deduce the condition for the solution?

Suppose that we have the differential equation $$u_t(x,t)=k^2u_{xx}(x,t), x \in (0,l), t>0$$ $$u(x,t): \text{ heat of rod at the position } x \ (0 \leq x \leq l )$$ If we have Dirichlet ...
0
votes
1answer
13 views

Verifying inhomogeneous solution of heat equation.

Here is my question. How (3.21) is induced? I tried several times but I could not get the answer.
0
votes
0answers
17 views

Conditions for always positive gradient of heat field in evolutionary thermo-elastic system

I am investigating stability and convergence of series of approximations for coupled thermoelasticity problem yielded by one-step recurrent time-integration scheme. I've managed to show that the ...
1
vote
0answers
49 views

solving PDE with state-dependent boundary conditions

I am interested in solving the following PDE (heat equation): $$\frac{\partial u}{\partial t} = \kappa \frac{\partial ^2 u}{\partial x^2}$$ In order to solve it, I discretize space uniformly into $N$ ...
0
votes
0answers
20 views

Two unknowns (heat transfer problem)

Hi I am attempting to work through a heat transfer problem and have reached the following formula, I am struggling to see how they have calculated J1 and J2 as seen below, any help would be ...
2
votes
1answer
57 views

Solution of $u_t=u_{xx}+xu_x$

I've been asked to solve the problem \begin{equation} \left\{\begin{array}{lc} u_t=u_{xx}+xu_x & \mbox{in }x\in\mathbb{R},t>0,\\ u(x,0)=g(x), & \mbox{on ...
0
votes
1answer
21 views

uniqueness heat equation

Consider the heat equation, $(1)$ $u_t=u_{xx}+f(x,t)$, $0<x<1$, $t>0$ $(2)$ $u(x,0)=\phi(x)$ $(3)$ $u(0,t)=g(t)$, $u(1,t)=h(t)$ When one wants to Show the uniqueness of solution of ...
0
votes
0answers
21 views

Finite Difference Method for Heat Equation with Neumann Boundary

I have read the book of Morton and Mayers. In chapter6, it said that the explicit finite difference scheme of a heat equation, $\frac{U^{n+1}_j-U^{n}_{j}}{\Delta ...
1
vote
0answers
19 views

Conservation of the $L^1$ norm in the Heat equation

I consider the homogeneous Heat equation in $\textbf{R}^d \times (0,T]$ for a fixed $T>0$ : \begin{cases} \partial_t f - \Delta f = 0 & \text{in } \textbf{R}^d \times (0,T] \\ f=f_0 & ...
0
votes
0answers
21 views

Parabolic Equation Solution Dependence on Coefficients

Given a heat equation of $u$ with spatial dependent diffusion coefficient $\alpha(x)\ge 0$ $$\frac{\partial u}{\partial t}=\Big(\frac{1}{2}\alpha_1^2\frac{\partial^2}{\partial ...
0
votes
1answer
54 views

Heat Equation Solution Dependence on Diffusion Coefficient

Given a heat equation of $u$ with spatial dependent diffusion coefficient $\alpha(x)\ge 0$ $$\frac{\partial u_\alpha}{\partial t}=\alpha(x) \frac{\partial^2 u_\alpha}{\partial x^2}$$ where $(t,x)\in ...
1
vote
0answers
23 views

Maximum Principle for the Derivatives of Parabolic PDE Solution

Is there a maximum principle for the spatial derivatives of the solution of a parabolic PDE with coefficients in front of the spatial partial derivatives depending on spatial variables only, similar ...
0
votes
1answer
31 views

Fourier series solution of the heat equation on $-2<x<2$

I have to solve the following boundary value problem: $u_t=u_{xx}$, $u(t,-2)=u(t,2)=0$ and $u(0,x)=f(x)$. I tried to solve the problem using the method of separation of variables. So assume ...
0
votes
1answer
34 views

Use energy method prove wave equation has a unique of solution.

