fixed broken drawing links
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@ -17,7 +17,7 @@ $\int _{-\infty} ^{\infty} \delta(t-a)f(t)\, dt=f(a)$
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properties:
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$\int_{{-\infty}}^{\infty} \delta(t-a)\, dt=1$
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![[Drawing 2023-10-25 13.16.20.excalidraw]]
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$\int _{-\infty} ^t \delta(t-a)\, dt=\begin{cases}0, & t<a \\ 1, & t\geq a\end{cases}=u(t-a)$
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$u'(t-a)=\delta(t-a)$
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@ -24,7 +24,7 @@ Sometimes the LT of a function does not exist. It can be undefined over an inter
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Def: $f(t)$ is piecewise continuous on an interval $I$ if $f(t)$ is continuous on $I$, except possibly at a <u>finite</u> number of points of <u>jump</u> discontinuity.
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What is continuity? the limit exists and equals the value at that point.
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So what is jump discontinuity? Look at the picture:
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![[Drawing 2023-10-11 13.17.32.excalidraw.png]]
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Def: $f(t)$ is of exponential order $\alpha$ if there exists positive constants $\ T,\ M$ such that $f(t)\leq Me^{\alpha t}$ for all $t\geq T$
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this is important so that: $f(t)e^{-st}\underset{ \text{as } t\to \infty }{ \nrightarrow } \infty$
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@ -37,7 +37,7 @@ $e^{5 t}$ is this of exponential order? Of course, $M=1,\alpha=5$
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what about $\sin(t)$? yes, sin is bounded between -1 and 1
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What about $e^{t^2}$? No, $t^2$ always outgrows $\alpha t$ eventually. This function does not have a LT.
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![[drawings/Drawing 2023-10-11 13.21.18.excalidraw.png]]
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## Properties
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@ -6,7 +6,7 @@ This lecture we will learn about periodic functions, specifically, non-sinusoida
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Definition:
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$f$ is periodic with period $T \quad (T>0)$ if:
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$$f(t)=f(t+T), \quad \forall\ t\in \mathbb{R}$$
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![[Drawing 2023-10-20 13.06.35.excalidraw.png]]
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We will now compute laplace transforms of these periodic functions. Computing DE's containing these periodic functions using something like #voparam would not be easy.
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Let's try taking the laplace of a periodic function $f(t)$:
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If we take the windowed version of the function (one period, where everywhere else is 0, ie:)
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@ -32,7 +32,7 @@ $$\mathcal{L}\{f\}=\mathcal{L}\{f_{T}\} \frac{1}{1-e^{-Ts}}$$
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handy formula! ^ will be used again.
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#ex
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Imagine another function: (image is of a square wave with a period of 2a, oscillates between 1 and 0, starts at 1 when t=0.)
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![[Drawing 2023-10-20 13.27.58.excalidraw.png]]
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Let's compute its LT:
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$\mathcal{L}\{f\}=\mathcal{L}\{f_{2a}\} \frac{1}{1-e^{-2as}}$
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$f_{2a}=u(t)-u(t-a)$ (this is the first period piece)
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@ -17,7 +17,7 @@ Theorem: With each $\sum_{n=0}^{\infty}a_{n}(x-x_{0})^n$ we can associate a radi
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The series is absolutely convergent
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for all $x$ such that $\mid x-x_{0}\mid<\rho$, and divergent for all $x$ where $\mid x-x_{0}\mid>\rho$
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"Who keeps stealing the whiteboard erases? (jokingly) It's a useless object, anyways"
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![[Drawing 2023-10-30 13.12.57.excalidraw.png]]
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how can we find $\rho$?
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Definition of ratio test: If $\lim_{ n \to \infty }\mid \frac{a_{n+1}}{a_{n}}\mid=L$
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then the radius of convergence $\rho$ is: $\rho=\frac{1}{L}$
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@ -189,7 +189,7 @@ if $p(x)$ and $q(x)$ are <u>analytic</u> functions in a vicinity of $x_{0}$ then
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we expect that the solution $y$ can be represented by a power series. This is true according to the following theorem:
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Theorem: If $x_{0}$ is an ordinary point then the differential equation above has two linearly independent solution of the form $\sum_{n=0} ^\infty a_{n}(x-x_{0})^n, \qquad\sum_{n=0}^\infty b_{n}(x-x_{0})^n$.
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The radius of convergence for them is at least as large as the distance between $x_{0}$ and the closest singular point (which can be real or complex).
