Solução de problemas de valor inicial usando os métodos de Runge-Kutta, Dormand-Prince e de Bulirsch-Stoer

Abstract

Several problems encountered in sciences and, particularly engineering, can be solved by mathematical modeling that results, in general, in differential equations. Ordinary differential equations are subsets of this universe, which constitute the formulation of initial value problems for a variety of processes and systems. The solution of these problems can be obtained by solving the differential equations that describe the dynamics of the system. One of the fundamental tools to obtain these solutions are the numerical methods, several of which have already been incorporated into some computational platform, such as, for example, Matlab. In this dissertation, the objective was to use the fixed-step Runge-Kutta methods and the Bulirsch-Stoer method, to solve three typical problems involving ordinary differential equations and to jointly compare the results with the Dormand-Prince method, incorporated into Matlab by ode45 subroutine. For the Runge-Kutta and Bulirsch-Stoer methods, two subroutines were written in Matlab language. Three problems were solved: 1) a simple ordinary differential equation with an analytical solution, used as a reference; 2) modeling of a vehicle suspension system; and 3) the Page equation. For problem 1, all methods provided an adequate solution with the appropriate integration step size. For problem 2, the Dormand-Prince method was relentless in keeping the solution stable even for larger step sizes. The Runge-Kutta and Bulirsch-Stoer methods, in this case, worked well for reduced steps. In problem 3, the method that provided an adequate stable solution was Bulirsch-Stoer. The others also work, for very small step sizes. Therefore, it was concluded that the appropriate method for solving initial value problems depends on the nature of the problem and the appropriate choice of the integration step-size.