Application of Metaheuristics in Solving Initial Value Problems (IVPs)

Abstract

Some differential equations admit analytic solutions given by explicit formulas. However, in most other case only approximated solutions can be found. Several methods are available in the literature to find approximate solutions to differential equations. Numerical methods form an important part of solving IVP in ODE, most especially in cases where there is no closed form of solutions. The present dissertation focus the attention toward solving IVP by transforming it to an optimization approach which can be solved through the application of non-standard methods called Metaheuristic. By transforming the IVP into an optimization problem, an objective function, which comprises both the IVP and initial conditions, is constructed and its optimum solutions represents an approximative solution of the IVP.The main contribution of the present thesis is divided in twofold. In the one hand, we consider IVPs as an optimization problem when the search of the optimum solution is performed by means of MAs including ABC, BA and FPA and a set of numerical methods including Euler methods, Runge–Kutta methods and predictor–corrector methods. On the other hand, we propose a new MA called Fractional L´evy Flight Bat Algorithm (FLFBA) (which is an improvement of the BA, based on velocity update through fractional calculus and local search procedure based on a L´evy distribution random walk). We illustrates its computational efficiency by comparing its performance with the previous methodds in solving the bacterial population growth models ( both the logistic growth model and the exponential growth model).