Evolutionary Computation Technique for Solving Riccati Differential Equation of Arbitrary Order
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.
[1] S. Bittanti, "History and Prehistory of the Riccati Equation" Proceedings
of the 35th Conference on Decision and Control Kobe, Japan December
1996.
[2] Reid W. T, "Riccati DifferentialEquations", Academic Press, 1972.
[3] Goldstine H. H, "A History of the Calculus of Variations from the 17th
through the 19th Century", Springer-Verlag,, 1980.
[4] Mtter S. K, "Filtering and Stochastic Control: a Historical Perspective",
in IEEE Control n Systems, pp. 67-76, June 1996.
[5] B.D Anderson, J.B. Moore, "Optimal Control-Linear Quadratic
Methods", Prentice-Hall, New Jersey, 1990.
[6] 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.
[7] 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.
[8] S. Momani, N. Shawagfeh, "Decomposition method for solving
fractional Riccati differential equations", Journal of Applied
Mathematics and Computation 182, pp 1083-1092, 2006.
[9] 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.
[1] S. Bittanti, "History and Prehistory of the Riccati Equation" Proceedings
of the 35th Conference on Decision and Control Kobe, Japan December
1996.
[2] Reid W. T, "Riccati DifferentialEquations", Academic Press, 1972.
[3] Goldstine H. H, "A History of the Calculus of Variations from the 17th
through the 19th Century", Springer-Verlag,, 1980.
[4] Mtter S. K, "Filtering and Stochastic Control: a Historical Perspective",
in IEEE Control n Systems, pp. 67-76, June 1996.
[5] B.D Anderson, J.B. Moore, "Optimal Control-Linear Quadratic
Methods", Prentice-Hall, New Jersey, 1990.
[6] 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.
[7] 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.
[8] S. Momani, N. Shawagfeh, "Decomposition method for solving
fractional Riccati differential equations", Journal of Applied
Mathematics and Computation 182, pp 1083-1092, 2006.
[9] 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.
@article{"International Journal of Information, Control and Computer Sciences:51952", author = "Raja Muhammad Asif Zahoor and Junaid Ali Khan and I. M. Qureshi", title = "Evolutionary Computation Technique for Solving Riccati Differential Equation of Arbitrary Order", abstract = "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 = "Riccati Equation, Non linear ODE, Fractional differential equation, Genetic algorithm.", volume = "3", number = "10", pages = "2346-7", }
