Hyperbolic Partial Differential Equation - Hyperbolic System and Conservation Laws

Hyperbolic System and Conservation Laws

There is a connection between a hyperbolic system and a conservation law. Consider a hyperbolic system of one partial differential equation for one unknown function . Then the system has the form

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

Now can be some quantity with a flux . To show that this quantity is conserved, integrate over a domain

If and are sufficiently smooth functions, we can use the divergence theorem and change the order of the integration and to get a conservation law for the quantity in the general form

which means that the time rate of change of in the domain is equal to the net flux of through its boundary . Since this is an equality, it can be concluded that is conserved within .

Read more about this topic:  Hyperbolic Partial Differential Equation

Famous quotes containing the words system, conservation and/or laws:

    “... in America ... children are instructed in the virtues of the system they live under, as though history had achieved a happy ending in American civics.”
    —Mary McCarthy (1912–1989)

    “A country grows in history not only because of the heroism of its troops on the field of battle, it grows also when it turns to justice and to right for the conservation of its interests.”
    —Aristide Briand (1862–1932)

    “Thus far women have been the mere echoes of men. Our laws and constitutions, our creeds and codes, and the customs of social life are all of masculine origin. The true woman is as yet a dream of the future. A just government, a humane religion, a pure social life await her coming.”
    —Elizabeth Cady Stanton (1815–1902)