World Academy of Science, Engineering and Technology
Published October 29, 2009 | Version 2829
Journal article Open

Evolutionary Computation Technique for Solving Riccati Differential Equation of Arbitrary Order

Description

In this article an evolutionary technique has been used for the solution of nonlinear Riccati differential equations of fractional order. In this method, genetic algorithm is used as a tool for the competent global search method hybridized with active-set algorithm for efficient local search. The proposed method has been successfully applied to solve the different forms of Riccati differential equations. The strength of proposed method has in its equal applicability for the integer order case, as well as, fractional order case. Comparison of the method has been made with standard numerical techniques as well as the analytic solutions. It is found that the designed method can provide the solution to the equation with better accuracy than its counterpart deterministic approaches. Another advantage of the given approach is to provide results on entire finite continuous domain unlike other numerical methods which provide solutions only on discrete grid of points.

Files

2829.pdf

Files (128.4 kB)

Name Size
md5:388cedcb48305868b250de0875c7a366
128.4 kB Preview Download

Additional details

References

  • S. Bittanti, "History and Prehistory of the Riccati Equation" Proceedings of the 35th Conference on Decision and Control Kobe, Japan December 1996.
  • Reid W. T, "Riccati DifferentialEquations", Academic Press, 1972.
  • Goldstine H. H, "A History of the Calculus of Variations from the 17th through the 19th Century", Springer-Verlag,, 1980.
  • Mtter S. K, "Filtering and Stochastic Control: a Historical Perspective", in IEEE Control n Systems, pp. 67-76, June 1996.
  • B.D Anderson, J.B. Moore, "Optimal Control-Linear Quadratic Methods", Prentice-Hall, New Jersey, 1990.
  • K. Diethelm, J. M. Ford, N. J. Ford, W. Weilbeer, "Pitfalls in fast numerical solvers for fractional differential equations" J. Comput. Appl. Math, 186 pp 482-503, 2006.
  • F. Mainardi, G. Pagnini, and R. Gorenflo, "Some aspects of fractional diffusion quations of single and distributed orders" Journal of Applied Mathematics and Computation, 187 No 1, pp 295-305, 2007.
  • S. Momani, N. Shawagfeh, "Decomposition method for solving fractional Riccati differential equations", Journal of Applied Mathematics and Computation 182, pp 1083-1092, 2006.
  • Duan Junsheng, An Jianye, and Xu Mingyu, "Solution of system of fractional differential equations by Adomian decomposition Method", Appl. Math. J. Chinese Univ. Ser. B, 22(1) pp 7-12, 2007. [10] Shaher Momani, and Zaid Odibat, "Comparision between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations", Computer & Mathematics with Applications, Vol 54, Issue 7-B pp 910-919. 2007 [11] L. Galeone, and R. Garrappa, "Fractional Adams-Moulton method" Mathematics and Computers in Simulation, vol. 79 issue 4 pp 1358- 1367, 2008. [12] G. Tsoulos and I. E. Lagaris, "Solving differential equations with genetic programming", Genetic programming and Evolvabe Machines, Vol. 7 No. 1 pp 33-54, 2006. [13] Paul E. MacNeil, "Genetic algorithms and solutions of an interesting differential equation", proceedings of the 10th annual conference on Genetic and evolutionary computation, pp 1711-1712, 2008. [14] Miller, K. S. and Ross, B., "An Introduction to the Fractional Calculus and Fractional Differential Equations" John Wiley and Sons, Inc., New York 1993. [15] Oldham, K. B. and Spanier, J., "The Fractional Calculus" Academic Press, New York 1974. [16] Daniel R. Rarisi et al. "Solving differential equations with unsupervised neural networks", Chemical engineering and processing, 42 pp 715-721 2003. [17] Lucie P. Aarts and Peter Van Der Veer, "Neural Network Method for solving the partial Differential Equations" Neural Processing Letters 14 pp 261-271, 2001. [18] A. Junaid, M. A. Z. Raja, and I. M. Qureshi, "Evolutionary computing approach for the solution of initial value problem in ordinary differential equation" Proceeding of World academy of science engineering and technology, vol. 55, pp 578-581 July 2009.