Partial Differential Equation Toolbox™ solves equations of the form
When the d coefficient is 0, but m is not, the documentation calls this a hyperbolic equation, whether or not it is mathematically of the hyperbolic form.
Using the same ideas as for the parabolic equation,
the numerical solution of
for x in Ω, where x represents a 2-D or 3-D point, with the initial conditions
for all x in Ω, and usual boundary conditions. In particular, solutions of the equation utt - cΔu = 0 are waves moving with speed .
Using a given mesh of Ω, the method of lines yields the second order ODE system
with the initial conditions
after we eliminate the unknowns fixed by Dirichlet boundary
conditions. As before, the stiffness matrix K and
the mass matrix M are assembled with the aid of
assempde from the problems
–∇ · (c∇u) + au = f and –∇ · (0∇u) + mu = 0.