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In this study a new method for finding exact solution of the Cauchy problem subject to a discontinuous initial profile for the two dimensional scalar conservation laws is suggested. For this aim, first, some properties of the weak solution of the linearized equation are investigated. Taking these properties into consideration an auxiliary problem having some advantages over the main problem is introduced. The proposed auxiliary problems also permit us to develop effective different numerical algorithms for finding the solutions. Some computer experiments are carried out.

EXACT SOLUTION OF THE CAUCHY PROBLEM FOR 2-D SCALAR CONSERVATION LAWS WITH CONSTANT COEFFICIENTS IN A CLASS OF DISCONTINUOUS FUNCTIONS