Construction of piecewise-analytical gas flows joined through shock waves near the axis or center of symmetry

Analytical solutions of a quasilinear system of equations with partial derivatives are constructed in the case where the initial data for different functions are specified on different surfaces and the resultant problem has singularities of the form $u/x$ and $w/x$. Conditions for existence and uniqueness of a solution in the form of formal power series for the problem posed and sufficient conditions for convergence of the series are indicated. A generalization of the problem considered is given. Results of the study are used to solve the problem of the focussing of a compression wave generated by a piston moving smoothly in a quiescent gas: a solution for $t=0$, including determination of the piston trajectory, and a solution for $t<0$, including unequivocal construction of the front of a reflected shock wave, are uniquely constructed from the distribution of gas-dynamic quantities for $t>0$. The solution of this problem is a generalization to the case of two independent variable self-similar Sedov's solutions.