Second-Order Linear PDEs

Second-Order Linear PDEs

• A general form is described as $Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y + Fu = G$, where A through G are functions of $(x, y)$.
  • If $B^2 - AC < 0$, the PDE is Elliptic, like Laplace's equation $\nabla^2 u = 0$.
  • If $B^2 - AC = 0$, the PDE is Parabolic, like the Heat equation $u_t = \nabla^2 u$.
  • If $B^2 - AC > 0$, the PDE is Hyperbolic, like the Wave equation $u_{tt} = \nabla^2 u$.

Examples

• Laplace Equation is shown as:
  • $\nabla^2 u = 0$
  • In 2D Cartesian coordinates: $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$
  • In 2D Polar coordinates: $\frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial u}{\partial r} \right) + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2} = 0$
• Heat Equation is shown as:
  • $\frac{\partial u}{\partial t} = \alpha \nabla^2 u$
  • In 1D: $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$
• Wave Equation is shown as:
  • $\frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u$
  • In 1D: $\frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$

Solving PDEs

• Separation of Variables is shown as:
  • Assume a solution of the form $u(x, t) = X(x)T(t)$.
  • Substitute into the PDE and separate variables.
  • Solve the resulting ODEs for $X(x)$ and $T(t)$.
  • Apply boundary conditions to determine constants.
  • Superimpose solutions to satisfy initial conditions, if needed.

Example: Heat Equation

• Consider the heat equation with Dirichlet boundary conditions:
  • $\frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2}$, $0 < x < L$, $t > 0$
  • $u(0, t) = u(L, t) = 0$
  • $u(x, 0) = f(x)$
• Let $u(x, t) = X(x)T(t)$.
  • The process is shown as:
    • $X(x)T'(t) = \alpha X''(x)T(t)$
    • $\frac{T'(t)}{\alpha T(t)} = \frac{X''(x)}{X(x)} = -\lambda$
    • $T'(t) + \alpha \lambda T(t) = 0$
    • $X''(x) + \lambda X(x) = 0$
• Solving for $X(x)$:
  • $X(x) = A \cos(\sqrt{\lambda}x) + B \sin(\sqrt{\lambda}x)$
• Applying boundary conditions:
  • $X(0) = X(L) = 0$
  • $X(0) = A = 0$
  • $X(L) = B \sin(\sqrt{\lambda}L) = 0$
  • $\sqrt{\lambda}L = n\pi$, $n = 1, 2, 3,...$
  • $\lambda_n = \left( \frac{n\pi}{L} \right)^2$
  • $X_n(x) = B_n \sin\left( \frac{n\pi x}{L} \right)$
• Solving for $T(t)$:
  • $T'(t) + \alpha \left( \frac{n\pi}{L} \right)^2 T(t) = 0$
  • $T_n(t) = C_n e^{-\alpha \left( \frac{n\pi}{L} \right)^2 t}$
• General solution:
  • $u(x, t) = \sum_{n=1}^{\infty} D_n \sin\left( \frac{n\pi x}{L} \right) e^{-\alpha \left( \frac{n\pi}{L} \right)^2 t}$
• Applying initial condition:
  • $u(x, 0) = f(x) = \sum_{n=1}^{\infty} D_n \sin\left( \frac{n\pi x}{L} \right)$
  • $D_n = \frac{2}{L} \int_0^L f(x) \sin\left( \frac{n\pi x}{L} \right) dx$