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Evolutionary Computation Technique For Solving Riccati Differential Equation Of Arbitrary Order

Authors: Raja Muhammad Asif Zahoor; Junaid Ali Khan; I. M. Qureshi;

Evolutionary Computation Technique For Solving Riccati Differential Equation Of Arbitrary Order

Abstract

{"references": ["S. Bittanti, \"History and Prehistory of the Riccati Equation\" Proceedings\nof the 35th Conference on Decision and Control Kobe, Japan December\n1996.", "Reid W. T, \"Riccati DifferentialEquations\", Academic Press, 1972.", "Goldstine H. H, \"A History of the Calculus of Variations from the 17th\nthrough the 19th Century\", Springer-Verlag,, 1980.", "Mtter S. K, \"Filtering and Stochastic Control: a Historical Perspective\",\nin IEEE Control n Systems, pp. 67-76, June 1996.", "B.D Anderson, J.B. Moore, \"Optimal Control-Linear Quadratic\nMethods\", Prentice-Hall, New Jersey, 1990.", "K. Diethelm, J. M. Ford, N. J. Ford, W. Weilbeer, \"Pitfalls in fast\nnumerical solvers for fractional differential equations\" J. Comput. Appl.\nMath, 186 pp 482-503, 2006.", "F. Mainardi, G. Pagnini, and R. Gorenflo, \"Some aspects of fractional\ndiffusion quations of single and distributed orders\" Journal of Applied\nMathematics and Computation, 187 No 1, pp 295-305, 2007.", "S. Momani, N. Shawagfeh, \"Decomposition method for solving\nfractional Riccati differential equations\", Journal of Applied\nMathematics and Computation 182, pp 1083-1092, 2006.", "Duan Junsheng, An Jianye, and Xu Mingyu, \"Solution of system of\nfractional differential equations by Adomian decomposition Method\",\nAppl. Math. J. Chinese Univ. Ser. B, 22(1) pp 7-12, 2007.\n[10] Shaher Momani, and Zaid Odibat, \"Comparision between the homotopy\nperturbation method and the variational iteration method for linear\nfractional partial differential equations\", Computer & Mathematics with\nApplications, Vol 54, Issue 7-B pp 910-919. 2007\n[11] L. Galeone, and R. Garrappa, \"Fractional Adams-Moulton method\"\nMathematics and Computers in Simulation, vol. 79 issue 4 pp 1358-\n1367, 2008.\n[12] G. Tsoulos and I. E. Lagaris, \"Solving differential equations with\ngenetic programming\", Genetic programming and Evolvabe Machines,\nVol. 7 No. 1 pp 33-54, 2006.\n[13] Paul E. MacNeil, \"Genetic algorithms and solutions of an interesting\ndifferential equation\", proceedings of the 10th annual conference on\nGenetic and evolutionary computation, pp 1711-1712, 2008.\n[14] Miller, K. S. and Ross, B., \"An Introduction to the Fractional Calculus\nand Fractional Differential Equations\" John Wiley and Sons, Inc., New\nYork 1993.\n[15] Oldham, K. B. and Spanier, J., \"The Fractional Calculus\" Academic\nPress, New York 1974.\n[16] Daniel R. Rarisi et al. \"Solving differential equations with\nunsupervised neural networks\", Chemical engineering and\nprocessing, 42 pp 715-721 2003.\n[17] Lucie P. Aarts and Peter Van Der Veer, \"Neural Network Method for\nsolving the partial Differential Equations\" Neural Processing Letters 14\npp 261-271, 2001.\n[18] A. Junaid, M. A. Z. Raja, and I. M. Qureshi, \"Evolutionary computing\napproach for the solution of initial value problem in ordinary differential\nequation\" Proceeding of World academy of science engineering and\ntechnology, vol. 55, pp 578-581 July 2009."]}

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.

Keywords

Genetic algorithm., Non linear ODE, Riccati Equation, Fractional differential equation

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This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
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