  A compact and accurate solution method is provided for problems whose
infinite power series solution diverges and/or whose series coefficients are
only known up to a finite order. The method only requires that either the power
series solution or some truncation of the power series solution be available
and that some asymptotic behavior of the solution is known away from the
series' expansion point. Here, we formalize the method of asymptotic
approximants that has found recent success in its application to thermodynamic
virial series where only a few to (at most) a dozen series coefficients are
typically known. We demonstrate how asymptotic approximants may be constructed
using simple recurrence relations, obtained through the use of a few known
rules of series manipulation. The result is an approximant that bridges two
asymptotic regions of the unknown exact solution, while maintaining accuracy
in-between. A general algorithm is provided to construct such approximants. To
demonstrate the versatility of the method, approximants are constructed for
three nonlinear problems relevant to mathematical physics: the Sakiadis
boundary layer, the Blasius boundary layer, and the Flierl-Petviashvili
monopole. The power series solution to each of these problems is underspecified
since, in the absence of numerical simulation, one lower-order coefficient is
not known; consequently, higher-order coefficients that depend recursively on
this coefficient are also unknown. The constructed approximants are capable of
predicting this unknown coefficient as well as other important properties
inherent to each problem. The approximants lead to new benchmark values for the
Sakiadis boundary layer and agree with recent numerical values for properties
of the Blasius boundary layer and Flierl-Petviashvili monopole.
