  We derive a combined analytical and numerical scheme to solve the
(1+1)-dimensional differential Kirchhoff system. Here the object is to obtain
an accurate as well as an efficient solution process. Purely numerical
algorithms typically have the disadvantage that the quality of solutions
decreases enormously with increasing temporal step sizes, which results from
the numerical stiffness of the underlying partial differential equations. To
prevent that, we apply a differential Thomas decomposition and a Lie symmetry
analysis to derive explicit analytical solutions to specific parts of the
Kirchhoff system. These solutions are general and depend on arbitrary
functions, which we set up according to the numerical solution of the remaining
parts. In contrast to a purely numerical handling, this reduces the numerical
solution space and prevents the system from becoming unstable. The differential
Kirchhoff equation describes the dynamic equilibrium of one-dimensional
continua, i.e. slender structures like fibers. We evaluate the advantage of our
method by simulating a cilia carpet.
