Combining Legendre's polynomials and genetic algorithm in the solution of nonlinear initial-value problems.(Research Article)(Report)

1. Introduction

Mathematical modeling is a constant in engineering daily life, and expressing phenomena by using differential equations is almost mandatory. Successful modeling implies ability to make simplifying assumptions but, even under these assumptions, the solution of differential equations implies a lot of analytical and numerical efforts [1],

Considering these difficulties and sometimes the impossibility of obtaining analytical solutions, several numerical methods were proposed and, nowadays, the great majority of these methods are part of symbolic and numeric softwares as MATLAB and Mathematica, for instance [2]. These methods are accurate and fast, but each one has its limitations, depending on the type of equation to be solved. The main difficulties appear when the mass-matrix is singular or the equation generates instability in the running process [3].

Here, trying to deal with this kind of problem for equations with continuous solutions, a method for initial-value problems in ordinary differential equations is developed, assuming that the space of solutions is a Hilbert space, equipped with a Legendre's polynomial basis [4].

The central idea is to look for the best polynomial coefficient combination, in order to satisfy the original differential equation in the whole interval, instead of searching a polynomial that better fits a set of points in the given interval.

Consequently, a new error criterion must be defined and a program by using differential evolution methods with elitism [5-7] is developed, based on this new criterion.

The determination of the search space developed here is performed by using sequential contractions, allowing fast evolutions with noticeable reductions in the error bands when compared with random search in large populations, as proposed by [5].

Several examples are solved by the method presented and by using the classical numerical methods, with the accuracy of the results being compared.

In Section 2, the error criterion is discussed and a combination of a global and local criteria is proposed, providing optimal fitting of the polynomial, regarding the function and its derivatives. Section 3 presents the analytical basis of the method, by using Legendre's polynomials to optimize the solutions of an initial-value problem.

The use of the differential genetic algorithm with elitism, guaranteeing stability, applied to the solution of ordinary differential equations, emphasizing the choice of the coefficients under the optimizing criteria developed in Section 2, is described in Section 4. Finally, in Section 5, some examples are presented and the numerical quality of the solutions is discussed.

2. Error Criterion

Usually, the adopted optimization criterion to approximate a function by a polynomial in a given interval is to minimize the square error integral [8]. This kind of approach minimizes the global error along the whole interval, but produces a large value for the local error at the starting and ending points of the interval.

Consequently, when this criterion is applied to an initial-value problem, the approximation does not produce the best result, as the error for the starting point of the interval remains large. To solve this problem, an error criterion considering the local and global results is proposed, minimizing the product of two factors: the maximum error at the starting point of the interval and the global integral error.

To give an idea of this procedure, the differential equation given by (2.1) with initial condition y(-1) = [e.sup.-1], that admits the solution given by (2.2) in the [- 1,1] interval, is studied.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.2)

If one tries to obtain, by using MATLAB, the 8th-order Legendre's polynomial that better fits to (2.2), the result, with double-precision format, is given by (2.3). Considering the substitution of y, given by (2.3), in the original differential equation (2.1), the integral error is 1.9.[10.sup.-8] for the whole interval

y = .0263[x.sup.8] - .1551[x.sup.6] + .4963[x.sup.4] - .9996[x.sup.2] + 1,0000. (2.3)

If one continues the process, with the result given by (2.3) as an initial solution candidate to the original equation (2.1), by using the method developed in the next Sections, a solution given by (2.4), with an integral error 8.7.[10.sup.-9] for the whole interval, is obtained

y = .0242[x.sup.8] - .1511[x.sup.6] + 0.4940[x.sup.4] - .9992[x.sup.2] + 1,0000. (2.4)

This result is related to the fact that the polynomial given by (2.3) is adequate to fit the function given by (2.2), without considering its derivatives. In order to adequate the solution to the differential equation, given by (2.1), the derivatives must be adjusted in the same way.

3. Analytic Foundations

In this section, it is considered the Hilbert space of piecewise continuous functions, defined in a closed interval [a,b], with a scalar product between f and g, belonging to the space, defined as [4]

<f,g> = [[integral].sup.b.sub.a] f g dx. (3.1)

Considering the real interval [-1,1], the Legendre's polynomials set is an orthogonal basis for the space of piecewise continuous function S, defined in this interval, that is, for any function f [member of] S

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3.2)

with [P.sub.k] being the k-degree normalized Legendre's polynomial and [a.sub.k] = <f, [P.sub.k)> [4,9].

It is important …