Hyperbolic Partial Differential Equation - Hyperbolic System of Partial Differential Equations

Hyperbolic System of Partial Differential Equations

Consider the following system of first order partial differential equations for unknown functions, where

(*) \quad \frac{\partial \vec u}{\partial t} + \sum_{j=1}^d \frac{\partial}{\partial x_j} \vec {f^j} (\vec u) = 0,

are once continuously differentiable functions, nonlinear in general.

Now define for each a matrix

A^j:= \begin{pmatrix} \frac{\partial f_1^j}{\partial u_1} & \cdots & \frac{\partial f_1^j}{\partial u_s} \\ \vdots & \ddots & \vdots \\ \frac{\partial f_s^j}{\partial u_1} & \cdots & \frac{\partial f_s^j}{\partial u_s} \end{pmatrix} ,\text{ for }j = 1, \ldots, d.

We say that the system is hyperbolic if for all the matrix has only real eigenvalues and is diagonalizable.

If the matrix has distinct real eigenvalues, it follows that it's diagonalizable. In this case the system is called strictly hyperbolic.

Read more about this topic:  Hyperbolic Partial Differential Equation

Famous quotes containing the words system, partial and/or differential:

    “A heresy can spring only from a system that is in full vigor.”
    —Eric Hoffer (1902–1983)

    “We were soon in the smooth water of the Quakish Lake,... and we had our first, but a partial view of Ktaadn, its summit veiled in clouds, like a dark isthmus in that quarter, connecting the heavens with the earth.”
    —Henry David Thoreau (1817–1862)

    “But how is one to make a scientist understand that there is something unalterably deranged about differential calculus, quantum theory, or the obscene and so inanely liturgical ordeals of the precession of the equinoxes.”
    —Antonin Artaud (1896–1948)