Heat Equation Questions and Answers

Heat Equation

$30.00

answered

Prove that $ \sum_{k=1}^{\infty} (-1)^k q^{k^2} \geq \frac{2}{\pi}\ \sum_{k=1}^{\infty} \frac{(-1)^k}{2k-1}{q}^{\frac{ (2k-1)^2}{4}}  $ for $0<q<1$ (updated bounty) $30.00 answered The title explains it. Prove that : $ \sum_{k=1}^{\infty} {(-1)^k\rm e}^{-\hat{t} k^2} \geq \frac{2}{\pi}\ \sum_{k=1}^{\infty} \frac{(-1)^k}{2k-1}{\rm e}^{\frac{-\hat{t} (2k-1)^2}{4}}  $ for $\hat{t}&...

Bob Lansdorp

$45.00

answered

Derive the solution $u(x,t)=\frac{x}{\sqrt{4 \pi}} \int_{0}^{t} \frac{1}{(t-s)^{3/2}}e^{\frac{-x^2}{4(t-s)}}g(s) \, ds$ for the heat equation $45.00 answered Given $g: [0,\infty) \rightarrow \R$, with $g(0)=0$, derive the formula for a solutions of the initial/boundary-value problem (Hint: Let $v(x,t) := u(x,t) - g(t)$  and extend $v$ to $\{x<0\}$ by...

Daniel

$2.00

answered

Why does this spatial discretization with n intervals have a position of (n-1)/n for each interval? $2.00 answered Hello, all. I am learning about how to descritize the Heat Equation in space. This case involves a rod along the xi-axis with some thermal conductivity, kappa.Now, I understand that one would descriti...

Exactplace

$5.00

answered

Integral of the fundamentla solution of the heat equation $5.00 answered Show that \int_{\mathbb{R}^n}\frac{1}{(4\pi t)^{\frac{n}{2}}}e^{-\frac{|x|^2}{4t}}dx=1, \forall t>0.

Poincare

$6.00

answered

Maximum principle for the heat equation involving an aditional linear term  $6.00 answered Let $\Omega$ be a bounded open region in $\mathbb{R}^n$, $\alpha >0$, $T>0$ and $f \in C^0 (\overline{\Omega})$ with $f \geq 0$ on $\Omega$. Also let $u \in C_1^2 (\Omega _T) \cap C^0 (\overline...

Daniel