The analytical solution does not, however, allow for objects breaking up in response to aerodynamic forces.

Many of the problems facing the finance community have no known analytical solution.

Their analytical solutions had been well-known challenges in this field for 50 years until 2008.

Such analytical solutions are limited to relatively simple flow systems and reactions.

But direct analytical solutions of the beam equation are possible only for the simplest cases.

In a few cases, an analytical solution of the rate equations may be determined, but this is relatively rare.

There is no simple analytical solution to this differential equation.

A complete analytical solution is provided for the tangential traction.

Sometimes, an analytical solution to a mathematical model is not known.

Although analytical solutions can only be found for specific and simple cases, they give a good insight to deal with more complex situations.