MATH529: L14 Laplace Transform solution of PDE examples

Overview

• Wave equation solution

• Heat equation

Wave equation

Apply to a PDE, $u_{tt}=a^2{u}_{xx}$ for $0<x<\pi$, $t>0$, with initial conditions $u(x,0)=-\sin 3x$, $u_t(x,0)=0$, boundary conditions $u(0,t)=u(\pi ,t)=0$.

• Apply Laplace transform, obtain a family of ODEs labeled by $s$

  ${\displaystyle \begin{array}{rcl}{s}^{2}U(x,s)-su(x,0)-\frac{\partial u}{\partial t}(x,0)& =& a^2\frac{{\partial }^2}{\partial x^2}U(x,s)\Rightarrow \\ a^2\frac{{\partial }^2}{\partial x^2}U(x,s)-s^{2}U(x,s)& =& s\sin 3x\end{array}}$

• Solve the ODE $\Rightarrow U(x,s)=c_1{e}^{sx/a}+c_2{e}^{-sx/a}-s\sin \left(3x\right)/(s^2+{\left(3a\right)}^2)$

Wave equation (cont)

Inverse Laplace transform and apply boundary conditions

Heat equation

• Apply to a PDE, $u_t=a^2{u}_{xx}$ for $0<x<\pi$, $t>0$, with initial conditions $u(x,0)=-\sin 3x$, boundary conditions $u(0,t)=u(\pi ,t)=0$.

  • Apply Laplace transform, obtain a family of ODEs labeled by $s$

    ${\displaystyle \begin{array}{rcl}sU(x,s)-u(x,0)& =& a^2\frac{{\partial }^2}{\partial x^2}U(x,s)\Rightarrow \\ a^2\frac{{\partial }^2}{\partial x^2}U(x,s)-sU(x,s)& =& \sin 3x\end{array}}$

  • Solve the ODE $\Rightarrow U(x,s)=c_1{e}^{\sqrt{s}x/a}+c_2{e}^{-\sqrt{s}x/a}-\sin \left(3x\right)/(s+{\left(3a\right)}^2)$

  • Inverse Laplace transform requires consideration of complex plane extension for general BCs. For homogeneous Dirichlet, $c_1=c_2=0$

  ${\displaystyle u(x,t)={ℒ}^{-1}[-\frac{\sin \left(3x\right)}{s+{\left(3a\right)}^2}]=-e^{-9a^{2}t}\sin \left(3x\right)}$