MATH529: L10 Orthogonal series expansions

Overview

• Orthogonal series

Robin boundary condition problem

• Consider heat conduction PDE $k{u}_{xx}=u_t$ for $0<x<1,t>0$, with BCs

  ${\displaystyle u(0,t)=0,hu(1,t)+u_x(1,t)=0,u(x,0)=1}$

• Separation of variables $u(x,t)=X\left(x\right)T\left(t\right)$ leads to Sturm-Liouville

  ${\displaystyle X^{\text{'}\text{'}}+{\alpha }^{2}X=0,X\left(0\right)=0,X^{\text{'}}\left(1\right)+hX\left(1\right)=0.}$

  Boundary conditions of form $a{u}_x+u=0$ are known as Robin BC's.

• Non-trivial solutions of Sturm-Liouville problem $X\left(x\right)=c_1\cos \alpha x+c_2\sin \alpha x$,

  ${\displaystyle X\left(0\right)=0\Rightarrow c_1=0,X^{\text{'}}\left(1\right)+hX\left(1\right)=0\Rightarrow \alpha \cos \alpha +h\sin \alpha =0\Rightarrow \tan \alpha =-\frac{\alpha }{h}}$

Orthogonal series

• The equation $\tan \alpha =-\alpha /h$ has an infinite number of roots denoted ${\alpha }_n$

• Note that in contrast to $\sin \beta L=0\Rightarrow {\beta }_n=n\pi /L$, the solutions ${\alpha }_n$ are not integer multiples of $\pi /L$, hence lead to the formation of a new series of orthogonal functions different from the Fourier series

  ${\displaystyle u(x,t)=\sum _{n=1}^{\infty }{A}_n{e}^{-k{\alpha }_n^{2}t}\sin {\alpha }_{n}x}$

• Apply initial condition to obtain $1={\sum }_{n=1}^{\infty }{A}_n\sin {\alpha }_{n}x$ with coefficients

  ${\displaystyle A_n=\frac{\underset{0}{\overset{1}{\int }}\sin {\alpha }_{n}xdx}{\underset{0}{\overset{1}{\int }}{\sin }^2{\alpha }_{n}xdx}}$

Neumann condition problem

• Consider wave equation $a^2{u}_{xx}=u_{tt}$ with BCs

  ${\displaystyle u(0,t)=0,u_x(1,t)=0,u(x,0)=x,u_t(x,0)=0}$

• Separation of variables leads to $X^{\text{'}\text{'}}+{\alpha }^{2}X=0,X\left(0\right)=0,X^{\text{'}}\left(1\right)=0$ and $T^{\text{'}\text{'}}+a^2{\alpha }^{2}T=0$. Apply BCs to $X\left(x\right)=c_1\cos \alpha x+c_2\sin \alpha x$

  ${\displaystyle X\left(0\right)=0\Rightarrow c_1=0,X^{\text{'}}\left(1\right)=0\Rightarrow \cos \alpha =0\Rightarrow {\alpha }_n=\frac{(2n-1)\pi }{2}}$

  Again, this an orthogonal but not Fourier series since ${\alpha }_n$ is not an integer multiple of a constant.

Orthogonal series

• Solution

  ${\displaystyle u(x,t)=\sum _{n=1}^{\infty }[A_n\cos a\left(\frac{2n-1}{2}\right)\pi t]\sin [\left(\frac{2n-1}{2}\right)\pi x]}$

• Apply initial conditions $u(x,0)=x,u_t(x,0)=f\left(t\right)$

  ${\displaystyle u(x,0)=x=\sum _{n=1}^{\infty }{A}_n\sin [\left(\frac{2n-1}{2}\right)\pi x]}$ ${\displaystyle A_n=\frac{\underset{0}{\overset{1}{\int }}x\sin [\left(\frac{2n-1}{2}\right)\pi x]dx}{\underset{0}{\overset{1}{\int }}{\sin }^2[\left(\frac{2n-1}{2}\right)\pi x]dx}}$

Multiple spatial dimensions

• Consider $k{u}_t=u_{xx}+u_{yy}={\nabla }^{2}u=▵u=div(\mathrm{grad}u)=\nabla \cdot \nabla u$

• Separation of variables $u(t,x,y)=T\left(t\right)X\left(x\right)Y\left(y\right)$

  ${\displaystyle u(x,y,t)=\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{A}_{mn}\exp [-k({\alpha }_m^2+{\beta }_n^2)]\sin ({\alpha }_{m}x)\sin ({\beta }_{n}y)}$