added lec 26

content/Partial differential equations (lec 26).md

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+
+#start of lec 26
+We start with some thermodynamics
+# Heat equation
+Heat equation not only describes thermodynamics but it can also model the diffusion of gasses.
+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 (🤯).
+![[Drawing 2023-11-08 13.07.19.excalidraw.png]]
+>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/ 
+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.
+Fourier figured out that:
+$\text{Heat flux} =   -k(x)a\frac{\partial u}{\partial x}(t,x) \Delta t$ 
+heat flux is in the positive x direction
+where du/dx is the opposite sign of the flux (because hot flows to cold.)
+where $u(t,x)$ is a function that describes the temperature in the tube.
+$-k(x+\Delta x)a\frac{\partial u}{\partial x}(t,x+\Delta x) \Delta t$
+$\Delta u=u(t+\Delta t,x)-u(t,x)$
+Amount of heat to change temperature over $\Delta t$ with $\Delta u$ is $\underbrace{ C(x) }_{ \text{specific heat capacity } }\underbrace{ \rho(x) }_{ \text{density} }\Delta u\underbrace{ a\Delta x }_{ \text{volume} }$
+
+$c(x)\rho(x)a\Delta x(u(t+\Delta t,x)-u(t,x))=a\Delta t(k(x+\Delta x) \frac{\partial u}{\partial x}(t_{1}x+\Delta x)-k(x) \frac{\partial u}{\partial x}(t,x))+Q(t,x)\Delta xa\Delta t$
+divide by $a\Delta x\Delta t$
+$c(x)\rho(x)\frac{(u(t+\Delta t,x)-u(t,x))}{\Delta t}=a\Delta t\left( k(x+\Delta x) \frac{\partial u}{\partial x}(t,x+\Delta x)-k(x) \frac{\partial u}{\partial x}(t,x) \right)+Q(t,x)$
+$\lim_{ \Delta x,\Delta t \to 0 }: c(x)\rho(x)\frac{ \partial u }{ \partial t }(t,x)=\frac{ \partial  }{ \partial x }\left( k(x)\frac{ \partial u }{ \partial x }(t,x) \right)+Q(t,x)$
+Any differential equation, to his knowledge, is to balance some volume and to take the limit to produce a pointwise / instantaneous balance (in the form of a differential equation)
+the Maxwell equations that describes magnetic and electric fields are a system of partial differential equations.
+thermodynamics can be very important for electrical engineers, for instance the heat produced in a transformer, or a battery. It has applications.
+$\frac{ \partial u }{ \partial t }=D \frac{ \partial^2 u }{ \partial x^2 }, \quad 0\leq x\leq L, \quad t>0$
+boundary conditions:
+$u(t,0)=u(t,L)=0 , \quad t>0$ (simple case)
+$u(0,x)=f(x) , \quad 0\leq x\leq L$
+These three equations form an IBVP (initial boundary value problem)
+we will work on this equation next lecture:
+$u(t,x)=T(t)X(x)$
+$T'x=DTx''$
+divide by DTx:
+$\frac{T'}{DT}=\frac{x''}{x}$
+on the left is a function of time only, on the right is a function of x only.
+$\frac{T'}{DT}=\frac{x''}{x}=-\lambda$ where $\lambda$ is a constant
+$x''+\lambda x=0$
+$u(t,0)=\cancel{ T(t) } \quad x(0)=0$
+$x(0)=x(L)=0$
+this is called an eigen value problem.
+
+#end of lec 26

content/_index.md

@@ -19,10 +19,11 @@ Good luck on midterms! <3 -Oct 18 2023
 [Laplace transform (lec 14-16)](laplace-transform-lec-14-16.html)
 [Solving IVP's using Laplace transform (lec 17-18)](solving-ivps-using-laplace-transform-lec-17-18.html)
 [(Heaviside) Unit step function (lec 18)](heaviside-unit-step-function-lec-18.html)
-[Periodic functions (lec 19)](periodic-functions-lec-19.html) (raw notes, not reviewed or revised yet.)
+[Periodic functions (lec 19)](periodic-functions-lec-19.html)
 [Convolution (lec 19-20)](convolution-lec-19-20.html) (raw notes, not reviewed or revised yet.)
 [Dirak δ-function (lec 21)](dirak-δ-function-lec-21.html) (raw notes, not reviewed or revised yet.)
 [Power series (lec 22-25)](power-series-lec-22-25.html) (raw notes, not reviewed or revised yet.)
+[Partial differential equations (lec 26)](partial-differntial-equations-lec-26.html) (raw notes, not reviewed or revised yet.)
 </br>
 [How to solve any DE, a flow chart](Solve-any-DE.png) (Last updated Oct 1st, needs revision. But it gives a nice overview.)
 [Big LT table (.png)](drawings/bigLTtable.png)