Approximate Solution of Riccati Differential Equation via Modified Greens Decomposition Method
Riccati differential equations (RDEs) plays important role in the various fields of defence, physics, engineering, medical science, and mathematics. A new approach to find the numerical solution of a class of RDEs with quadratic nonlinearity is presented in this paper. In the process of solving the pre-mentioned class of RDEs, we used an ordered combination of Green’s function, Adomian’s polynomials, and Pade` approximation. This technique is named as green decomposition method with Pade` approximation (GDMP). Since, the most contemporary definition of Adomian polynomials has been used in GDMP. Therefore, a specific class of Adomian polynomials is used to advance GDMP to modified green decomposition method with Pade` approximation (MGDMP). Further, MGDMP is applied to solve some special RDEs, belonging to the considered class of RDEs, absolute error of the obtained solution is compared with Adomian decomposition method (ADM) and Laplace decomposition method with Pade` approximation (LADM-Pade`). As well, the impedance of the method emphasised with the comparative error tables of the exact solution and the associated solutions with respect to ADM, LADM-Pade`, and MGDMP. The observation from this comparative study exhibits that MGDMP provides an improved numerical solution in the given interval. In spite of this, generally, some of the particular RDEs (with variable coefficients) cannot be easily solved by some of the existing methods, such as LADM-Pade` or Homotopy perturbation methods. However, under some limitations, MGDMP can be successfully applied to solve such type of RDEs.
Reid, W.T. Riccati differential equations. Academic Press. 1972.
Dehghan, M. & Taleei, A. A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients. Comput. Phys. Commun., 2010, 181, 43–51. https://doi.org/10.1016/j.cpc.2009.08.015
Mukherjee, S. & Roy, B. Solution of Riccati equation with variable coefficient by differential transform method. Int. J. Nonlinear Sci., 2012, 14, 251–256.
Kafi, R. Al; Abdillah, B. & Mardiyati, S. Approximate solution of Riccati differential equations and DNA Repair model with Adomian decomposition method. J. Phys. Conf. Ser., 2018, 1090, 012017. https://doi.org/10.1088/1742-6596/1090/1/012017
Palumbo, N.F. & Jackson, T.D. Integrated missile guidance and control: A state dependent Riccati differential equation approach. In Proceedings of the IEEE International Conference on Control Applications (Cat. No.99CH36328) (IEEE), 1999, 243–248. https://doi.org/10.1109/CCA.1999.806207
Dharmaiah, V. Thory of ordinary differential equations. Delhi: PHI Learning Private Limited, 2013.
Adomian, G. Solving frontier problems of physics: The Decomposition Method. Dordrecht: Springer Netherlands, 2013. https://doi.org/10.1007/978-94-015-8289-6
Cherruault, Y. Convergence of Adomian’s method. Kybernetes, 1989, 18, 31–38. https://doi.org/10.1108/eb005812
Abbasbandy, S. Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian’s decomposition method. Appl. Math. Comp., 2006, 172, 485–490. https://doi.org/10.1016/J.AMC.2005.02.014
Abbasbandy, S. Iterated He’s homotopy perturbation method for quadratic Riccati differential equation. Appl. Math. Comp., 2006, 175, 581–589. https://doi.org/10.1016/J.AMC.2005.07.035
Abbasbandy, S. A new application of He’s variational iteration method for quadratic Riccati differential equation by using Adomian’s polynomials. J. Comp. Appl. Math., 2007, 207, 59–63. https://doi.org/10.1016/J.CAM.2006.07.012
Tsai, P.Y. & Chen, C.K. An approximate analytic solution of the nonlinear Riccati differential equation. J. Franklin Inst., 2010, 347, 1850–1862. https://doi.org/10.1016/j.jfranklin.2010.10.005
Arya, M. & Ujlayan, A. Solution of Riccati differential equation with Green’s function and Padè approximation technique. Adv. Differ. Equations Control Process, 2019, 21, 31–52. https://doi.org/10.17654/DE021010031
Alkresheh, H.A. New classes of Adomian polynomials for the Adomian decomposition method. Int. J. Eng. Science Invention, 2016, 5, 37-44.
Adomian, G. A review of the decomposition method and some recent results for nonlinear equations. Comp. Math. Appl., 1991, 22, 101–127. https://doi.org/10.1016/0898-1221(91)90220-X
Biazar, J.; Babolian, E. & Islam, R. Solution of the system of ordinary differential equations by Adomian decomposition method. Appl. Math. Comp., 2004, 147, 713–719. https://doi.org/10.1016/S0096-3003(02)00806-8
Biazar, J.; Babolian, E. & Islam, R. Solution of a system of Volterra integral equations of the first kind by Adomian Method. Appl. Math. Comp., 2003, 139, 249–258. https://doi.org/10.1016/S0096-3003(02)00173-X
Babolian, E.; Biazar, J. & Vahidi, A.R. Solution of a system of nonlinear equations by Adomian decomposition method. Appl. Math. Comp., 2004, 150, 847–854. https://doi.org/10.1016/S0096-3003(03)00313-8
Hatami, M.; Ganji, D.D.; Sheikholeslami, M.; Hatami, M.; Ganji, D.D. & Sheikholeslami, M. Introduction to differential transformation method. Diff. Transform. Method Mech. Eng. Probl., 2017, 1–54. https://doi.org/10.1016/B978-0-12-805190-0.00001-2
Luo, X.G. A two-step Adomian decomposition method. Appl. Math. Comp., 2005, 170, 570–583. https://doi.org/10.1016/J.AMC.2004.12.010
Wazwaz, A.M. A reliable modification of Adomian decomposition method. Appl. Math. Comp., 1999, 102, 77–86. https://doi.org/10.1016/S0096-3003(98)10024-3
Chisholm, A. & Common, J.S. Padé approximation and its applications. L. Wuytack editorial. Berlin, Heidelberg: Springer Berlin Heidelberg, 1979. https://doi.org/10.1007/BFb0085571
Baker, G.A. & Morris, P. The convergence of sequences of Padé approximants. J. Math. Anal. Appl., 1982, 87, 382–394. https://doi.org/10.1016/0022-247X(82)90131-7
Agarwal, R. & Regan, D. An introduction to ordinary differential equations. 2008.
Deo, S.G. & Raghavendra, V. Ordinary differential equations and stability theory, 2005.
El-Tawil, M.A.; Bahnasawi, A.A. & Abdel-Naby, A. Solving Riccati differential equation using Adomian’s decomposition method. Appl. Math. Comp., 2004, 157, 503–514. https://doi.org/10.1016/J.AMC.2003.08.049
Where otherwise noted, the Articles on this site are licensed under Creative Commons License: CC Attribution-Noncommercial-No Derivative Works 2.5 India