From cef502af341568e0e380e2f5c554e7bf7475d765 Mon Sep 17 00:00:00 2001 From: Sasserisop Date: Thu, 21 Dec 2023 16:44:30 -0700 Subject: [PATCH] few fixes and added temporary message on index page --- content/Heat equation (lec 30-33).md | 2 +- content/Periodic functions (lec 19).md | 2 +- content/Wave equation (lec 33-36).md | 4 ++-- content/_index.md | 3 +++ 4 files changed, 7 insertions(+), 4 deletions(-) diff --git a/content/Heat equation (lec 30-33).md b/content/Heat equation (lec 30-33).md index c968cff..e88f89c 100644 --- a/content/Heat equation (lec 30-33).md +++ b/content/Heat equation (lec 30-33).md @@ -194,7 +194,7 @@ $a_{0}=\frac{2}{\pi}\int _{0} ^{\pi/2} 1\, dx=1$ $a_{n}=\frac{2}{\pi}\int _{0} ^{\pi/2} 1\cdot\cos(nx) \, dx=\frac{2}{n\pi}\sin(nx)|_{0} ^\frac{\pi}{2}=\frac{2}{n\pi}\sin\left( \frac{n\pi}{2} \right) \quad n=1,2,\dots$ We can take out the zero terms: $a_{2k}=0 \qquad a_{2k-1}=\frac{2}{(2k-1)\pi}(-1)^{k+1}$ -$u(t,x)=\frac{1}{2}+\frac{2}{\pi}\sum_{k=1}^\infty \frac{(-1)^{k+1}}{2k-1}e^{-(2k-1)^2t}\cos((2k-1)x)$ +$$u(t,x)=\frac{1}{2}+\frac{2}{\pi}\sum_{k=1}^\infty \frac{(-1)^{k+1}}{2k-1}e^{-(2k-1)^2t}\cos((2k-1)x)$$ We are done, and we didn't need to take out the zero terms but if you want to be diligent, then there you go. Plot (only showing 100 harmonics, that's why the red line looks a lil' wiggly): ![plot](drawings/insulatedheat.png) diff --git a/content/Periodic functions (lec 19).md b/content/Periodic functions (lec 19).md index 68371a0..e929937 100644 --- a/content/Periodic functions (lec 19).md +++ b/content/Periodic functions (lec 19).md @@ -43,7 +43,7 @@ $$\mathcal{L}\{f\}=\frac{1}{s(1+e^{-as})}$$ #ex #IVP #periodic #second_order_nonhomogenous #LT #partial_fractions Solve for $y(t)$ in the following second order periodic equation: -$$y''+3y'+2y=f(t) \qquad y(0)=y'(0)=0 \quad a=1$$where $f(t)$ is from the previous example and $a$ is the width of $\frac{1}{2}$ of a period in the function $f(t)$ +$$y''+3y'+2y=f(t) \qquad y(0)=y'(0)=0 \quad a=1$$where $f(t)$ is from the previous example and $a$ is the width of $\frac{1}{2}$ of a period in the function $f(t)$. This means $f(t)$ has a period of $T=2$ Hit it with the LT! $s^2Y+3sY+2Y=\mathcal{L}\{f\}= \frac{1}{s(1+e^{-1s})}$ $(s+1)(s+2)Y= \frac{1}{s(1+e^{-1s})}$ diff --git a/content/Wave equation (lec 33-36).md b/content/Wave equation (lec 33-36).md index e5b95f7..fc527c8 100644 --- a/content/Wave equation (lec 33-36).md +++ b/content/Wave equation (lec 33-36).md @@ -55,11 +55,11 @@ btw this equation models the electromagnetic field, to some approximation. the lowest mode is called the fundamental mode, the following terms after are called harmonics. If two instruments play the same note (same fundamental frequency), they sound different! and that's because of their difference in harmonics. The modes are standing waves in the string. -"my claim, and this is not just my claim [...], is that any object, including social objects, can be described by waves. [...] Everything is a wave." +"My claim, and this is not just my claim [...], is that any object, including social objects, can be described by waves. [...] Everything is a wave." You can model elementary particle behaviours with solitons (non linear waves.) In life in the real world, all waves have finite speed. So that's why its important to learn the wave equation. It's the prototype to waves. -"waves are the fundamental object. [...]. So that's why it's important, these are the fundamental objects of nature here." +"Waves are the fundamental object. [...]. So that's why it's important, these are the fundamental objects of nature here."
So, let's try an example using our formula above: $f(x)=\begin{cases}x, & 0\leq x\leq \frac{\pi}{2}\\ \ \pi-x, & \frac{\pi}{2} +Good luck on the final! <3 +If we do bad on the exam, Petar will come after us with the Dirac delta 🤜💥 +
[Separable equations (lec 1)](separable-equations-lec-1.html) [Homogenous equations (lec 2)](homogenous-equations-lec-2.html) [Linear equations (lec 2-3)](linear-equations-lec-2-3.html)