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    Entropy Solution of a First Order Hyperbolic Type Equations with a Non-Convex State Function in a Class Of Discontinuous Functions
    (Beykent Üniversitesi, 2012) Resulov, Mahir; Sahin, Ethem Ilhan
    In 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.
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    Exact Solutions of the Cauchy Problem for 2-D Scalar Conservation Laws
    (Beykent Üniversitesi, 2012) Resulov, Mahir; Sentürk, Kenan
    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.
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    Solutions of a 1-D Conservation Laws without Convexity
    (Beykent Üniversitesi, 2012) Resulov, Mahir; Sinsoysal, Bahaddin
    In this paper the exact solution for Cauchy problem of first order nonlinear partial equation with piece-wise initial condition described scalar conservation laws without convexity of the state function. In particular, the state functions having four and one point of inflection are considered. The structure of solutions is investigated.