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Hyperbolic Equations

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, hyperbolic implements 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 c.

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 the function assempde from the problems

–∇ · (cu) + au = f and –∇ · (0∇u) + mu = 0.

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