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![[Drawing 2023-10-30 13.12.57.excalidraw.png]]
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## Examples for calculating $\rho$
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#ex
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@ -218,7 +218,7 @@ $y''+\frac{x}{x^2+1}y'+\frac{y}{x^2+1}=0$
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remember singular points can be complex the two singular points are:
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$x^2=1=0 \qquad x=\pm i$
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now we have to compute the two distances of these singular points to x=1
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![[Drawing 2023-11-03 13.40.54.excalidraw.png]]
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To calculate distance: $\alpha_{1}+\beta_{1}i, \qquad \alpha_{2}+\beta_{2}i$
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$\sqrt{ (\alpha_{1}-\alpha_{2})^2+(\beta_{1}-\beta_{2})^2 }$
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$\rho\geq \sqrt{ 1^2+1^2 }=\sqrt{ 2 }$
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@ -75,7 +75,7 @@ To recap, taking a frictionless mass spring system which is initially at rest, a
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>Sometimes I feel this effect when I'm sitting in the back of the bus, where the bus is stopped at a red light. I wonder if this modulated vibration pattern is attributed to this exact phenomenon.
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# Shortcut for solving DE of a mass-spring system
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![[Drawing 2023-10-06 13.24.11.excalidraw]]
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$my''+by'+ky=mg+F$
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move $mg$ to LHS and replace $y$ with $y_{new}$ (remember, the $\frac{mg}{k}$ is a constant, its derivative is 0):
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$m\left( y-\frac{mg}{k} \right)''+b\left( y-\frac{mg}{k} \right)'+k\left( y-\frac{mg}{k} \right)=F$
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@ -5,7 +5,7 @@ Second order equations arise from very simple problems many engineers face, for
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#second_order
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### $$a_{2}(t)y''+a_{1}(t)y'+a_{0}(t)y=f(t)$$
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To motivate our interest: #fix
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![[Drawing 2023-09-15 13.32.48.excalidraw]]
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$ma=my''=-by'-ky$
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Look how a second order equation describes the motion of a mass-spring system!
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> Circuits that contains resistors, capacitors and inductors also behaves with this equation as well if you ignore the external magnetic fields around the circuit.
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@ -4,9 +4,7 @@ We start with some thermodynamics
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# Heat equation
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Heat equation not only describes thermodynamics but it can also model the diffusion of gasses. It is a partial differential equation.
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Strikingly, it can also model option prices in the stock market. However, using it as a strategy to make money is not so simple, because if it worked then everyone would try to use it to make money, which would cause the overall strategy to be less effective as the option prices start to get priced to accommodate for the prediction (🤯).
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![[Drawing 2023-11-08 13.07.19.excalidraw.png]]
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>I'm sorry the image doesn't display properly :( I'm trying to get images to work on my notes. For now you can see the relevant .png files in the github repo under content/drawings/
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We assume that the tube is perfectly insulating along its surface, this helps reduce the problem into a one dimensional problem. Heat can only travel inside and along the x axis.
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Fourier figured out that:
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$\text{Heat flux} = -k(x)a\frac{\partial u}{\partial x}(t,x) \Delta t$
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@ -1,7 +1,7 @@
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#ex #SoLE
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Lets start modelling some electric circuits again:
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![[Drawing 2023-10-25 13.43.26.excalidraw]]
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The circuit is switched on (battery in series) at $t=0$ and is then switched off (battery is bypassed) at $t=1$
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Applying KVL:
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$0.2I_{1}'+0.1I_{3}'+2I_{1}=g(t) \qquad \text{where } g(t)=\begin{cases}6, & 0\leq t\leq 1 \\0, & 1<t\end{cases}$
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@ -331,7 +331,7 @@ mark {
