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:

    “... the yearly expenses of the existing religious system ... exceed in these United States twenty millions of dollars. Twenty millions! For teaching what? Things unseen and causes unknown!... Twenty millions would more than suffice to make us wise; and alas! do they not more than suffice to make us foolish?”
    —Frances Wright (1795–1852)

    “Both the man of science and the man of art live always at the edge of mystery, surrounded by it. Both, as a measure of their creation, have always had to do with the harmonization of what is new with what is familiar, with the balance between novelty and synthesis, with the struggle to make partial order in total chaos.... This cannot be an easy life.”
    —J. Robert Oppenheimer (1904–1967)

    “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)