A new efficient method for solving quadratic Riccati differential equation

  • Authors

    • Belal Batiha Higher Colleges of Technology (HCT), Abu Dhabi Men's College, United Arab Emirates (UAE)
    2015-01-03
    https://doi.org/10.14419/ijamr.v4i1.4113
  • Quadratic Riccati equation, Variational iteration method, Multistage variational iteration method.

  • Abstract

    A new efficient method called the multistage variational iteration method (MVIM) is applied to the solution of quadratic Riccati differential equations. A comparison between MVIM solution with classical variational iteration method (VIM) and exact solution has been made and show that the MVIM is a powerful method to the solution of nonlinear differential equations.

  • References

    1. [1] Reid WT. Riccati differential equations (Mathematics in science and engineering, Vol. 86). New York: Academic Press.; 1972.

      [2] Anderson BD, Moore JB. Optimal control-linear quadratic methods. Prentice-Hall, New Jersey 1999.

      [3] Lasiecka I, Triggiani R. Differential and algebraic Riccati equations with application to boundary/point control problems: continuous theory and approximation theory (Lecture notes in control and information sciences, Vol. 164). Berlin: Springer.; 1991.

      [4] Dubois F, Saidi A. Unconditionally stable scheme for Riccati equation. ESAIM Proceedings 8 (2000) 39-52.

      [5] Bahnasawi AA, El-Tawil MA, Abdel-Naby A. Solving Riccati differential equation using Adomian's decomposition method. Appl Math Comput 157 (2004) 503-514.
      http://dx.doi.org/10.1016/j.amc.2003.08.049

      [6] Tan Y, Abbasbandy S. Homotopy analysis method for quadratic Riccati differential equation. Commun Nonlin Sci Numer Simul 13 (3) (2008) 539-546.
      http://dx.doi.org/10.1016/j.cnsns.2006.06.006

      [7] Geng, F. A modified variational iteration method for solving Riccatic differential equations. Comp.Math. with Application 6 (2010) 1868-1872.
      http://dx.doi.org/10.1016/j.camwa.2010.07.017

      [8] He JH. A new approach to nonlinear partial differential equations. Commun Nonlin Sci Numer Simul 2 (1997) 230-235.
      http://dx.doi.org/10.1016/S1007-5704(97)90007-1

      [9] He JH. Approximate analytical solution of Blasius' equation. Commun Nonlin Sci Numer Simul 3 (1998) 260-263.
      http://dx.doi.org/10.1016/S1007-5704(98)90046-6

      [10] He JH. Approximate analytical solution for seepage OW with fractional derivatives in porous media. Comput Meth Appl Mech Engrg 167 (1998) 57-68.
      http://dx.doi.org/10.1016/S0045-7825(98)00108-X

      [11] He JH. Variational iteration method-a kind of non-linear analytical technique: some examples. Int J Non-Linear Mech 34 (1999) 699-708.
      http://dx.doi.org/10.1016/S0020-7462(98)00048-1

      [12] He JH. Variational iteration method for autonomous ordinary differential systems. Appl Math Comput 114 (2000) 115-123.
      http://dx.doi.org/10.1016/S0096-3003(99)00104-6

      [13] He JH. A simple perturbation approach to Blasius equation. Appl Math Comput 140 (2003) 217-222.
      http://dx.doi.org/10.1016/S0096-3003(02)00189-3

      [14] He JH, Wan YQ, Guo Q. An iteration formulation for normalized diode characteristics. Int J Circ Theory Appl 32 (2004) 629-632.
      http://dx.doi.org/10.1002/cta.300

      [15] He JH. Some asymptotic methods for strongly nonlinear equations. Int J Modern Phys B 20 (2006) 1141-1199.
      http://dx.doi.org/10.1142/S0217979206033796

      [16] He JH. Non-perturbative methods for strongly nonlinear problems. Berlin: dissertation.de-Verlag im Internet GmbH.; 2006.

      [17] He JH. Variational iteration method-some recent results and new interpretations. J Comput Appl Math 207 (1) (2007) 3-17.
      http://dx.doi.org/10.1016/j.cam.2006.07.009

      [18] Soliman AA. A numerical simulation and explicit solutions of KdV-Burgers' and Lax's seventh-order KdV equations. Chaos Solitons Fractals 29 (2) (2006) 294-302
      http://dx.doi.org/10.1016/j.chaos.2005.08.054

      [19] Abulwafa EM, Abdou MA, Mahmoud AA. The solution of nonlinear coagulation problem with mass loss. Chaos Solitons Fractals 29 (2) (2006) 313-330.
      http://dx.doi.org/10.1016/j.chaos.2005.08.044

      [20] Momani S, Abuasad S. Application of He's variational iteration method to helmholtz equation. Chaos Solitons Fractals 27 (2006) 1119-1123.
      http://dx.doi.org/10.1016/j.chaos.2005.04.113

      [21] Odibat ZM, Momani S. Application of variational iteration method to nonlinear differential equations of fractional order. Int J Nonlinear Sci Numer Simul 7 (2006) 27-34.
      http://dx.doi.org/10.1515/IJNSNS.2006.7.1.27

