Complex systems often result in intractable mathematical models when classical methodological reduction methods are used in modeling. As a result, reduction methods more holistic in nature are normally used to avoid modeling on a component scale. In this dissertation, several new reduction methods are proposed with the intent of extending the application of classical methodological reduction methods to complex systems. As a motivating example, a large scale self-similar potential-driven tree network is used as a model complex system.
In the linear case, the self-similarity present in the physical system is translated to a self-similarity in the mathematical model. This is in turn used to analytically reduce an otherwise intractable DAE system to a much simpler ODE. It is also shown that for very large systems, it can sometimes be advantageous to approximate the system as infinite in scale. In the non-linear case, two numerical algorithms are presented to simplify dynamic analysis of piping networks. These methods are based on Chorin’s multi-step projection method for solving the Navier-Stokes equations. In addition to the self-similar tree network, other self-similar network structures are considered. In particular, grid-like networks are considered, and potential reduction methods are proposed.
Finally, in the course of studying the self-similar potential-driven tree network the appearance of fractional-order derivatives is noted several times. Based on this observation, fractional-order system identification is proposed as an extension of typical black-box reduction methods and experimental data acquired from a shell-and-tube heat exchanger is used to demonstrate its usefulness.
While the analysis presented is in terms of potential-driven transport networks, it should be noted that the specific examples used were chosen for their simplicity. But the methods used should be more generally applied to the reduction of self-similar complex systems, and even more generally, to large equation sets and DAE systems. Futhermore, the repeated appearance of fractional-order operators, and as a special case, fractional-order derivatives and integrals, suggest a very rich relationship between self-similar complex systems and fractional calculus.