With the help of relation Eqs

With the help of relation (3.5), Eqs. (3.7) translates aswithOn using the basic state condition (3.6), the system of Eqs. (3.8), (3.9) yield the condition which relates the functions f() and g() aswhere the constant α is given asUsing the boundary conditions (2.21) and (3.10), Eqs. (2.22), (2.23) may be solved in the region .
The basic state solutions are computed from the basic relations (3.4) for three values of /U0=0.25,0.50,0.75 and presented in the tables (1)–(3).

Three examples
The differential equations (2.22) and (2.23) and the boundary conditions (2.21) and (3.8) are linear. The solution of the plane piston problem may be reduced to the following two elementary solutionsTherefore, the solution of the plane piston problem is the linear combination of the two elementary solutionswhere Ω is constant and defined asThe effect of the applied gravity on the distribution of functions f(η) and g(η) in a non-ideal flow field is presented in Figs. 1–3 for the values of Tables 1–3. In a weak gravitational field the internal and, sometimes, kinetic aniracetam of the gas will exhaust to overcome the applied gravity and the process is slowed down due to increase in the value of parameter of non-idealness as compared to what it would in an ideal gas. The propagation velocity of the shock wave is . The strength of the shock wave is increased if U1 is positive. Further, the expression (3.9) shows that U1 has the same sign as u1 and therefore f(η). Figs. 1–3 shows that f()≈0 for the all cases. Also, the monotonic decreasing values of g(η) from the piston η=1 to shock wave η= decreases more rapidly with an increase in the value of the parameter of non-idealness () which implies that the internal energy of the gas between the piston and shock wave exhausts more rapidly in a non-ideal gas. Further, the effect of non-idealness of the gas is to decrease the values of f(η) from piston to the shock which shows that the process of acceleration/deceleration is slowed down due to non-idealness of the gas.

Strong shock wave approximation
For the case of strong shock wave, the flow region becomes narrow, and . Consequently the system of Eqs. (2.22) and (2.23) take the following formEliminating ∂g(η)/∂η from the above equation we find the differential equation in terms of f(η) as followsThe general solution of the above differential equation is determined easily asUsing the above relation in Eq. (5.1) we determine the value of g(η) as followswhere β=1/a0, constants c1and c2 are obtained by the boundary conditions (2.21) and (3.10) asandWith the help of the original relation (2.20), the profile of the first order solution are given asWe now discuss the case of strong shock wave, i.e., /U0→0, which gives =(γ+1)/2. For γ=5/3, we havewhere the constant c1and c2 are given numerically asThe value of c1 and c2 appearing in the Eqs. (5.11) and (5.12) increases with an increase in the value of consequently the negative value of f(η) increases, and consequently the strength of the shock wave decreases. Also, the monotonic decreasing function g(η) shows further decreasing trend with an increase in the value of , therefore, the gas internal energy exhausts which agrees with the earlier results discussed in [7]. Since (−1) is much smaller than one. The analytical results presented here give the approximate solution and describes qualitatively the basic features of the influence of applied gravity in a non-ideal gas. Likewise, we derive the analytical solutions near the piston, which are expected to be more accurate. Eqs. (5.1) and (5.2) are reduced toThe gas motion will be accelerated near the piston if g(1)>0, on the other hand decelerated if g(1)<0. From the above relations it is clear that the internal energy always decreases in the region near the piston. Also, the non-idealness of the flow field contributes to decrease the internal energy further.
Results and discussion
The hyperbolic system of Eqs. (2.1), (2.2), (2.3) possesses three families of characteristics dx/dt=u, the trajectory of fluid particle, and dx/dt=u±a, the outgoing and incoming wavelets. With the help of expansion (2.11) the characteristic relations have been changed as follows