Problem: Use energy method prove wave equation has a unique of solution. $\begin{align}U_{tt}+U_t-C^2U_{xx}=0 \\ U(a,t)=U(b,t)=0 \\ U(x,0)=f(x) \\ u_t(x,0)=g(x) \end{align} \quad \text{with} \quad ...
1
vote
0answers
18 views

uniform absolute convergence for function series

Consider the Theta function $\theta(x,t)=\sum\limits_{m=-\infty}^{\infty}K(x+2m,t)$, where $K(x,t)=\frac{e^{-\frac{x^2}{4t}}}{\sqrt{4 \pi t}}$. The $\theta$ function is seen in the solution of the ...
3
votes
1answer
31 views

Heat Equation on a disc

Question: The temperature distribution $u(r, θ, t)$ in a circular disc of unit radius $(0 ≤ r < 1, 0 ≤ θ < 2π)$ evolves according to the heat equation with unit thermal diffusivity $$u_t = ...
2
votes
1answer
37 views

fundamental solution for the heat equation in a rod, $0<x<1$

It is known that the function, called fundamental solution, $K(x,t)=\frac{e^{-\frac{x^2}{4t}}}{\sqrt{4 \pi t}}$ satisfies the equation $u_t=u_{xx}$. I want to find the solution of the problem D.E: ...
1
vote
0answers
21 views

Reducing heat equation into nondimensional form

I want to get nondimensional form of heat equation $u_t=a(x,t)u_{xx}$. For the case of $a(x,t)=a(t)$, by setting $A(t)=\int_0^ta(\eta)d\eta$ and $t=\phi(\tau)$, where $\phi$ is the inverse mapping ...
3
votes
1answer
46 views

Prove that $\{S(t)\}_{t \ge 0}$ is not a contraction semigroup on $L^\infty(\mathbb{R}^n)$

Define for $t > 0$ $$[S(t)g](x) = \int_{\mathbb{R}^n} \Phi(x-y,t)g(y) \, dy \quad (x \in \mathbb{R}^n),$$ where $g : \mathbb{R}^n \to \mathbb{R}$ and $\Phi$ is the fundamental solution of the ...
4
votes
2answers
75 views

heat equation with convection and forcing function

How can I solve this: $u_t - x u_x -x^2 u_{xx} = \ln{x}$ $u(x,0) = \sin ((\pi/2) \ln{x})$ $u(1,t) = 0 \quad u_x(e,t)=0$ What I have so far: Since we have homogenous BC consider no forcing term to ...
0
votes
1answer
21 views

heat equation in three dimensions with non homogeneous bc

I'd like to solve the heat equation for a cylinder in 3D, in cylindrical coordinates with no azimuthal dependence. The equation is homogeneous but the bc at the cylinder wall has an arbitrary ...
0
votes
1answer
49 views

Prove that $\{S(t)\}_{t \ge 0}$ is a contration semigroup on $L^2(\mathbb{R}^n)$

Define for $t > 0$ $$[S(t)g](x) = \int_{\mathbb{R}^n} \Phi(x-y,t)g(y) \, dy \quad (x \in \mathbb{R}^n),$$ where $g : \mathbb{R}^n \to \mathbb{R}$ and $\Phi$ is the fundamental solution of the ...
0
votes
0answers
31 views

Rearranging initial data of the heat equation.What result do we obtain for the solution?

Suppose we have the heat equation on the whole space or on some open and bounded set U with Dirichlet boundary conditions and some initial data $f$ for $t=0$.If we consider a rearrangement of the ...
0
votes
0answers
43 views

Solution to the heat equation using a finite difference scheme

I have used a difference scheme and Fourier Analysis to find an expression for the solution to the heat Equation. My problem is plotting my solution. $w_{k,j,m}=(1-\Delta t\mu_k)^msin(k\pi x_j)$ ...
0
votes
1answer
70 views

Solve Heat Equation using Fourier Transform (non homogeneous)

