Exact Solutions of Generalized Riemann Problem for Nonhomogeneous Shallow Water Equations

Riemann problems for generalized gas dynamics

“…Thus, in view of ( 20), (21), and (45), system (1) can be written in terms of Riemann invariants 𝑊 (1) and 𝑊 (2) as…”

“…17 Among others, Radha et al 18 used the differential constraints method to completely characterize the Riemann problem for nonconstant initial data for rate-type materials. Furthermore, the method of differential constraints was also used to determine the exact solution of quasilinear systems depending on various applications like p-systems with relaxation conditions (Curró, and Manganaro 19 ), for Chaplygin gas model (Kumar and Radha 20 ), for nonhomogeneous shallow water equations (Sueet et al 21 ), and for the homogeneous p-system (Manganaro et al 22 ). Recently, Meleskho et al 23 discussed the generalized simple wave solution for a magnetic fluid using the differential constraint method and also applied it to systems of equations written in Riemann invariants.…”

“…used the differential constraints method to completely characterize the Riemann problem for nonconstant initial data for rate‐type materials. Furthermore, the method of differential constraints was also used to determine the exact solution of quasilinear systems depending on various applications like p‐systems with relaxation conditions (Curró, and Manganaro 19 ), for Chaplygin gas model (Kumar and Radha 20 ), for nonhomogeneous shallow water equations (Sueet et al 21 22 …”

Study of wave propagation in arterial blood flow under symmetry analysis

“…where Λ represents the jump across Z x and defined as Λ = 𝜂r (2) and 𝜂 is the amplitude and ∇ = ( 𝜕 𝜕A , 𝜕 𝜕u ) . Now, using (20), (22), and ( 23) in (24), we obtained the Bernoulli type of equation for the amplitude 𝜂 as where…”

“…This approach for deriving the optimal system is also pursued by Sekhar and Satapathy. 17 For the exact solution of PDEs arising in many physical phenomena, using symmetry analysis and Riemann problem, we refer Bira et al 18,21 and Sahoo et al, 19,20 whereas the evolutionary behavior of weak discontinuities and elementary wave interaction for quasi-linear hyperbolic systems, we refer Raja et al, 22 Satapathy and Sekhar, 23 Zeidan and Bira, 24 and Sil and Raja Sekhar. 25 The structure of this article is as follows: Section 2 deals with a brief discussion on the model along with the derivation of the symmetry group of transformations.…”