Abstract
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.
Keywords
Scalar conservation laws state function without convexity convex and concave hull weak solution
References
- Collins, P. Fluids Flow in Porous Materials. 1964.
- Goritskii, A.A., Krujkov, S.N., Chechkin, G.A. A First Order Quasi-Linear Equations with Partial Differential Derivatives. Pub. Moskow University, Moskow, 1997.
- Kin, Y.J., Lee, Y., Structure of Fundamental Solutions of a Conservation Laws without Convexity, Applied Mathematics, vol.8, pp. 1-20, 2008.
- Krushkov, S.N., First Order Quasilinear Equations in Several Independent Variables, Math. USSS Sb., 10, pp.217-243, 1970.
- Lax, P.D. The Formation and Decay of Shock Waves, Amer. Math Monthly, 79, pp. 227-241, 1972.
- 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.
- Oleinik, O.A., Discontinuous Solutions of Nonlinear Differential Equations, Usp.Math. Nauk, 12, pp. 3-73, 1957.
- 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.
- Rasulov, M.A., Conservation Laws in a Class of Discontinuous Functions, Seckin, Istanbul, 2011 (in Turkish).
- 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.
- Smoller, J.A., Shock Wave and Reaction Diffusion Equations, Springer-Verlag, New York Inc., 1983.
- Toro, E.F, Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer-Verlag, Berlin Heidelberg, 1999.
- Whitham, G.B. Linear and Nonlinear Waves, Wiley Int., New York, 1974.
Abstract
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.
Keywords
Scaler korunum kanunları bükeyliği olmayan durum fonksiyonu konveks ve konkav katman zayıf çözüm.
References
- Collins, P. Fluids Flow in Porous Materials. 1964.
- Goritskii, A.A., Krujkov, S.N., Chechkin, G.A. A First Order Quasi-Linear Equations with Partial Differential Derivatives. Pub. Moskow University, Moskow, 1997.
- Kin, Y.J., Lee, Y., Structure of Fundamental Solutions of a Conservation Laws without Convexity, Applied Mathematics, vol.8, pp. 1-20, 2008.
- Krushkov, S.N., First Order Quasilinear Equations in Several Independent Variables, Math. USSS Sb., 10, pp.217-243, 1970.
- Lax, P.D. The Formation and Decay of Shock Waves, Amer. Math Monthly, 79, pp. 227-241, 1972.
- 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.
- Oleinik, O.A., Discontinuous Solutions of Nonlinear Differential Equations, Usp.Math. Nauk, 12, pp. 3-73, 1957.
- 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.
- Rasulov, M.A., Conservation Laws in a Class of Discontinuous Functions, Seckin, Istanbul, 2011 (in Turkish).
- 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.
- Smoller, J.A., Shock Wave and Reaction Diffusion Equations, Springer-Verlag, New York Inc., 1983.
- Toro, E.F, Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer-Verlag, Berlin Heidelberg, 1999.
- Whitham, G.B. Linear and Nonlinear Waves, Wiley Int., New York, 1974.
Details
| Primary Language | English |
|---|---|
| Subjects | Engineering |
| Journal Section | Research Article |
| Authors | |
| Publication Date | May 6, 2015 |
| IZ | https://izlik.org/JA29EU43JH |
| Published in Issue | Year 2012 Volume: 5 Issue: 1-2 |