MATH529: L06 Heat Equation

Overview

• Problem statement

• Separation of variables

• General solution

• Boundary value problems:

  • fixed temperature at ends

  • insulated boundaries

Problem statement

• Rod of length $L$ with initial temperature $f\left(x\right)$ and zero temperature at ends

• Denote temperature at time $t$, position $x$ by $u(x,t)$

• Heat equation

  ${\displaystyle \frac{\partial u}{\partial t}=k\frac{{\partial }^{2}u}{\partial x^2},k\mathrm{conductivity}}$

• Initial condition

  ${\displaystyle u(x,t=0)=f\left(x\right)}$

• Boundary conditions

  • fixed temperature: $u(x=0,t)=u_0$, $u(x=L,t)=u_1$, (e.g., $u_0=u_1=0$)

  • heat flux: $u_x(x=0,t)=q_0$, $u_x(x=L,t)=q_1$, (e.g., $q_0=q_1=0$)

Separation of variables

• $u_t=k{u}_{xx}$, $u(x,0)=f\left(x\right)$, $u(0,t)=0$, $u(L,t)=0$. (Dirichlet problem)

• $u(x,t)=X\left(x\right)T\left(t\right)\Rightarrow$ $X{T}^{\text{'}}=k{X}^{\text{'}\text{'}}T$

  ${\displaystyle \frac{1}{k}\frac{T^{\text{'}}}{T}=\frac{X^{\text{'}\text{'}}}{X}=-\lambda }$

  • $\lambda =0\Rightarrow X^{\text{'}\text{'}}=0\Rightarrow X\left(x\right)=c_1+c_{2}x$, $T^{\text{'}}=0\Rightarrow T\left(t\right)=c_3$.

    $u(x,t)=X\left(x\right)T\left(t\right)=(c_1+c_{2}x)c_3=b_1+b_{2}x$

    $u(0,t)=b_1=0$, $u(L,t)=b_{2}L=0\Rightarrow b_2$. Only solution $u(x,t)=0$ $✠$

  • $\lambda =-{\alpha }^2<0\Rightarrow X^{\text{'}\text{'}}-{\alpha }^{2}X=0\Rightarrow X\left(x\right)=c_4\cosh (\alpha x)+c_5\sinh (\alpha x)$

    $T^{\text{'}}=k{\alpha }^{2}T\Rightarrow T\left(t\right)=c_6\exp [k{\alpha }^{2}t]$ (blow-up) $✠$

  • $\lambda ={\alpha }^2>0\Rightarrow$$X^{\text{'}\text{'}}+{\alpha }^{2}X=0\Rightarrow X\left(x\right)=c_7\cos (\alpha x)+c_8\sin (\alpha x)$,

    $T^{\text{'}}=-k{\alpha }^{2}T\Rightarrow T\left(t\right)=c_6\exp [-k{\alpha }^{2}t]$

Eigenvalues of the solution

• $u_t=k{u}_{xx}$, $u(x,0)=f\left(x\right)$, $u(0,t)=0$, $u(L,t)=0$. (Dirichlet problem)

• $u(x,t)=X\left(x\right)T\left(t\right)\Rightarrow \frac{1}{k}\frac{T^{\text{'}}}{T}=\frac{X^{\text{'}\text{'}}}{X}=-\lambda =-{\alpha }^2$

• Only possible choice of $\lambda ={\alpha }^2>0\Rightarrow$

  ${\displaystyle u(x,t)=[b_1\cos (\alpha x)+b_2\sin (\alpha x)]\exp [-k{\alpha }^{2}t]}$

  • $u(0,t)=0\Rightarrow b_1\exp [-k{\alpha }^{2}t]=0\Rightarrow b_1=0$

  • $u(L,t)=0\Rightarrow$$b_2\sin (\alpha L)\exp [-k{\alpha }^{2}t]=0\Rightarrow b_2\sin (\alpha L)=0$

    • $b_2=0\Rightarrow u(x,t)=0✠$

    • $\sin (\alpha L)=0\Rightarrow \alpha L=n\pi \Rightarrow {\alpha }_n=\frac{n\pi }{L}$ all lead to a possible solution

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

Fourier sine series solution

• $u(x,t=0)=f\left(x\right)={\sum }_{n=1}^{\infty }{b}_n\sin ({\alpha }_{n}x)$,

  ${\displaystyle b_n=(f\left(x\right),\sin \left({\alpha }_{n}x\right))=\frac{2}{L}\underset{0}{\overset{L}{\int }}f\left(x\right),\sin \left({\alpha }_{n}x\right)dx}$

• Solution is

  ${\displaystyle u(x,t)=\sum _{n=1}^{\infty }{b}_n\sin ({\alpha }_{n}x)\exp [-k{\alpha }_n^{2}t]?}$

Heat flux boundary conditions

• $u_t=k{u}_{xx}$, $u(x,0)=f\left(x\right)$, $u_x(0,t)=0$, $u_x(L,t)=0$. (Neumann problem)

• Only possible choice of $\lambda ={\alpha }^2>0\Rightarrow$

  ${\displaystyle u(x,t)=[b_1\cos (\alpha x)+b_2\sin (\alpha x)]\exp [-k{\alpha }^{2}t]}$

  • $u_x(0,t)=0\Rightarrow \alpha [b_2]\exp [-k{\alpha }^{2}t]=0\Rightarrow b_2=0$

  • $u_x(L,t)=0\Rightarrow$$\alpha [-b_1\sin (\alpha L)]\exp [-k{\alpha }^{2}t]=0\Rightarrow {\alpha }_n=\frac{n\pi }{L}$

    ${\displaystyle u_n(x,t)=\sum _{n=1}^{\infty }{b}_n\cos ({\alpha }_{n}x)\exp [-k{\alpha }_n^{2}t]+b_0}$ ${\displaystyle b_n=(f\left(x\right),\cos \left({\alpha }_{n}x\right))=\frac{2}{L}\underset{0}{\overset{L}{\int }}f\left(x\right)\cos \left({\alpha }_{n}x\right)dx}$ ${\displaystyle b_0=(f\left(x\right),\cos \left({\alpha }_{n}x\right))=\frac{1}{L}\underset{0}{\overset{L}{\int }}f\left(x\right)dx}$

General form of boundary condition for heat equation: Robin

• $u_t=k{u}_{xx}$, $u(x,0)=f\left(x\right)$,

  $u(0,t)+a{u}_x(0,t)=b$, $u(L,t)+c{u}_x(L,t)=d$. (constant value Robin problem)

  $a,b,c,d\in ℝ$ , always has a solution

• Variable value Robin problem $a\left(t\right),b\left(t\right),c\left(t\right),d\left(t\right)$, sometimes has a solution