MATH529: L05 Boundary Value Problems in Rectangular Coordinates

Overview

• Separable partial differential equations (PDEs)

• Classical PDEs and BVPs

  • Heat equation $u_t=k{u}_{xx}$

  • Wave equation $u_{tt}=a^2{u}_{xx}$

  • Laplace equation $u_{xx}+u_{yy}=0$

Partial Differential Equations (PDEs)

• Recall that an $n^{\mathrm{th}}-\mathrm{order}$ ODE is stated as

  ${\displaystyle \begin{array}{llll}\mathrm{Explicit}:& y^{\left(n\right)}=f(x,y,y^{\text{'}},\dots ,y^{(n-1)})& \mathrm{Implicit}:& F(x,y,y^{\text{'}},\dots ,y^{\left(n\right)})=0\end{array}}$

• A partial differential equation (PDE) arises for multiple independent variables

• Linear PDE for $u(x,y)$

  ${\displaystyle A\frac{{\partial }^{2}u}{\partial x^2}+B\frac{{\partial }^{2}u}{\partial x\partial y}+C\frac{{\partial }^{2}u}{\partial y^2}+D\frac{\partial u}{\partial x}+E\frac{\partial u}{\partial y}+Fu=G}$

  (1)

  with $A,\dots ,G$ functions of $x,y$

• General solution of (1) is quite difficult, but there exist particular solution procedures of wide utility

Separation of Variables

• Assume that $u(x,y)$ can be written as a product $u(x,y)=X\left(x\right)Y\left(y\right)$

  ${\displaystyle \begin{array}{lllll}\frac{\partial u}{\partial x}=X^{\text{'}}Y,& \frac{\partial u}{\partial y}=X{Y}^{\text{'}},& \frac{{\partial }^{2}u}{\partial x^2}=X^{\text{'}\text{'}}Y,& \frac{{\partial }^{2}u}{\partial x\partial y}=X^{\text{'}}{Y}^{\text{'}}& \frac{{\partial }^{2}u}{\partial y^2}=X{Y}^{\text{'}\text{'}}\end{array}.}$

• Useful notation

  ${\displaystyle u_x=\begin{array}{lllll}\frac{\partial u}{\partial x},& u_y=\frac{\partial u}{\partial y},& u_{xx}=\frac{{\partial }^{2}u}{\partial x^2},& u_{xy}=\frac{{\partial }^{2}u}{\partial x\partial y},& u_{yy}=\frac{{\partial }^{2}u}{\partial y^2}.\end{array}}$

• Example: $u_{xx}=4u_y$. Assuming $u(x,y)=X\left(x\right)Y\left(y\right)$ leads to $X^{\text{'}\text{'}}Y=4X{Y}^{\text{'}}$

  ${\displaystyle \frac{X^{\text{'}\text{'}}}{4X}=\frac{Y^{\text{'}}}{Y}=-\lambda ,}$

  with $\lambda$ a constant

ODEs from PDEs and separation of variables

• $\frac{X^{\text{'}\text{'}}}{4X}=\frac{Y^{\text{'}}}{Y}=-\lambda \Rightarrow X^{\text{'}\text{'}}+4\lambda X=0$ and $Y^{\text{'}}+\lambda Y=0$

• Cases:

  • $\lambda =0$. $X^{\text{'}\text{'}}=0\Rightarrow X\left(x\right)=c_1+c_{2}x$, $Y^{\text{'}}=0\Rightarrow Y\left(y\right)=c_3$

  • $\lambda =-{\alpha }^2<0$. $X^{\text{'}\text{'}}-4{\alpha }^{2}X=0\Rightarrow X\left(x\right)=c_4\cosh \left(2\alpha x\right)+c_5\sinh \left(2\alpha x\right)$. $Y^{\text{'}}-{\alpha }^{2}Y=0\iff Y^{\text{'}}={\alpha }^{2}Y\Rightarrow Y\left(y\right)=c_6\exp \left({\alpha }^{2}y\right)$

  • $\lambda ={\alpha }^2>0$. $X^{\text{'}\text{'}}+4{\alpha }^{2}X=0\Rightarrow X\left(x\right)=c_7\cos \left(2\alpha x\right)+c_8\sin \left(2\alpha x\right)$.

