Memetic Heuristic Computation for Nonlinear Problems in Engineering

Abstract

This dissertation presents new heuristic computational schemes for solving the nonlinear problems in engineering that are governed by nonlinear ordinary differential equations (NODEs) and nonlinear partial differential equations (NPDEs). The heuristic schemes comprising of Evolutionary Algorithms (EAs) and a linear combination of some basis functions are presented for solving NODEs. The approximate solution of NODEs is deduced as a linear combination of some basis functions with unknown parameters. Three different basis functions including log sigmoid, Bernstein polynomials, and polynomial basis have been used for the approximate modeling. A fitness function is used to convert the NODE into an equivalent global error minimization problem. Two popular EAs including Genetic Algorithm (GA) and Differential Evolution (DE), and local search techniques, such as, Interior Point Algorithm (IPA) and Active Set Algorithm (ASA) are used to solve the minimization problem and to obtain the unknown parameters. The memetic algorithm schemes combining GA with IPA (GA-IPA) and GA with ASA (GA-ASA) are also explored. The schemes have been tested on various nonlinear problems including Bratu problem, Duffing van der pol oscillator, Michaelis- Menten biochemical reaction system, and power-law fin-type problem. An elegant hybridization of Exp-function method with nature inspired computing (NIC) has been presented for the numerical solution of NPDEs. Exp-function method is used to express the travelling wave solution of the given NPDE. The NPDE is converted into an optimization problem. Two popular NIC techniques including GA and particle swarm optimization (PSO) are used to solve the optimization problem. The scheme has been successfully tested on some important NPDEs including generalized Burger-Fisher, Burger-Huxley, and Fisher equations. The proposed numerical solutions are found in a good agreement with the exact solutions and quite competent with those reported by some well-known classical methods like adomian decomposition method (ADM), variational iteration method (VIM), and homotopy perturbation method (HPM). It is also observed that the memetic algorithm schemes are good choice for the optimization of such problem. The presented schemes are simple as well as efficient, and they provide the numerical solution not only at the grid points but also at any value in the solution domain.