Geodesic
A first-order partial differential equation of the common type with different ways of initial data
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 20 (2018) no. 2, pp. 132-147.

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The Cauchy problem for a quasi-linear first order partial differential equation is studied for different cases of initial data. In the first case, the line carrying the initial data is specified parametrically; in the second case, this line is described in Cartesian coordinates and has an infinite length; in the third case, the line is specified in Cartesian coordinates and its length is finite. In each case, the local resolvability conditions are formulated for the considered quasi-linear equation and it is shown that the solution has the same smoothness as the function defining the initial conditions. To study the above problems the method of additional argument was used. Using this method, some system of integral equations is solved, and the solution of this system gives the solution of the Cauchy problem for the original equation.
Keywords: quasi-linear first order partial differential equation, Cauchy problem, method of an additional argument, local resolvability, integral equation. 
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S. N. Alekseenko; L. E. Platonova. A first-order partial differential equation of the common type with  different ways of initial data. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 20 (2018) no. 2, pp. 132-147. http://geodesic.mathdoc.fr/item/SVMO_2018_20_2_a0/

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