definition of left and right limits. definition of piecewise continuous function.
definition of Fourier cosine series \(f\left ( x\right ) =\frac{a_{0}}{2}+\sum _{n=1}^{\infty }a_{n}\cos \left ( n\frac{2\pi }{T}x\right ) =\frac{a_{0}}{2}+\sum _{n=1}^{\infty }a_{n}\cos \left ( nx\right ) \) for \(0<x<\pi \).
Examples of Fourier cosine series
definition of Fourier sine series \(f\left ( x\right ) =\sum _{n=1}^{\infty }b_{n}\sin \left ( n\frac{2\pi }{T}x\right ) =\frac{a_{0}}{2}+\sum _{n=1}^{\infty }b_{n}\sin \left ( nx\right ) \) for \(0<x<\pi \).
Examples of Fourier sine series
Fourier series For period \(T=2\pi \) \begin{align*} f\left ( x\right ) & \approx \frac{a_{0}}{2}+\sum _{n=1}^{\infty }a_{n}\cos \left ( n\frac{2\pi }{T}x\right ) +b_{n}\sin \left ( n\frac{2\pi }{T}x\right ) \qquad -\pi <x<\pi \\ & \approx \frac{a_{0}}{2}+\sum _{n=1}^{\infty }a_{n}\cos \left ( nx\right ) +b_{n}\sin \left ( nx\right ) \end{align*}
Where
\begin{align*} a_{n} & =\frac{1}{\pi }\int _{-\pi }^{\pi }f\left ( x\right ) \cos \left ( nx\right ) dx\qquad n=0,1,2,\cdots \\ b_{n} & =\frac{1}{\pi }\int _{-\pi }^{\pi }f\left ( x\right ) \sin \left ( nx\right ) dx\qquad n=1,2,\cdots \end{align*}
If \(f\left ( x\right ) \) is even then \(b_{n}=0\) and if \(f\left ( x\right ) \) is odd, then \(a_{n}=0\).
Fourier series examples.
Shows how F.S. on \(-L<x<L\,\) can be obtained from know F.S. on \(-\pi <x<\pi \). Not clear why example 2 on page 22 replaces \(a=\frac{1}{\pi }\).
\begin{align*} f_{+}^{\prime }\left ( x_{0}\right ) & =\lim _{\substack{x\rightarrow x_{0}\\x>x_{0}}}\frac{f\left ( x\right ) -f\left ( x_{0}^{+}\right ) }{x-x_{0}}\\ f_{-}^{\prime }\left ( x_{0}\right ) & =\lim _{\substack{x\rightarrow x_{0}\\x>x_{0}}}\frac{f\left ( x\right ) -f\left ( x_{0}^{+}\right ) }{x-x_{0}} \end{align*}
Smooth function is one who is continuous and its derivative is also continuous. For example \(f\left ( x\right ) =x^{2}\) is smooth, but \(f\left ( x\right ) =\left \vert x\right \vert \) is not smooth.
Piecewise smooth function is one which \(f\left ( x\right ) \) and \(f^{\prime }\left ( x\right ) \) are piecewise continuous.
Bessel’s inequalities
\begin{align*} \frac{a_{0}^{2}}{2}+\sum _{n=1}^{\infty }a_{n}^{2} & \leq \frac{2}{\pi }\int _{0}^{\pi }\left [ f\left ( x\right ) \right ] ^{2}dx\\ \lim _{n\rightarrow \infty }a_{n} & =0\\ \sum _{n=1}^{\infty }b_{n}^{2} & \leq \frac{2}{\pi }\int _{0}^{\pi }\left [ f\left ( x\right ) \right ] ^{2}dx\\ \lim _{n\rightarrow \infty }b_{n} & =0 \end{align*}
Lemma 1 If \(f\left ( x\right ) \) is P.W.C. on \(0<x<\pi \) then
\[ \lim _{N\rightarrow \infty }\int _{0}^{\pi }f\left ( x\right ) \sin \left ( \left ( N+\frac{1}{2}\right ) x\right ) dx=0 \]
Lemma 2 If \(g\left ( x\right ) \) is P.W.C. on \(0<x<\pi \) and that \(g_{+}^{\prime }\left ( 0\right ) \) exist, then
\[ \lim _{N\rightarrow \infty }\int _{0}^{\pi }g\left ( x\right ) \frac{\sin \left ( \left ( N+\frac{1}{2}\right ) x\right ) }{2\sin \frac{x}{2}}dx=\frac{\pi }{2}g\left ( 0^{+}\right ) \]
Where \(\frac{\sin \left ( \left ( N+\frac{1}{2}\right ) x\right ) }{2\sin \frac{x}{2}}\) is called the Dirichlet kernel \(D_{N}\left ( x\right ) \).
\begin{align*} D_{N}\left ( x\right ) & =\frac{1}{2}+\sum _{n=1}^{N}\cos \left ( nx\right ) \\ D_{N}\left ( x\right ) & =\frac{\sin \left ( \left ( N+\frac{1}{2}\right ) x\right ) }{2\sin \frac{x}{2}}\\ \int _{0}^{\pi }D_{N}\left ( x\right ) dx & =\frac{\pi }{2} \end{align*}
If \(f\left ( x\right ) \) is P.W.C. on \(-\pi <x<\pi \) and \(f\left ( x\right ) \) is periodic on all of \(x\) with period \(2\pi \) then at each \(x\) where \(f_{+}^{\prime }\left ( x\right ) \) and \(f_{-}^{\prime }\left ( x\right ) \) both exist, then \(f\left ( x\right ) \) converges to the average of \(f\left ( x\right ) \) at \(x\) which is \(\frac{f\left ( x^{+}\right ) +f\left ( x^{-}\right ) }{2}\). Proof is long.