I know how to solve heat equation where it's like $u_t=k.u_{xx}$ (using Fourier Transform or using Separation of Variables) but this exercise is really difficult for me. I have this: ...
1
vote
1answer
26 views

Heat Equation: Polynomial Inital, Neumann Boundary

Consider the heat equation solution $h(x,t)$ where $x \in [0,1]$ and $t \geq 0$ with initial condition $h(x,t=0)=f(x)$ and Neumann boundary $\frac{d}{dx} h(x=0,t)=0$ and $\frac{d}{dx} h(x=1,t)=0$. ...
1
vote
0answers
36 views

Laplace transform of solution to heat equation

in the book: Partial Differential Equations by Lawrence C. Evans (second edition) on page 203 the author uses the laplace transform to get an solution to the resolvent equation, which I do not really ...
0
votes
0answers
17 views

Energy estimate for non classic heat equation

Let $$u_t = au_{xx}+ cu_x, a>0$$ With $$u(x,0)=f(x), u(0,t)=u(\pi,t)=0$$ Find a bound for $c$ so that the solution doesn't grow for each $f(x)$. My attempt (all integrals are from $0$ to $\pi$): ...
1
vote
0answers
23 views

Modeling point source heat subject to flow

Modeling the 2D diffusion of heat from a point source submerged in flowing water. The system would be a stream (I.e. And open channel velocity profile and not flow through a pipe) I would use the ...
0
votes
0answers
11 views

Solution for a heat equation

I'm not being able to find a solution $u = u(x,t)$ for $u_t = u_{xx}$ in $[0,1]\times[0,+\infty)$ with boundary conditions $u(0,t)=u(1,t)=t$ for $t \geq 0$ and $u(x,0)=0$ for $0 \leq x \leq 1$. I ...
0
votes
1answer
24 views

Dilation properties of heat equation. How can a be treated as a constant, when it is dependent on time?

I have a question about the dilation Properties of the heat Equation. It is proven that each dilation $v(x, t) = u(\sqrt{ax}, at)$, by using the Chain rule, and treating a as a constant. That I ...
9
votes
0answers
157 views

Energy in the heat equation.

Before getting to question, some background. Let $u(x,t)$ be the temperature in a laterally insulated rod of length $L$, at position $x$ and time $t$. The temperature satisfies the heat equation ...
1
vote
0answers
43 views

Discrete Laplacian

I have the following question and I can't figure out how to do the proof. Could you give me some hints in both directions of the equivalence? Suppose $A$ is a bounded subset of $\mathbb{Z}^d$. Then ...
0
votes
1answer
13 views

Radially symmetric heat equation with Gaussian initial conditions

$\rho(\textbf x,t)$ obeys the heat equation i.e. $\rho_t=D \Delta \rho$ with initial conditions $\rho (\textbf x, 0) = \rho_0 e^{-r^2/a^2}$. I have tried separation of variables but this only gives an ...
6
votes
1answer
67 views

Can anyone solve a stochastic differential equation - related to neuroscience research?

I'm a neuroscience grad student, and I'm hoping one of ya'll could help me solve this problem regarding particle diffusion. It relates to my research on molecular-level neural plasticity, but I've ...
1
vote
0answers
43 views

Diffusion of a chemical species inside a Y-shaped tube

I'm trying to model diffusion of a chemical species X inside a Y-shaped tube, whose diameter (thickness) is constant everywhere. The diffusion constant of X is $D$ ($\mu$m$^2$/s), so the concentration ...
1
vote
0answers
33 views

Solve 1D wave equation on half-line using method of images

I'm trying to solve $\theta_t - D\theta_{xx} = f(x,t)$ on the half-line $0 < x < \infty$ for $0< t < \infty$ given boundary and initial conditions $\theta(0,t) = h(t)$, $\theta(x,0) = ...
2
votes
1answer
49 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 ...
3
votes
1answer
62 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
37 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
67 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
63 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
40 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
54 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
150 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
22 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
70 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)= ...