APPLICATION OF FRACTIONAL CALCULUS TO ENGINEERING: A NEW COMPUTATIONAL APPROACH

Abstract

In this dissertation, a new heuristic computational intelligence technique has been developed for the solution for fractional order systems in engineering. These systems are provided with generic ordinary linear and nonlinear differential equations involving integer and non-integer order derivatives. The design scheme consists of two parts, firstly, the strength of feed-forward artificial neural network (ANN) is exploited for approximate mathematical modeling and secondly, finding the optimal weights for ANN. The exponential function is used as an activation function due to availability of its fractional derivative. The linear combination of these networks defines an unsupervised error for the system. The error is reduced by selection of appropriate unknown weights, obtained by training the networks using heuristic techniques. The stochastic techniques applied are based on nature inspired heuristics like Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) algorithm. Such global search techniques are hybridized with efficient local search techniques for rapid convergence. The local optimizers used are Simulating Annealing (SA) and Pattern Search (PS) techniques. The methodology is validated by applying to a number of linear and nonlinear fraction differential equations with known solutions. The well known nonlinear fractional system in engineering based on Riccati differential equations and Bagley- Torvik Equations are also solved with the scheme. The comparative studies are carried out for training of weights for ANN networks with SA, PS, GA, PSO, GA hybrid with SA (GA-SA), GA hybrid with PS (GA-PS), PSO hybrid with SA (PSO-SA) and PSO hybrid with PS (PSO-PS) algorithms. It is found that the GA-SA, GA-PS, PSO-SA and PSO-PS hybrid approaches are the best stochastic optimizers. The comparison of results is made with available exact solution, approximate analytic solution and standard numerical solvers. It is found that in most of the cases the design scheme has produced the results in good agreement with state of art numerical solvers. The advantage of our approach over such solvers is that it provides the solution on continuous time inputs with finite interval instead of predefine discrete grid of inputs. The other perk up of the scheme in its simplicity of the concept, ease in use, efficiency, and effectiveness.