Consider a sphere of radius $a$ that is heated uniformly and then immersed in cold water. The temperature in the sphere can be modelled by $\theta(r,t)$ where $\theta(r,t)$ satisfies $$\frac{\partial \theta}{\partial t}= \frac{D}{r^2}\frac{\partial}{\partial r}\left(r^2 \frac{\partial \theta}{\partial r} \right)$$ where $0<r<a, t>0, \theta(a,t)=0,\theta(r,0)=1$. I’m given that the solution is $$\theta(r,t)=\frac{1}{r}\sum_{n=1}^{\infty}a_ne^{\left(\frac{\pi n}{a}\right)^2Dt}\sin\left(\frac{\pi n}{a}r\right)$$ where $$a_n=\frac{-2a(-1)^n}{\pi n}$$ I’m […]

We know that a solution to the Cauchy problem on $\mathbb{R}$ : $u_{xx}=u_t$ with condition $u|_{t=0}=\varphi(x)$ is of the form $$u(x,t)=\dfrac{1}{2\sqrt{\pi t}}\int_{-\infty}^{\infty}\exp\left({\dfrac{-|x-y|^2}{4t}}\right)\varphi(y)dy$$ I want to know what happen as $t\to\infty$ in the case that $\varphi(x)$ is a bounded continuous function that satisfies boundary condition at infinity $\lim_{x\to\infty}\varphi(x)=A$ and $\lim_{x\to-\infty}\varphi(x)=B$ Any hint on how to use […]

So consider the heat equation on a rod of length $L$, $u_t (x,t) = c^2 u_{xx} (x,t)$, $\forall (x,t) \in [0,L]$ x $\mathbb{R}^+ $, and the energy at time $t$ defined as, $$E(t)=\frac{1}{2}\int_{0}^{L} u(x,t)^2 dx.$$ How would I show that $E(t) \geq 0$ for every $t \in \mathbb{R}^+$, and that $$ E'(t) = -c^2 \int_{0}^{L} […]

Given $g : [0,\infty) \to \mathbb{R}$, with $g(0)=0$, derive the formula $$u(x,t)=\frac{x}{\sqrt{4\pi}}\int_0^t \frac 1{(t-s)^{3/2}}e^{-\frac{x^2}{4(t-s)}}g(s)\,ds$$ for a solution of the initial/boundary-value problem \begin{cases}u_t-u_{xx}=0 & \text{in }\mathbb{R}_+ \times (0,\infty) \\ \qquad \quad \, u=0 & \text{on } \mathbb{R}_+ \times \{t=0\} \\ \qquad \quad \, u= g & \text{on }\{x=0\} \times [0,\infty). \end{cases} (Hint: Let $v(x,t):=u(x,t)-g(t)$ and extend […]

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 problem $(1)-(3)$, s/he can use so-called energy method or use maximum principle. My Question: What is the difference between these method? Is the class of solutions change for each method? Thanks in […]

I am confused about different cases of boundary condition for half space heat equation problem and I describe each case next. The basic problem setup is to solve $$ \mathrm{u}_{t} = \mathrm{u}_{xx}\,,\qquad x > 0\,,\quad 0 < t \leq T $$ with initial data $\mathrm{u}\left(\, 0,x\,\right) = f(x)$, consider $f(x)$ to be continuous for all […]

The question was asked here ($u$ is a $C^2$ solution of $u_t – \Delta u = f(u)$ and $u=0$ on $\partial\Omega \times (0,\infty)$. Show if $u(x,0) \geq 0$, then $u(x,t) \geq 0$) However, my question is if $u(x,0)\leq C$ for all $x \in \Omega$, then how to show that $u(x,t)\leq Ce^{Mt}$ for all $x \in […]

Usually second-order linear PDE’s are classified as elliptic, parabolic, or hyperbolic (or ultrahyperbolic) depending on the eigenvalues of the coefficient matrix. The three cases correspond to the three most famous second-order PDE’s: Elliptic – Laplace’s equation $\nabla^2 u = 0$. Parabolic – the heat equation $u_t = \nabla^2 u$. Hyperbolic – the wave equation $u_{tt} […]

A theorem of PDEs sais that the following Cauchy problem for the heat equation \begin{align*} & \partial_t u = \partial_{x}^2 u, \quad (t,x) \in \mathbb{R_+} \times \mathbb{R}, \\ & u|_{ t = 0 } = u_0 \in S'( \mathbb{R} ) \end{align*} admits a unique solution in $ C( \mathbb{R_+}, S'( \mathbb{R} ) ) $, where […]

Consider the Cauchy-Dirichlet problem for the heat equation in a non-cylindrical region $\Omega \subset \mathbf{R}^+ \times \mathbf{R}$: $\Omega = \{ (t,x): \; 0 \leq t \leq 1, \; x \leq (1-t)^\alpha \}$, $\partial_t \theta(t,x) = \frac{1}{2} \partial_{xx} \theta(t,x), \qquad (t,x) \in \Omega$, $\theta(0,x) = h(x)$ where $h(x)$ is $C^\infty$, dominated by $e^{-x^2}$ and $h(x) \to […]

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