Exact analysis of a nonlinear partial differential equation of gas dynamics
A new second-order nonlinear partial differential equation is derived from one-dimensional unsteady non-isentropic gas-dynamic equations through the introduction of three 'potential' functions. Appropriate boundary conditions at the shock and at the piston in terms of the new functions are obtained. The nonlinear partial differential equation is analysed in great detail. Intermediate integrals and generalized Riemann invariants are discovered. Using the classical Lie group method, the direct similarity method due to Clarkson and Kruskal (1989), and equation-splitting etc., large families of new solutions are found. The direct similarity method is found to yield the most general results. Solutions with shocks (both finite and strong) are constructed to illustrate the applicability of the solutions.