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background-color: var(--highlighter-marker-color); }
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img {
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max-width: 100%;
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height: auto; }
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@ -1,458 +0,0 @@
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textarea,
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::selection {
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background: var(--selected-text-background-color); }
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/**
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Bear base styles
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(Retrieved from official app, all credits belong to Bear Team)
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*/
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||||
|
||||
.address a {
|
||||
color: #545454;
|
||||
background-image: linear-gradient(to bottom, rgba(0, 0, 0, 0) 50%, #0da35e 50%);
|
||||
background-repeat: repeat-x;
|
||||
background-size: 2px 2px;
|
||||
background-position: 0 1.05em; }
|
||||
|
||||
.address svg {
|
||||
position: relative;
|
||||
top: 0.2em;
|
||||
display: inline-block;
|
||||
margin-right: 0.2em; }
|
||||
|
||||
.color-preview {
|
||||
display: inline-block;
|
||||
width: 1em;
|
||||
height: 1em;
|
||||
border: solid 1px rgba(0, 0, 0, 0.3);
|
||||
border-radius: 50%;
|
||||
margin-right: 0.1em;
|
||||
position: relative;
|
||||
top: 0.2em;
|
||||
white-space: nowrap; }
|
||||
|
||||
.color-code {
|
||||
margin-right: 0.2em;
|
||||
font-family: "Menlo-Regular";
|
||||
font-size: 0.9em; }
|
||||
|
||||
.color-hash {
|
||||
opacity: 0.4; }
|
||||
|
||||
.ordered-list-number {
|
||||
color: var(--accent-text-color);
|
||||
text-align: right;
|
||||
display: inline-block;
|
||||
min-width: 1em; }
|
||||
|
||||
.arrow svg {
|
||||
position: relative;
|
||||
top: 0.08em;
|
||||
display: inline-block;
|
||||
margin-right: 0.15em;
|
||||
margin-left: 0.15em; }
|
||||
|
||||
.arrow svg #rod {
|
||||
stroke: #545454; }
|
||||
|
||||
.arrow svg #point {
|
||||
fill: #545454; }
|
||||
|
||||
mark {
|
||||
color: inherit;
|
||||
display: inline;
|
||||
padding: 0.2em 0.5em;
|
||||
background-color: var(--highlighter-marker-color); }
|
||||
|
||||
img {
|
||||
max-width: 100%;
|
||||
height: auto; }
|
||||
|
||||
/**
|
||||
Custom styles
|
||||
*/
|
||||
li > p {
|
||||
display: inline-block;
|
||||
margin-left: 16px; }
|
||||
|
||||
hr {
|
||||
background: var(--divider-color);
|
||||
margin-top: 2.6em;
|
||||
margin-bottom: 12px; }
|
||||
|
||||
h4 {
|
||||
font-family: "AvenirNext-Medium";
|
||||
color: var(--title-text-color);
|
||||
font-size: 1.05em;
|
||||
margin-bottom: .47143em; }
|
||||
|
||||
* + p,
|
||||
* + ul,
|
||||
* + ol,
|
||||
* + blockquote {
|
||||
/*margin-top: 1.6em;*/ }
|
||||
|
||||
svg + ul,
|
||||
svg + ol {
|
||||
margin-top: 0; }
|
||||
|
||||
* + h2,
|
||||
* + h3,
|
||||
* + h4 {
|
||||
margin-top: 2.8em; }
|
||||
|
||||
h1, h2, h3, h4, h5, h6 {
|
||||
position: relative; }
|
||||
|
||||
h1:before,
|
||||
h2:before,
|
||||
h3:before,
|
||||
h4:before,
|
||||
h5:before,
|
||||
h6:before {
|
||||
position: absolute;
|
||||
left: -2.2em;
|
||||
color: var(--heading-indicator);
|
||||
font-size: 12px; }
|
||||
|
||||
h1:before {
|
||||
content: "H1"; }
|
||||
|
||||
h2:before {
|
||||
content: "H2"; }
|
||||
|
||||
h3:before {
|
||||
content: "H3"; }
|
||||
|
||||
h4:before {
|
||||
content: "H4"; }
|
||||
|
||||
h5:before {
|
||||
content: "H5"; }
|
||||
|
||||
h6:before {
|
||||
content: "H6"; }
|
||||
|
||||
.note-wrapper {
|
||||
margin-top: 2em; }
|
||||
|
||||
.highlighted {
|
||||
background: var(--selected-text-background-color); }
|
||||
|
||||
.hashtag {
|
||||
cursor: pointer;
|
||||
margin-bottom: 4px;
|
||||
position: relative;
|
||||
margin-right: 2px; }
|
||||
|
||||
.hashtag:before {
|
||||
content: "#"; }
|
||||
|
||||
* + table {
|
||||
margin-top: 12px; }
|
||||
|
||||
table {
|
||||
border-radius: 4px;
|
||||
border: 1px solid var(--separator-color);
|
||||
border-collapse: inherit;
|
||||
overflow: hidden;
|
||||
width: 100%; }
|
||||
|
||||
table th {
|
||||
font-family: "Avenir-Medium", "AvenirLTStd-Medium"; }
|
||||
|
||||
table td,
|
||||
table th {
|
||||
padding: 0.3em 0.8em; }
|
||||
|
||||
table tr:nth-child(odd) {
|
||||
background-color: var(--selected-text-inactive-background-color); }
|
||||
|
||||
pre,
|
||||
p code,
|
||||
li code {
|
||||
border: 1px solid var(--separator-color);
|
||||
padding: 10px;
|
||||
font-size: 12px;
|
||||
background: white;
|
||||
overflow: hidden; }
|
||||
|
||||
li code,
|
||||
p code {
|
||||
padding: 2px; }
|
||||
|
||||
li img {
|
||||
width: 122px; }
|
||||
|
||||
.thumbnail {
|
||||
width: 122px;
|
||||
height: 76px;
|
||||
border: 1px solid var(--separator-color);
|
||||
object-fit: cover; }
|
||||
|
||||
.turbolinks-progress-bar {
|
||||
visibility: hidden; }
|
||||
|
|
@ -1 +0,0 @@
|
|||
{"Target":"css/style.css","MediaType":"text/css","Data":{}}
|
||||
|
|
@ -405,7 +405,7 @@ mark {
|
|||
}
|
||||
|
||||
img {
|
||||
max-width: 100%;
|
||||
max-width: 70%;
|
||||
height: auto
|
||||
}
|
||||
|
||||
|
|
|
|||
Loading…
Reference in New Issue