# Documentation

## Hyperbolic Equations

Partial Differential Equation Toolbox™ solves equations of the form

$m\frac{{\partial }^{2}u}{\partial {t}^{2}}+d\frac{\partial u}{\partial t}-\nabla ·\left(c\nabla u\right)+au=f.$

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

$m\frac{{\partial }^{2}u}{\partial {t}^{2}}-\nabla \cdot \left(c\nabla u\right)+au=f,$

for x in Ω, where x represents a 2-D or 3-D point, with the initial conditions

$\begin{array}{c}u\left(x,0\right)={u}_{0}\left(x\right)\\ \frac{\partial u}{\partial t}\left(x,0\right)={v}_{0}\left(x\right)\end{array}$

for all x in Ω, and usual boundary conditions. In particular, solutions of the equation utt - cΔu = 0 are waves moving with speed $\sqrt{c}$.

Using a given mesh of Ω, the method of lines yields the second order ODE system

$M\frac{{d}^{2}U}{d{t}^{2}}+KU=F$

with the initial conditions

$\begin{array}{c}{U}_{i}\left(0\right)={u}_{0}\left({x}_{i}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\\ \frac{d}{dt}{U}_{i}\left(0\right)={v}_{0}\left({x}_{i}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\forall i\end{array}$

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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