Exact Solutions of Generalized Riemann Problem for Nonhomogeneous Shallow Water Equations
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DOI: 10.1007/s13226-020-0460-2
EXACT SOLUTIONS OF GENERALIZED RIEMANN PROBLEM FOR NONHOMOGENEOUS SHALLOW WATER EQUATIONS Sueet Millon Sahoo, T. Raja Sekhar and G. P. Raja Sekhar Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India e-mails: [email protected]; [email protected] (Received 31 May 2018; after final revision 14 March 2019; accepted 27 June 2019) In this paper, we consider quasilinear hyperbolic system of balance laws describing one-dimensional nonhomogeneous shallow water equations with generalized Riemann initial data. We obtain exact solutions to the shallow water equations with friction by using differential constraint method. A special case of the obtained solution provides well known rarefaction wave to the homogeneous case of the governing equations. We construct a convenient example for the generalized Riemann problem and study the behavior of the solution profiles. Key words : Generalized Riemann problem; exact solutions; differential constraint method; nonhomogeneous shallow water equations. 2010 Mathematics Subject Classification : 35L60, 35L45, 35Q53, 74G05.
1. I NTRODUCTION System of quasilinear partial differential equations are generally used to describe various physical phenomena in fields like marine engineering, fluid dynamics, plasma physics, chemistry, biology, physics and many other application areas. Exact solutions of such system of equations are important to understand of nonlinear phenomena in various fields of science, especially in physics. Many powerful and systematic methods have been proposed during the last few decades, by several mathematicians such as symmetry analysis, perturbed method, vanishing viscosity method and so on. Apart from these methods, differential constraint method is one of the most powerful method which gives a systematic procedure to find a class of exact solutions for first order hyperbolic system of balance
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S. M. SAHOO, T. RAJA SEKHAR AND G. P. RAJA SEKHAR
laws. Janenko [1] pioneered the idea of differential constraint method and a survey of the same can be found in the book by Sirdov et al. [2]. The procedure of differential constraint method involves two steps. The first step is to find a compatible system of equations corresponding to the given system via differential constraints. The second step is to construct solutions of this compatible system which is overdetermined. Because the solution has to satisfy the differential constraints, it makes easier to construct particular solutions of governing system [3]. Here our main attention is to determine simple wave solutions to generalized Riemann problem (GRP) for hyperbolic system of first order PDEs involving source-like terms. The GRP is the special case of Cauchy problem for quasilinear hyperbolic system of balance laws in one space dimension which is having single discontinuity. The initial data consists of piecewise smooth functions on both sides of the discontinuity. While classical Riemann problems can be solved exactly for many re
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