  We construct a supersonic-sonic smooth patch solution for the two dimensional
steady Euler equations in gas dynamics. This patch is extracted from the Frankl
problem in the study of transonic flow with local supersonic bubble over an
airfoil. Based on the methodology of characteristic decompositions, we
establish the global existence and regularity of solutions in a partial
hodograph coordinate system in terms of angle variables. The original problem
is solved by transforming the solution in the partial hodograph plane back to
that in the physical plane. Moreover, the uniform regularity of the solution
and the regularity of an associated sonic curve are also verified.
