The concept of differentiation and integration to no integer order, dates from 1695, when Leibniz mentions it in a letter to L'Hopital. Since then, many scientists developed the area and notable contributions have been made, both in theory and in applications [3,4,5,7,8,9,10].

The integer-order differential nature of the potential expressions motivated several authors to propose its generalization in a fractional calculus (FC) perspective. Therefore, a fractional multipole produces at point a potential .

Inspired by the integer-order recursive approximation of fractional-order transfer functions presented previously, we developed a GA for implementing a fractional order potential. In this line of thought, we developed a one-dimensional GA that places recursively n charges    odd   even at the symmetrical positions (with exception of that corresponds to the centre of the n-array of charges where there is a single charge ).

Our goal is to compare the approximate potential , resulting from a number of charges and the corresponding locations, with the desired reference potential . Therefore, in this study the experiments consist of executing the GA several times, in order to generate a combination of positions and charges that lead to an electrical potential with a fractional slope similar to the desired reference potential.

The results show a good fit between the two functions and, moreover, that it is possible to find more than one 'good' solution. Nevertheless, for a given application, a superior precision may be required and, in that case, a larger number of charges must be used. We verify that the position of the charges varies significantly with the number of charges used in the algorithm. Therefore, the charge versus location pattern is not clear and its comparison with a fractal recursive layout is still under investigation.

We conclude that the GA reveals a good balance between the accuracy of the results and the computation time. The GA approach constitutes a step towards the development of a systematic design technique and, consequently, several of its aspects must be further evaluated. Research on the approximation feasibility and its convergence, error variation with the range and the number of charges, improvement when adopting an extended library of primitives rather than, merely, point charges and its extension to the three-dimensional space is presently under development.

1

D.E. Goldberg, "Genetic Algorithms in Search Optimization and Machine Learning", Addison-Wesley, 1989.

2

Z. Michalewicz, "Genetic Algorithms + Data Structures = Evolution Programs", Springer-Verlag, 1996.

3

N. Engheta, "On Fractional Calculus and Fractional Multipoles in Electromagnetism", IEEE Transactions on Antennas and Propagation, Vol. 44, No. 4, 554-566, April 1996. doi:10.1109/8.489308

4

J.T. Machado, Isabel Jesus, Alexandra Galhano, Albert W. Malpica, Fernando Silva, József K. Tar, "Fractional Order Dynamics In Classical Electromagnetic Phenomena", in "Fifth EUROMECH Nonlinear Dynamics Conference - ENOC 2005", Eindhoven, 1322-1326, 7- 12 August 2005.

5

J.T. Machado, Isabel Jesus, Alexandra Galhano, "A Fractional Calculus Perspective in Electromagnetics", in "ASME - Int. Design Eng. Technical Conf. & Computers and Information in Eng. Conf. - 5th Int. Conf. on Multibody Systems, Nonlinear Dynamics and Control", U.S.A., Sept. 2005.

6

M. Mitchell, "An Introduction to Genetic Algorithms", MIT Press, 1998.

7

Keith B. Oldham and Jerome Spanier, "The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order", Academic Press, 1974.

8

A. Oustaloup, "La Commande CRONE: Commande Robuste d'Ordre Non Entier", Hermes, 1991.

9

K. Küpfmüller, "Einführung in die Theoretische Elektrotechnik", Springer-Verlag, Berlin, 1939.

10

L. Bessonov, "Applied Electricity for Engineers", MIR Publishers, Moscow, 1968.

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