ABSTRACT
In the previous chapters, traditional branches of advanced applied mathematics have been summarized and our focus was on computer aided solutions to those problems. Over the last few decades, many new applied math topics emerged which are referred to as nontraditional mathematics in this book. For instance, fuzzy logic and fuzzy inference are presented and used for imitating imprecise human thinking and linguistic behaviors. The artificial neural networks are established based on the mathematical model imitating the neural network of biological systems. The genetic algorithm-based optimization procedures are proposed based on the principles of survival of the fittest. These new branches of applied mathematics are promising areas of research in science and engineering offering important tools for real-life problems. In Section 10.1, classical set theory, fuzzy set and fuzzy inference are presented with their implementations in MATLAB. Section 10.2 introduces artificial neural network in general and feedforward neural network in particular with back-propagation algorithms, where MATLAB solutions for network construction, training and generalization are presented using data fitting problems for illustration. In Section 10.3, evolution type optimization algorithms, including genetic algorithms and particle swarm optimization methods, are introduced. The global solutions to optimization problems are also explored. Waveletbased methods and solutions are given in Section 10.4 using signal de-noising as an application example. Fundamental introduction to rough set and rough set-based attribute reduction is given in Section 10.5 with real-life examples. In Section 10.6, a comprehensive introduction to fractional-order calculus and numerical solutions to fractional-order ordinary differential equations is given, where from a programming point of view, the design and application of the classes and objects dedicated for fractional-order calculus are demonstrated thoroughly.
