Spherical explosion with central energy source

We present a novel semi-analytic solution describing the propagation of a spherical blast wave driven by a central energy source. The initial density profile has a power-law function of the distance from the center and the energy is injected only into the central region at a rate given by a power-law function of time. This solution is composed of three regions separated by the contact surface and the shock front. The innermost region is assumed to be uniform and the outside of the contact surface includes the shocked matter described by self-similar solutions. We analytically derive the applicable range of parameters of this solution from requirements to satisfy boundary conditions. We perform numerical simulations for flows with various values of parameters, some of which reside out of the thus derived applicable range, and compare with the semi-analytic solutions.