      [22] Bildik N, Konuralp A. The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations. Int J Nonlinear Sci Numer Simul 7 (2006) 65-70.
      http://dx.doi.org/10.1515/IJNSNS.2006.7.1.65

      [23] Dehghan M, Tatari M. The use of He's variational iteration method for solving a Fokker-Planck equation. Phys Scr 74 (2006)310-316.
      http://dx.doi.org/10.1088/0031-8949/74/3/003

      [24] Wazwaz AM. The variational iteration method for rational solutions for KdV, K(2,2), Burgers, and cubic Boussinesq equations. J Comput Appl Math 207(1) (2007) 18-23 .
      http://dx.doi.org/10.1016/j.cam.2006.07.010

      [25] Tamer A. Abassy, Magdy A. El-Tawil, H. El Zoheiry. Toward a modi_ed variational iteration method. J Comput Appl Math. 207(1) (2007) 137-147 .
      http://dx.doi.org/10.1016/j.cam.2006.07.019

      [26] Abbasbandy S. A new application of He's variational iteration method for quadratic Riccati differential equation by using Adomian's polynomials. J Comput Appl Math. 207(1) (2007) 59-63. .
      http://dx.doi.org/10.1016/j.cam.2006.07.012

      [27] Junfeng L. Variational iteration method for solving two-point boundary value problems. J Comput Appl Math. 207(1) (2007) 92-95 .
      http://dx.doi.org/10.1016/j.cam.2006.07.014

      [28] Batiha B, Noorani M S M, Hashim I, Ismail E S. The multistage variational iteration method for a class of nonlinear system of ODEs. Phys. Scr. 76 (2007) 388-392
      http://dx.doi.org/10.1088/0031-8949/76/4/018

      [29] Goh S.M, Noorani M.S.M., Hashim I. Prescribing a multistage analytical method to a preypredator dynamical system. Physics Letters A 373 (2008) 107-110.
      http://dx.doi.org/10.1016/j.physleta.2008.11.009

      [30] Goh S.M, Noorani M.S.M., Hashim I. A new application of variational iteration method for the chaotic Rossler system. Chaos, Solitons and Fractals 42 (2009) 1604-1610.
      http://dx.doi.org/10.1016/j.chaos.2009.03.032

      [31] Goh S.M, Noorani M.S.M., Hashim I. E_cacy of variational iteration method for chaotic Genesio system-Classical and multistage approach. Chaos, Solitons and Fractals 40 (2009) 2152-2159.
      http://dx.doi.org/10.1016/j.chaos.2007.10.003

      [32] Inokuti M. Sekine H, Mura T. General use of the Lagrange multiplier in nonlinear mathematical physics. In: Nemat-Nassed S, editor. Variational method in the mechanics of solids. Pergamon Press.; 1978.

      [33] Abdou MA, Soliman AA. Variational iteration method for solving Burger's and coupled Burger's equations. J Comput Appl Math 181 (2005) 245-251.
      http://dx.doi.org/10.1016/j.cam.2004.11.032

      [34] Moghimi M, Hejazi F. Variational iteration method for solving generalized Burger-Fisher and Burger equations. Chaos Solitons Fractals, 33(5) (2007) 1756-1761. .
      http://dx.doi.org/10.1016/j.chaos.2006.03.031

      [35] Abdulaziz O, Noor NFM, Hashim I, Noorani MSM. Further accuracy tests on Adomian decomposition method for chaotic systems. Chaos, Solitons and Fractals 36 (2008) 1405-11.
      http://dx.doi.org/10.1016/j.chaos.2006.09.007

      [36] Hashim I, Noorani MSM, Ahmad R, Bakar SA, Ismail ES, Zakaria AM, et al. Accuracy of the Adomian decomposition method applied to the Lorenz system. Chaos, Solitons and Fractals 28 (2006) 1149-1158.
      http://dx.doi.org/10.1016/j.chaos.2005.08.135

      [37] Noorani MSN, Hashim I, Ahmad R, Bakar SA, Ismail ES, Zakaria AM, et al. Comparing numerical methods for the solutions of the Chen system. Chaos, Solitons and Fractals 32 (4) (2007)1296-1304.
      http://dx.doi.org/10.1016/j.chaos.2005.12.036

      [38] Polyanin AD, Zaitsev VF. Handbook of exact solutions for ordinary differential equations. Florida: CRC.; 2003.

      [39] Batiha B, Noorani M S M, Hashim I. Application of variational iteration method to a general Riccati equation. International Mathematical Forum, 2 (56) (2007) 2759-2770.

  • Downloads

  • How to Cite

    Batiha, B. (2015). A new efficient method for solving quadratic Riccati differential equation. International Journal of Applied Mathematical Research, 4(1), 24-29. https://doi.org/10.14419/ijamr.v4i1.4113

    Received date: 2015-01-03

    Accepted date: 2015-01-03

    Published date: 2015-01-03