The Perturbed Riemann Problem for a Macroscopic Production Model with Chaplygin Gas
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The Perturbed Riemann Problem for a Macroscopic Production Model with Chaplygin Gas Pengyan Wang1 · Chun Shen1 Received: 10 April 2020 / Revised: 12 August 2020 / Accepted: 17 August 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract The exact Riemann solutions for a macroscopic production model under the equation of state given by the Chaplygin gas are solved explicitly for all possible Riemann initial data. It is discovered interestingly that a composite hyperbolic wave is involved in Riemann solution under some specially designated initial conditions, which is made up of a rarefaction wave and a delta contact discontinuity attached on the wave front of the rarefaction wave. Furthermore, the constructions of global solutions to the perturbed Riemann problem for this system are also displayed in completely explicit forms when the initial data are taken to be three piecewise constant states under some suitable restrictive conditions by using the method of characteristics. During the process of constructing global solutions, the interactions of elementary waves are studied in detail. Moreover, it is proved rigorously that Riemann solutions are stable with respect to the specific small perturbations of Riemann initial data. Keywords Riemann problem · Wave interaction · Chaplygin gas · Macroscopic production model Mathematics Subject Classification 35L65 · 35L67 · 76N15
1 Introduction In recent years, the macroscopic production models have become one of the important research topics in the study of production planning and control in the manufacturing
Communicated by Norhashidah Hj. Mohd. Ali. This work is partially supported by Shandong Provincial Natural Science Foundation (ZR2019MA058).
B 1
Chun Shen [email protected] School of Mathematics and Statistics Science, Ludong University, Yantai, Shandong Province 264025, People’s Republic of China
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P. Wang and C. Shen
industry. In the previous stage, the fluid-like continuous models [1,2] have been introduced to model high-volume product flows, which can be used to depict the product flow by using discrete event simulations in an accumulated way. The most widely used model is a scalar conservation law under appropriate assumptions on the flux function which is called as a clearing function for the product flow. In general, a given clearing function describes the averaged sample data, but cannot illustrate the data diffusion. In order to solve this problem, the following macroscopic production model [3] consisting of two conservation laws is proposed in the form
ρ t +(ρu)x = 0, ρu 1 + P(ρ) + ρu 2 1 + P(ρ) = 0, t
(1.1)
x
in which ρ and u are used to represent the product density and velocity, respectively, which are often required to be non-negative. In addition, t and x express the time and the production stage (or the degree of completion) to describe the work-in-progress and the pressure term P refers to the anticipated factor in the production line which is usually given by the equation o
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