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Partial Differential Equation Toolbox™ solves equations of the form

$$m\frac{{\partial}^{2}u}{\partial {t}^{2}}+d\frac{\partial u}{\partial t}-\nabla \xb7\left(c\nabla u\right)+au=f$$

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 *u _{tt}* -

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

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

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