Exact solutions of generalized Riemann problem for rate-type material

Riemann problem for rate‐type materials with nonconstant initial conditions

“…1 ≡ 1. Thus, on solving (3.3) and (3.5), equation (3.1) admits the following solutions, which recovers the solution obtained in 22 that follows a different line of approach:…”

“…Similarly, with the assumption that , we have . Thus, on solving () and (), equation () admits the following solutions, which recovers the solution obtained in 22 that follows a different line of approach: …”

“…1,2 A variety of mathematical methods for finding exact solutions to such systems have been proposed over the years (see previous works [3][4][5][6] ). The approach based on the use of differential constraints, proposed by Janenko 7 (see also Rozhdestvenski and Janenko 8 and Sidorov et al 9 ), has been of considerable interest in recent years (see previous studies [10][11][12][13][14][15][16][17][18][19][20][21][22][23] ). Based on Lie symmetry analysis, an approximate rarefaction wave-type solution to the Riemann problem with nonclassical discontinuous initial data for a system of balance laws describing rate-type materials was presented in Conforto et al 6 ; here, the initial data for the variable u are discontinuous whereas the initial data for the variable v are constants.…”

Study of wave propagation in arterial blood flow under symmetry analysis

“…where C 1 is arbitrary constant; which in turn produces the solution of system (3) as Further, from (19), we study the average velocity profile of the blood flow and cross-sectional area of the artery for different values of stiffness s. The arterial stiffness is influenced by many factors like biological aging, smoking, and poor cardiovascular health, which we studied graphically. With the increasing s (increasing age), the thickness of the artery increases; consequently, the average velocity in the blood flow increases as it observed in Figure 1A, whereas in Figure 1B, it is noticed that the cross-sectional area of the arteries decreases with increasing stiffness of the artery s, ensuing high pressure in the artery wall, which has a higher risk of hypertension as well as other illnesses including chronic renal disease and stroke.…”

“…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.…”