An Effective Perturbation Iteration Algorithm for Solving Riccati Differential Equations

Abstract

In the following study, a novel approach called the Perturbation Iteration Algorithm PIA has been proposed and subsequently adopted for deriving and solving the Riccati differential equation. This new Perturbation Iteration Method is efficient and has no requirement of a small parameter assumption as its earlier classical counterparts do. Some examples have been presented to exhibit how simply and efficiently the proposed method works. After deriving the exact solution of the Riccati equation, the capability and the simplicity of the proposed technique is clarified. A percentage error for each example has also been presented.

References

• Reid, W. T. (1972) Riccati Differential Equations. Academic Press, New York.
• Carinena, J. F, Marmo, G. , Perelomov, A. M. & Ranada M. F. (1998) Related operators and exact solutions of Schrodinger equations. International Journal of Modern Physics, 13. pp 4913-4929
• Scott, M. R. (1973) Invariant Imbedding and its Applications to Ordinary Differential Equations. Addison-Wesley, Boston.
• Tan, Y. & Abbasbandy, S. (2008) Homotopy analysis method for quadratic Riccati differential equation. Commun. Nonlinear Sci. Numer. Simul. , 13. pp 539-546
• Abbasbandy, S. (2007) A new application of Hes variational iteration method for quardratic Riccati differential equation by using Adomians polynomials. Journal of Computer Application, 207. pp 59–63
• El-Tawil, M. A. , Bahnasawi, A. A. & Abdel-Naby A. (2004) Solving Riccati differential equation using Adomians decomposition method. Applied Mathematics and Computation, 157. pp 503-514
• Abbasbandy, S. (2006) Iterated Hes homotopy perturbation method for quadratic Riccati differential equation. Applied Mathematics and Computation, 175. pp 581-589
• Abbasbandy, S. (2006) Homotopy Perturbation method for quadratic Riccati differential equation and comparision with Adomains Decomposition method. Applied Mathematics and Computation, 172. pp 485–90
• Geng,F. , Lin, Y. & Cui, M. (2009) A piecewise variational iteration method for Riccati differential equations. Comput. Math. Appl. , 58. pp 2518-2522
• Gulsu, M. & Sezer, M. (2006) On the solution of the Riccati equation by the Taylor matrix method. Applied Mathematics and Computation, 176. pp 414–421
• Akyuz-Dacoglu, A. & Sezer, M. (2003) Chebyshev polynomial solutions of systems of high-order linear differential equations with variable coefficients. Applied Mathematics and Computation, 144. pp 237-247
• Nayfeh, A. H. (1973) it Perturbation Methods. Wiley- Interscience, New York.
• Skorokhod, A. V. , Hoppensteadt, F. C. & Salehi, H. (2002) Random Perturbation Methods with Applications in Science and Engineering. Springer, New York.
• Pakdemirli, M. , Aksoy, Y. & Boyaci, H. (2011) A new perturbation-iteration approach for first order differential equations. Mathematical and Computational Applications, 16(4). pp 890-899
• Aksoy, Y. & Pakdemirli, M. (2010) New perturbationiteration solutions for Bratu-type equations. Computers and Mathematics with Applications, 59(8). pp -2808
• Aksoy, Y. Pakdemirli, M. & Abbasbandy, S. (2012) New perturbation-iteration solutions for nonlinear heat transfer equations. International Journal of Numerical Methods for Heat and Fluid Flow, 22(7). pp 814-28

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