The Method of Characteristics and Classical Solutions of First Order Pdes
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Date
2014-08-08
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Addis Ababa University
Abstract
The Method of Characteristics is a Powerful Method that Allows one to reduce
any Rst-order linear PDE to an ODE, Which Can be Subsequently Solved Using
ODE techniques and it can be generalized to quasilinear equations as well.
the principal results of this paper are: 1) The Cauchy problem
@u
@t
+ f(t; x; u;
@u
@x
) = 0; inft > 0; x 2 Rng;
u(0; x) = (x); onft = 0; x 2 Rng
has locally a unique C2-solution.
2) If the Jacobian (Dx=Dy)(t; y) of the mapping x = x(t; y) vanishes some-
where, it is impossible to extend the C2-solution beyond a point where the
Jacobian vanishes.
3) Suppose that the characteristic curves do not meet in a neighborhood of
the point where the Jacobian vanishes. Then the solution keeps being of
class C1, but not of class C2, in the neighborhood of the point.
Description
Keywords
Method, Characteristics, Classical Solutions, Order Pdes