What a PDE is, and how to classify one

A partial differential equation is a relation between an unknown function of two or more variables and its partial derivatives. We write \( u_t = \partial u/\partial t \), \( u_{xx} = \partial^2 u/\partial x^2 \), and \( \Delta u = u_{xx} + u_{yy} + u_{zz} \) for the Laplacian. Four equations carry most of this course: the transport equation \( u_t + c\,u_x = 0 \), the heat equation \( u_t = k\,u_{xx} \), the wave equation \( u_{tt} = c^2 u_{xx} \) and Laplace's equation \( \Delta u = 0 \). A classical solution is a function with enough continuous derivatives that the equation holds at every point of the domain. The example below checks that \( u = x^2 - y^2 \) is one for Laplace's equation: \( u_{xx} = 2 \) and \( u_{yy} = -2 \) cancel.

The order is the highest derivative that appears. The equation is linear when it has the form \( L[u] = f \) with \( L[\alpha u + \beta v] = \alpha L[u] + \beta L[v] \); then, when \( f = 0 \), any combination of solutions is again a solution (superposition), and nearly every explicit method in this course depends on that. Nonlinear equations come in grades. Burgers' equation \( u_t + u\,u_x = 0 \) is quasilinear (linear in its highest derivatives, with coefficients depending on \( u \)). The Navier-Stokes equations \( u_t + (u\cdot\nabla)u - \nu\Delta u + \nabla p = 0 \), \( \nabla\cdot u = 0 \) are semilinear: the second-order term \( \nu\Delta u \) is linear and the nonlinearity involves only first derivatives.

A linear second-order equation in two variables, \( A u_{xx} + 2B u_{xy} + C u_{yy} + \text{(lower order)} = 0 \), is classified by the sign of \( B^2 - AC \): negative means elliptic (Laplace: \( A = C = 1, B = 0 \)), zero means parabolic (heat, written in \( x, t \): \( A = k \), \( B = C = 0 \)), positive means hyperbolic (wave: \( A = c^2, C = -1 \)). The coefficients may depend on position, so the type can change inside one domain: the Tricomi equation \( y\,u_{xx} + u_{yy} = 0 \) is elliptic where \( y > 0 \) and hyperbolic where \( y < 0 \), and it models flow passing through the speed of sound.

Picture it: drop a spike of data into each type. An elliptic equation spreads its influence over the whole domain at once and returns a smooth surface, like a drum skin pushed at one point. A parabolic equation smooths the spike into a bump that widens as time runs forward. A hyperbolic equation sends the spike off along two lines at finite speed, sharp as it started.

Think it: the type tells you which data to give. Elliptic problems want boundary values, parabolic problems want an initial state plus boundary values, and hyperbolic problems want an initial position and velocity. Give the wrong data (initial position and velocity to Laplace's equation, say) and the problem becomes ill-posed, which is the subject of the well-posedness lesson.