Nothing new here. Seems same as last one. If \(f\left ( x\right ) \) is PWC and \(f^{\prime }\left ( x\right ) \) is PWC, and \(f\left ( x\right ) \) is periodic, then F.S. of \(f\left ( x\right ) \) converges to mean of \(f\left ( x\right ) \) at each point \(x\).
Examples on the Fourier theorem
Nothing new here.
If \(f\left ( x\right ) \) is continuous on \(-\pi <x<\pi \) (notice it has to be continuous, not PWC) and if \(f\left ( -\pi \right ) =f\left ( \pi \right ) \) and \(f^{\prime }\left ( x\right ) \) is PWC on \(-\pi <x<\pi \) then
\[ \sum _{n=1}^{\infty }a_{n}^{2}+b_{n}^{2}\]
converges. Proof is given. And
\[ \sum _{n=1}^{N}\alpha _{n}^{2}+\beta _{n}^{2}\leq \frac{1}{\pi }\int _{-\pi }^{\pi }\left [ f^{\prime }\left ( x\right ) \right ] ^{2}dx\qquad N=1,2,3,\cdots \]
Where \begin{align*} f^{\prime }\left ( x\right ) & =\frac{\alpha _{0}}{2}+\sum _{n=1}^{\infty }\alpha _{n}\cos \left ( nx\right ) +\beta _{n}\sin \left ( nx\right ) \\ \alpha _{0} & =0\\ \alpha _{n} & =nb_{n}\\ \beta _{n} & =na_{n} \end{align*}
M test is used to check if series is U.C. (uniform convergent). If we can find \(\sum _{n=1}^{\infty }M_{n}\) which is convergent and \(M_{n}\) is positive constant, and where \(\left \vert f_{n}\left ( x\right ) \right \vert \leq M_{n}\) for each \(n\) in \(a<x<b\), then series \(\sum _{n=1}^{\infty }f_{n}\left ( x\right ) \) is U.C.
Theorem If \(f\left ( x\right ) \) is continuous on \(-\pi \leq x\leq \pi \) and \(f\left ( -\pi \right ) =f\left ( \pi \right ) \) and \(f^{\prime }\left ( x\right ) \) is PWC, then \(f\left ( x\right ) \) both absolutely and uniformly convergent,
Not on exam.
Same conditions as section 17 theorem. If \(f\left ( x\right ) \) is continuous on \(-\pi \leq x\leq \pi \) and \(f\left ( -\pi \right ) =f\left ( \pi \right ) \) and \(f^{\prime }\left ( x\right ) \) is PWC, then F.S. of \(f\left ( x\right ) \) can be differentiated term by term.
As long as \(f\left ( x\right ) \) is PWC, we can integrate F.S. term by term.
\[ Au_{xx}+Bu_{xy}+Cu_{yy}+Du_{x}+Eu_{y}+Fu=G \]
And definitions.
Flux is \(\Phi =-K\frac{du}{dn}\) where \(K\) is thermal conductivity. Flux is amount of heat passing in normal direction per unit area in one second. Derivation of heat PDE
\[ u_{t}=ku_{xx}\]
where \(k\) is thermal diffusivity \(k=\frac{K}{\sigma \delta }\) where \(\sigma \) is specific heat and \(\delta \) is density of material.
Nothing much here.
Just need to know the equations. Will be given in exam.
Not in exam
Just need to know Neumann and Dirichlet.
Do not think this will be on exam.
Derivation of \(y_{tt}=a^{2}y_{xx}\) using physics. Will not be on exam.
Generalization of section 28.
To derive solution to \(y_{tt}=a^{2}y_{xx}\), use \(u=x+at,v=x-at\) and the PDE becomes \(y_{uv}=0\) which has solution \(y=\Phi \left ( u\right ) +\Psi \left ( v\right ) \) or
\[ y\left ( x,t\right ) =\Phi \left ( x+at\right ) +\Psi \left ( x-at\right ) \]
Where initial conditions are \(y\left ( x,0\right ) =f\left ( x\right ) ,y_{t}\left ( x,0\right ) =g\left ( x\right ) \) then the solution becomes
\[ y\left ( x,t\right ) =\frac{1}{2}\left ( f\left ( x+at\right ) +f\left ( x-at\right ) \right ) +\frac{1}{2a}\int _{x-at}^{x+at}g\left ( s\right ) ds \]
\[ L\left ( c_{1}u_{1}+c_{2}u_{2}\right ) =c_{1}Lu_{1}+c_{2}Lu_{2}\]
Suppose each function \(u_{i}\) satisfies a linear homogeneous differential equation or boundary value problem \(Lu=0\), then \(\sum _{n=1}^{\infty }u_{n}\) also satisfies the same equation.
Some examples. Go over.
Show how to solve \(X^{\prime \prime }+\lambda X=0\) for different boundary conditions.
Applying Eigenvalues and eigenfunctions to heat PDE on rod.
Applying Eigenvalues and eigenfunctions to wave PDE On string \(u_{tt}=a^{2}u_{xx}\) with fixed on ends and have initial conditions.
Not on exam