SOLUTION OF A 1-D CONSERVATION LAWS WITHOUT CONVEXITY

Öz

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. 

Anahtar Kelimeler

• Scalar conservation laws,state function without convexity,convex and concave hull,weak solution

SOLUTION OF A 1-D CONSERVATION LAWS WITHOUT CONVEXITY

Öz

Bu makalede bükeyliği olmayan durum fonksiyonuna sahip birinci mertebeden nonlineer kısmi türevli diferansiyel denklem için yazılmış parçalı sürekli başlangıç koşullu Cauchy probleminin gerçek çözümleri elde edilmiştir. Özel olarak, sırasıyla dört ve bir dönüm noktalarına sahip durum fonksiyonları ele alınmış ve çözümün yapısı incelenmiştir.

Anahtar Kelimeler

• Scaler korunum kanunları, bükeyliği olmayan durum fonksiyonu, konveks ve konkav katman, zayıf çözüm.

Kaynakça

1. Collins, P. Fluids Flow in Porous Materials. 1964.
2. Goritskii, A.A., Krujkov, S.N., Chechkin, G.A. A First Order Quasi-Linear Equations with Partial Differential Derivatives. Pub. Moskow University, Moskow, 1997.
3. Kin, Y.J., Lee, Y., Structure of Fundamental Solutions of a Conservation Laws without Convexity, Applied Mathematics, vol.8, pp. 1-20, 2008.
4. Krushkov, S.N., First Order Quasilinear Equations in Several Independent Variables, Math. USSS Sb., 10, pp.217-243, 1970.
5. Lax, P.D. The Formation and Decay of Shock Waves, Amer. Math Monthly, 79, pp. 227-241, 1972.
6. Lax, P.D. Weak Solutions of Nonlinear Hyperbolic Equations and Their Numerical Computations, Comm. of Pure and App. Math, Vol VII, pp 159-193, 1954.
7. Oleinik, O.A., Discontinuous Solutions of Nonlinear Differential Equations, Usp.Math. Nauk, 12, pp. 3-73, 1957.
8. Rasulov, M.A. On a Method of Solving the Cauchy Problem for a First Order Nonlinear Equation of Hyperbolic Type with a Smooth Initial Condition, Soviet Math. Dok. 43, No.1, 1991.

9. Rasulov, M.A., Conservation Laws in a Class of Discontinuous Functions, Seckin, Istanbul, 2011 (in Turkish).
10. Rasulov, M.A.,.. On a Method of Calculation of the First Phase Saturation During the Process of Displacement of Oil by Water from Porous Medium. App. Mathematics and Computation, vol. 85, Issue l, pp. l-16, 1997.
11. Smoller, J.A., Shock Wave and Reaction Diffusion Equations, Springer-Verlag, New York Inc., 1983.
12. Toro, E.F, Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer-Verlag, Berlin Heidelberg, 1999.
13. Whitham, G.B. Linear and Nonlinear Waves, Wiley Int., New York, 1974.