The Method Of Characteristics And Classical Solutions Of First Order Pdes

The Method of Characteristics is a Powerful Method that Allows one to reducernany Rst-order linear PDE to an ODE, Which Can be Subsequently Solved UsingrnODE techniques and it can be generalized to quasilinear equations as well.rnthe principal results of this paper are: 1) The Cauchy problemrn@urn@trn+ f(t; x; u;rn@urn@xrn) = 0; inft > 0; x 2 Rng;rnu(0; x) = (x); onft = 0; x 2 Rngrnhas locally a unique C2-solution.rn2) If the Jacobian (Dx=Dy)(t; y) of the mapping x = x(t; y) vanishes some-rnwhere, it is impossible to extend the C2-solution beyond a point where thernJacobian vanishes.rn3) Suppose that the characteristic curves do not meet in a neighborhood ofrnthe point where the Jacobian vanishes. Then the solution keeps being ofrnclass C1, but not of class C2, in the neighborhood of the point.

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