    $Y^{\text{'}}+{\alpha }^{2}Y=0\iff Y^{\text{'}}=-{\alpha }^{2}Y\Rightarrow Y\left(y\right)=c_9\exp (-{\alpha }^{2}y)$

• Form $u(x,y)=X\left(x\right)Y\left(y\right)$ for each case:

  • $\lambda =0$. $u(x,y)=c_3(c_1+c_{2}x)=b_1+b_{2}x$

  • $\lambda =-{\alpha }^2<0$. $u(x,y)=[b_4\cosh \left(2\alpha x\right)+b_5\sinh \left(2\alpha x\right)]$$\exp \left({\alpha }^{2}y\right)$

  • $\lambda ={\alpha }^2>0.$ $u(x,y)=[b_7\cos \left(2\alpha x\right)+b_8\sin \left(2\alpha x\right)]$$\exp (-{\alpha }^{2}y)$

• Linear PDE, multiple solutions from $u(x,y)=X\left(x\right)Y\left(y\right)$, $u={\sum }_{k=1}^{\infty }{u}_k$

Classification of linear second-order PDEs

• The PDE

  ${\displaystyle A\frac{{\partial }^{2}u}{\partial x^2}+B\frac{{\partial }^{2}u}{\partial x\partial y}+C\frac{{\partial }^{2}u}{\partial y^2}+D\frac{\partial u}{\partial x}+E\frac{\partial u}{\partial y}+Fu=G}$

  is:

  • hyperbolic if $B^2-4AC>0$

  • parabolic if $B^2-4AC=0$

  • elliptic if $B^2-4AC<0$

• The above terms arise from classification of quadratic forms

  ${\displaystyle \left[\begin{array}{ll}{\partial }_x& {\partial }_y\end{array}\right]\left[\begin{array}{ll}A& B/2\\ B/2& C\end{array}\right]\left[\begin{array}{l}{\partial }_x\\ {\partial }_y\end{array}\right]u+D\frac{\partial u}{\partial x}+E\frac{\partial u}{\partial y}+Fu=G}$ ${\displaystyle \det (\left[\begin{array}{ll}A& B/2\\ B/2& C\end{array}\right]-\lambda \left[\begin{array}{ll}1& 0\\ 0& 1\end{array}\right])=}$

Classical PDEs

• Heat equation $u_t=k{u}_{xx}$, parabolic, $s=-{\xi }^2$

• Wave equation $u_{tt}=a^2{u}_{xx}$, hyperbolic, $s^2=a^2{\xi }^2\iff s^2-a^2{\xi }^2=0$

• Laplace equation $u_{xx}+u_{yy}=0$, elliptic, ${\xi }^2+{\eta }^2=0$

Boundary conditions

• Initial conditions: given at a single point, typically for time variable

  • ODE: $y^{\left(n\right)}\left(t\right)=f(t,y,y^{\text{'}},\dots ,y^{(n-1)})$, $y\left(0\right)=y_0,\dots ,y^{(n-1)}\left(0\right)=y_0^{(n-1)}$

  • Time component of PDE: $u_t=u_{xx}$, $u(x,t=0)=u_0\left(x\right)$

• Boundary conditions: given at multiple points, typically for space variables

  • ODE (Dirichlet): $y^{\text{'}\text{'}}\left(x\right)=f(x,y,y^{\text{'}})$, $y\left(a\right)=y_a$, $y\left(b\right)=y_b$

  • ODE (Neumann): $y^{\text{'}\text{'}}\left(x\right)=f(x,y,y^{\text{'}})$, $y^{\text{'}}\left(a\right)=y_a^{\text{'}}$, $y^{\text{'}}\left(b\right)=y_b^{\text{'}}$

  • ODE (Robin): $y^{\text{'}\text{'}}\left(x\right)=f(x,y,y^{\text{'}})$, $A_{1}y\left(a\right)+B_1{y}^{\text{'}}\left(a\right)=C$, ...

  • PDE space component (Dirichlet): $u_t=u_{xx}$, $u(a,t)=f\left(t\right)$,$u(b,t)=g\left(t\right)$

  • PDE space component (Neumann): $u_t=u_{xx}$, $u_x(a,t)=f\left(t\right)$, ...

  • PDE space component (Robin): $u_t=u_{xx}$, $A_{1}u(a,t)+B_1{u}^{\text{'}}(a,t)=f\left(t\right)$,…