Resulov, MahirSahin, Ethem Ilhan2015-04-032015-04-032012BEYKENT UNIVERSITY JOURNAL OF SCIENCE AND ENGINEERING Volume 5(1-2), 2012, 155-1811307-3818In this paper, a method for obtaining an exact and numerical solution of the Cauchy and first type initial boundary value problems for a first order partial differential equation with a non convex state function is suggested. For this purpose, we introduce an auxiliary problem since it has some advantages over the main problem, and it is equivalent to the main problem in a definite sense. Using this auxiliary problem, we propose a efficient method for finding the location of shock which appears in the solution of main problem and its evolution in time. The suggested auxiliary problem permit also us to prove of convergence in mean of a numerical solution to the exact solution of the main problem. Moreover, using the auxiliary problem we can write the higher order numerical scheme with respect to the time variable. Some results of the comparison of the exact and numerical solutions are illustrated.enRiemann's and Buckley-Leverett's problemsnon-convex state functionconvex and concave hullnumerical solution in a class of discontinuous functionsArticle