Riemann Problem for First-Order Partial Equations Without the Convexity of a State Functions

  • Mahir RasulovEmail author
  • S. Ozgur Ulas
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9045)


In this work, the exact solution of the Riemann problem for first-order nonlinear partial equation with non-convex state function in \(Q_T=\{(x,t)|x\in I=\left( -\infty ,\ \ \infty \right) ,\ t\in \left[ 0,T\right) \}\subset R^2\) is found. Here \(F\in C^2{(Q}_T)\ \) and \(\ F^{''}(u)\) change their signs, that is F(u) has convex and concave parts. In particular, the state function \(F\left( u\right) =-{\cos u\ }\) on \(\ \left[ \frac{\pi }{2},\frac{3\pi }{2}\right] \) and \(\ \left[ \frac{\pi }{2},\frac{5\pi }{2}\right] \) is discussed. For this, when it is necessary, the auxiliary problem which is equivalent to the main problem is introduced. The solution of the proposed problem permits constructing the weak solution of the main problem that conserves the entropy condition. In some cases, depending on the nature of the investigated problem a convex or a concave hull is constructed. Thus, the exact solutions are found by using these functions.


First order nonlinear partial differential equations Riemann problem Characteristics Weak solution Shock wave Convex and concave hull 


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Mathematics and ComputingBeykent UniversityIstanbulTurkey
  2. 2.Institute of SciencesBeykent UniversityTaksimTurkey

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