Introduction to partial differential equations
When quantities vary in both space and time, like temperature along a rod or the shape of a vibrating string, we need partial differential equations.
What a PDE is
A partial differential equation involves an unknown function of several variables and its partial derivatives. The three classic second-order PDEs are the heat equation \(u_t = ku_{xx}\), the wave equation \(u_{tt} = c^2u_{xx}\), and Laplace's equation \(u_{xx} + u_{yy} = 0\). They describe diffusion, vibration, and steady states.
The heat equation
\(u_t = ku_{xx}\) says temperature rises where the temperature profile is concave up (cooler regions surrounded by warmer ones) and falls where it is concave down. Heat flows from hot to cold, smoothing out differences. It also models diffusion of chemicals and the spread of probability in random walks.
u_t = k\,u_{xx}The wave equation
\(u_{tt} = c^2u_{xx}\) describes vibrating strings, sound, and light, with waves travelling at speed \(c\). Unlike heat, waves do not smooth out: they carry shapes along and reflect at boundaries. D'Alembert showed every solution on an infinite string is \(u = F(x - ct) + G(x + ct)\), a right-moving plus a left-moving wave.
u_{tt} = c^2 u_{xx}Laplace's equation
\(u_{xx} + u_{yy} = 0\) describes steady states: temperature after it stops changing, electrostatic potential, and ideal fluid flow. Its solutions, harmonic functions, have no local maxima or minima inside a region; the extremes occur on the boundary.
Boundary and initial conditions
A PDE alone has infinitely many solutions. Problems specify initial conditions (the starting temperature or string shape and velocity) and boundary conditions: Dirichlet conditions fix values (an end held at 0 degrees), and Neumann conditions fix fluxes (an insulated end has \(u_x = 0\)).
Classifying PDEs
Second-order linear PDEs are classified as parabolic (heat), hyperbolic (wave), or elliptic (Laplace). The type determines which conditions make a well-posed problem and how information spreads: instantly and smoothly (parabolic), at finite speed (hyperbolic), or as a global balance (elliptic).
Superposition again
For linear homogeneous PDEs, sums of solutions are solutions. This allows building complicated solutions from simple ones, the key idea behind separation of variables and Fourier series in the next lesson.
Common mistakes
Confusing \(u_t\) and \(u_{tt}\), which separates heat from waves. Forgetting boundary conditions. Assuming the heat equation preserves sharp features; it smooths them immediately.
Key takeaways
PDEs involve partial derivatives in several variables. The heat equation diffuses and smooths, the wave equation transports and reflects, and Laplace's equation describes steady states. Initial and boundary conditions (Dirichlet or Neumann) make problems well posed, and linearity allows superposition.
Flashcard
Equation with partial derivatives
Check your understanding
Which equation describes a vibrating string?