Before solving a PDE you must read its two most important labels — its order
and whether it is linear. Together they decide which methods are even available.
We write subscripts for
partial derivatives:
u_t = \partial u/\partial t,
u_{xx} = \partial^2 u/\partial x^2, and so on.
The order is the highest derivative that appears. The heat equation
u_t = \alpha\,u_{xx} is second order (because of
u_{xx}); the transport equation
u_t + c\,u_x = 0 is first order.
Linear, and the operator viewpoint
A PDE is linear if the unknown u and its derivatives
appear only to the first power, never multiplied together and never inside a nonlinear function.
The cleanest way to say this is with a linear operator
L — a rule that eats a function and returns one — such that the
equation reads L[u] = g and
L[\alpha u + \beta v] = \alpha\,L[u] + \beta\,L[v].
For the heat equation, L[u] = u_t - \alpha\,u_{xx} is linear. The
equation is homogeneous when g = 0 and
inhomogeneous (forced) otherwise. Examples:
- u_t = \alpha\,u_{xx} — linear, homogeneous, second order.
- u_{tt} = c^2 u_{xx} + f(x,t) — linear, inhomogeneous (a forcing term f).
- u_t + u\,u_x = 0 — nonlinear: the product u\,u_x breaks linearity (this is Burgers' equation).
- Order = the highest partial derivative present.
- Linear = u and its derivatives appear to the first power only, with coefficients that may depend on the independent variables but not on u; equivalently L[\alpha u + \beta v] = \alpha L[u] + \beta L[v].
- Homogeneous = the right-hand side is zero (L[u] = 0); otherwise it is forced.
Linearity is the property the whole course exploits — it is what lets solutions be added.