From 040639499846ce1c85f9192f3867b43586c05549 Mon Sep 17 00:00:00 2001 From: Sasserisop Date: Fri, 20 Oct 2023 19:24:29 -0600 Subject: [PATCH] added lec 19 --- content/Convolution (lec 19).md | 15 + .../Drawing 2023-10-20 13.06.35.excalidraw.md | 4439 +++++++++++++++++ .../Drawing 2023-10-20 13.27.58.excalidraw.md | 2799 +++++++++++ content/Periodic functions (lec 19).md | 54 + content/_index.md | 6 +- 5 files changed, 7310 insertions(+), 3 deletions(-) create mode 100644 content/Convolution (lec 19).md create mode 100644 content/Drawing 2023-10-20 13.06.35.excalidraw.md create mode 100644 content/Drawing 2023-10-20 13.27.58.excalidraw.md create mode 100644 content/Periodic functions (lec 19).md diff --git a/content/Convolution (lec 19).md b/content/Convolution (lec 19).md new file mode 100644 index 0000000..ad20c31 --- /dev/null +++ b/content/Convolution (lec 19).md @@ -0,0 +1,15 @@ +# Convolution +A convolution is an operation of function, we take two functions, convolute them and get a new function. +Definition of convolution between f and g: +$$(f*g)(t):=\int _{0} ^t f(t-v)g(v)\, dv$$ +property 1) $f*g=g*f$ +proof: +$f*g=\int _{0} ^t f(t-v)g(v)\, \underset{ t-v=u }{ dv }=-\int _{t} ^0 f(u)g(t-u) \, du$ +$=\int _{0} ^t g(t-u)f(u)\, du=g*f \quad \Box$ + +property 2) $(f+g)*h=f*h+g*h$ +property 3) $(f*g)*h=f*(g*h)$ +property 4) $f*0=0$ +property 5) $\mathcal{L}\{f*g\}=F(s)G(s)$ +he will see us tomorrow at 10oclock. ;) +#end of lec 19 \ No newline at end of file diff --git a/content/Drawing 2023-10-20 13.06.35.excalidraw.md b/content/Drawing 2023-10-20 13.06.35.excalidraw.md new file mode 100644 index 0000000..ef52648 --- /dev/null +++ b/content/Drawing 2023-10-20 13.06.35.excalidraw.md @@ -0,0 +1,4439 @@ +--- + +excalidraw-plugin: parsed +tags: [excalidraw] + +--- +==⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== + + +# Text Elements +%% +# Drawing +```json +{ + "type": "excalidraw", + "version": 2, + "source": 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\quad (T>0)$ if: +$$f(t)=f(t+T), \quad \forall\ t\in \mathbb{R}$$ +![[Drawing 2023-10-20 13.06.35.excalidraw]] +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. +If we take the windowed version of the function (one period, where everywhere else is 0, ie:) + +$f_{T}(t)=\begin{cases}f(t)\ ,\ & 0\leq t\leq T \\0\ ,\ & \text{otherwise}\end{cases}$ + +we can "glue together" many of these windows together to rebuild our $f(t)$, like this: +$f(t)=f_{T}(t)+f_{T}(t-T)u(t-T)+f_{T}(t-2T)u(t-2T)+\dots$ +$\mathcal{L}\{f\}=\mathcal{L}\{f_{T}\}+\mathcal{L}\{f(t-T)u(t-T)\}+\dots$ +recall the formula from last lec: $\mathcal{L}\{u(t-a)f(t-a)\}=e^{-as}F(s)$ +then: +$\mathcal{L}\{f\}=\mathcal{L}\{f_{T}\}(1+e^{-TS}+e^{-2TS}+e^{-3TS}+\dots)$ +$\mathcal{L}\{f\}=\mathcal{L}\{f_{T}\}(1+e^{-TS}+(e^{-TS})^{2}+(e^{-TS})^{3}+\dots)$ +This is a geometric series! $1+r+r^2+\dots$ +Geometric series are convergent when $|r|<1$ +and equal to: $\frac{1}{1-r}$ +in this case, $r=e^{-Ts}$ +so: +$$\mathcal{L}\{f\}=\mathcal{L}\{f_{T}\} \frac{1}{1-e^{-Ts}}$$ +handy formula! ^ will be used again. +#ex +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.) +![[Drawing 2023-10-20 13.27.58.excalidraw]] +$\mathcal{L}\{f\}=\mathcal{L}\{f_{2a}\} \frac{1}{1-e^{-2as}}$ +$f_{2a}=u(t)-u(t-a)$ (this is the first period piece) +$\implies \mathcal{L}\{f_{2a}\}=\mathcal{L}\{u(t)\}-\mathcal{L}\{u(t-a)\}=\frac{1}{s}- \frac{e^{-as}}{s}$ +plug back in: +$\mathcal{L}\{f\}=\mathcal{L}\{f_{2a}\} \frac{1}{1-e^{-2as}}=\frac{1}{s}\cancel{ (1-e^{-as}) } \frac{1}{\cancel{ (1-e^{-as}) }(1+e^{-as})}$ +$$\mathcal{L}\{f\}=\frac{1}{s(1+e^{-as})}$$ + +#ex +$y''+3y'+2y=f(t)$ where $f(t)$ is from the previous example +$y(0)=y'(0)=0,\ a=1$ (a is width of 1/2 period in the function f(t)) +$s^2Y+3sY+2Y=\mathcal{L}\{f\}= \frac{1}{s(1+e^{-s})}$ +$Y(s)=\frac{1}{s(s+1)(s+2)} \frac{1}{1+e^{-s}}$ +$=F(s) \frac{1}{1+e^{-s}}$ + +$\mathcal{L}^{-1}\{F\}=\mathcal{L}^{-1}\{\frac{1}{2} \frac{1}{s}+\frac{1}{2} \frac{1}{s+2}-\frac{1}{s+1}\}$ +$=\frac{1}{2}+\frac{1}{2}e^{-2t}-e^{-t}$ +$y(t)= \dots$ +we want to use formula from earlier +so we need to change $F(s) \frac{1}{1+e^{-s}}$ +to: $F(s) \frac{1e^{-s}}{1-e^{-2s}}$ +$y(t)=p(t)$, periodic with period of 2 ($T=2$) + +$$f_{2a}(t)=\mathcal{L}^{-1}\{F(s)-F(s)e^{-s}\}=\frac{1}{2}+\frac{1}{2}e^{-2t}-e^{-t}-\left( \frac{1}{2}+\frac{1}{2} e^{-2(t-1)}-e^{-(t-1)})u(t-1 \right)$$ + + diff --git a/content/_index.md b/content/_index.md index c37c09f..e554b37 100644 --- a/content/_index.md +++ b/content/_index.md @@ -19,8 +19,8 @@ Good luck on midterms! <3 -Oct 18 2023 [Laplace transform (lec 14-16)](laplace-transform-lec-14-16.html) (raw notes, not reviewed or revised yet.) [Solving IVP's using Laplace transform (lec 17-18)](solving-ivps-using-laplace-transform-lec-17-18.html) (raw notes, not reviewed or revised yet.) [(Heaviside) Unit step function (lec 18)](heaviside-unit-step-function-lec-18.html) (raw notes, not reviewed or revised yet.) - - +[Periodic functions (lec 19)](periodic-functions-lec-19.html) (raw notes, not reviewed or revised yet.) +[Convolution (lec 19)](convolution-lec-19.html)
-[How to solve any DE, a flow chart](Solve-any-DE.png) +[How to solve any DE, a flow chart](Solve-any-DE.png) (Last updated Oct 1st, needs revision. But it gives a nice overview.)
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