1.1 Partial differential equations
Definition 1.1.1. A partial differential equation (PDE) of order \(k\) for a function \(u\) of \(x = (x_1, \dots , x_n)\) is a relation \(F(x, u, Du, D^2u, \dots , D^ku) = 0\) among \(u\) and its partial derivatives up to order \(k\). It is
- (i)
- linear if \(F\) is affine in \(u\) and all its derivatives, with coefficients depending only on \(x\);
- (ii)
- semilinear if it is linear in the highest-order derivatives with coefficients depending only on \(x\);
- (iii)
- quasilinear if it is linear in the highest-order derivatives, with coefficients that may depend on \(x\), \(u\) and lower derivatives;
- (iv)
- fully non-linear otherwise.
A classical solution on an open set \(\Omega \) is a function with continuous derivatives up to order \(k\) satisfying the equation at every point of \(\Omega \).
Example 1.1.2. With \(t\) for time and \(\Delta = \partial _1^2 + \dots + \partial _n^2\) the Laplacian:
- (a)
- the transport equation \(u_t + c\cdot \nabla u = 0\) (linear, first order);
- (b)
- the heat or diffusion equation \(u_t = \kappa \Delta u\) (linear, second order);
- (c)
- the wave equation \(u_{tt} = c^2\Delta u\) (linear, second order);
- (d)
- Laplace’s equation \(\Delta u = 0\) and Poisson’s equation \(\Delta u = f\) (linear, second order);
- (e)
- Burgers’ equation \(u_t + uu_x = \nu u_{xx}\) (quasilinear; inviscid if \(\nu = 0\));
- (f)
- the eikonal equation \(|\nabla u| = 1\) of geometrical optics (fully non-linear, first order);
- (g)
- the Korteweg–de Vries equation \(u_t + 6uu_x + u_{xxx} = 0\) for shallow-water waves (quasilinear, third order).
Linear homogeneous equations obey the superposition principle: a linear combination of solutions is a solution. Most of the methods of this course exploit it. A PDE alone has infinitely many solutions, far more than an ODE: where an ODE solution depends on finitely many constants, a PDE solution typically depends on arbitrary functions. Additional conditions, initial and boundary conditions, single out one, and Hadamard’s requirements for a well-posed problem, existence, uniqueness and continuous dependence on the data, are the organising principle of the theory.
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