Table of Contents Author Guidelines Submit a Manuscript
International Journal of Differential Equations
Volume 2015, Article ID 580741, 7 pages
http://dx.doi.org/10.1155/2015/580741
Research Article

Numerical Solution of Riccati Equations by the Adomian and Asymptotic Decomposition Methods over Extended Domains

1Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, P.O. Box 41335-1914, Rasht, Iran
2Department of Mathematics, Rasht Branch, Islamic Azad University, P.O. Box 41335-3516, Rasht, Iran
3Department of Mathematics, Guilan Science and Research Branch, Islamic Azad University, Rasht, Iran

Received 8 July 2015; Accepted 27 August 2015

Academic Editor: Timothy R. Marchant

Copyright © 2015 Jafar Biazar and Mohsen Didgar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. W. T. Reid, Riccati Differential Equations, Academic Press, New York, NY, USA, 1972. View at MathSciNet
  2. E. S. Fraga, The Schrödinger and Riccati Equations, vol. 70 of Lecture Notes in Chemistry, Springer, Berlin, Germany, 1999.
  3. R. Shankar, Principles of Quantum Mechanics, Plenum, New York, NY, USA, 1980. View at Publisher · View at Google Scholar
  4. A. Khare and U. Sukhatme, Supersymmetry in Quantum Mechanics, World Scientific, Singapore, 2001.
  5. M. I. Zelekin, Homogeneous Spaces and Riccati Equation in Variational Calculus, Factorial, Moscow, Russia, 1998, (Russian).
  6. V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons, Springer Series in Nonlinear Dynamics, Springer, Berlin, Germany, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  7. I. L. Buchbinder, S. D. Odintsov, and I. L. Shapiro, Effective Action in Quantum Gravity, IOP Publishing, 1992.
  8. K. Milton, S. D. Odintsov, and S. Zerbini, “Bulk versus brane running couplings,” Physical Review D, vol. 65, Article ID 065012, 2002. View at Publisher · View at Google Scholar
  9. H. C. Rosu and F. A. de la Cruz, “One-parameter Darboux-transformed quantum actions in thermodynamics,” Physica Scripta, In press.
  10. J. F. Cariñena, G. Marmo, A. M. Perelomov, and M. F. Rañada, “Related operators and exact solutions of Schrödinger equations,” International Journal of Modern Physics A, vol. 13, no. 28, pp. 4913–4929, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  11. B. D. Anderson and J. B. Moore, Optimal Control-Linear Quadratic Methods, Prentice-Hall, Upper Saddle River, NJ, USA, 1999.
  12. I. Lasiecka and R. Triggiani, Differential and Algebraic Riccati Equations with Application to Boundary/Point Control Problems: Continuous Theory and Approximation Theory, vol. 164 of Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  13. G. Adomian, Nonlinear Stochastic Operator Equations, Academic Press, Orlando, Fla, USA, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  14. G. Adomian and R. Rach, “On linear and nonlinear integro-differential equations,” Journal of Mathematical Analysis and Applications, vol. 113, no. 1, pp. 199–201, 1986. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic, Dordrecht, The Netherlands, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  16. G. Adomian and R. Rach, “On the solution of algebraic equations by the decomposition method,” Journal of Mathematical Analysis and Applications, vol. 105, no. 1, pp. 141–166, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  17. A. M. Wazwaz, Partial Differential Equations and Solitary Waves Theory, Higher Education Press, Beijing, China, Springer, Berlin, Germany, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  18. A.-M. Wazwaz, Linear and Nonlinear Integral Equations: Methods and Applications, Higher Education Press, Beijing, China, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  19. H. Bulut and D. J. Evans, “On the solution of the Riccati equation by the decomposition method,” International Journal of Computer Mathematics, vol. 79, no. 1, pp. 103–109, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  20. M. A. El-Tawil, A. A. Bahnasawi, and A. Abdel-Naby, “Solving Riccati differential equation using Adomian's decomposition method,” Applied Mathematics and Computation, vol. 157, no. 2, pp. 503–514, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  21. P.-Y. Tsai and C.-K. Chen, “An approximate analytic solution of the nonlinear Riccati differential equation,” Journal of the Franklin Institute, vol. 347, no. 10, pp. 1850–1862, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. A. R. Vahidi and M. Didgar, “An improved method for determining the solution of Riccati equations,” Neural Computing and Applications, vol. 23, no. 5, pp. 1229–1237, 2013. View at Publisher · View at Google Scholar · View at Scopus
  23. A. R. Vahidi, Z. Azimzadeh, and M. Didgar, “An efficient method for solving Riccati equation using homotopy perturbation method,” Indian Journal of Physics, vol. 87, no. 5, pp. 447–454, 2013. View at Publisher · View at Google Scholar · View at Scopus
  24. A. R. Vahidi, M. Didgar, and R. C. Rach, “An improved approximate analytic solution for Riccati equations over extended intervals,” Indian Journal of Pure and Applied Mathematics, vol. 45, no. 1, pp. 27–38, 2014. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  25. G. Adomian, “An investigation of the asymptotic decomposition method for nonlinear equations in physics,” Applied Mathematics and Computation, vol. 24, no. 1, pp. 1–17, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  26. G. Adomian, “An adaptation of the decomposition method for asymptotic solutions,” Mathematics and Computers in Simulation, vol. 30, no. 4, pp. 325–329, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  27. G. Adomian, “Solving the nonlinear equations of physics,” Computers & Mathematics with Applications, vol. 16, no. 10-11, pp. 903–914, 1988. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. R. Rach and J.-S. Duan, “Near-field and far-field approximations by the Adomian and asymptotic decomposition methods,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5910–5922, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  29. K. Haldar and B. K. Datta, “Integrations by asymptotic decomposition,” Applied Mathematics Letters, vol. 9, no. 2, pp. 81–83, 1996. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  30. R. C. Rach, “A new definition of the Adomian polynomials,” Kybernetes, vol. 37, no. 7, pp. 910–955, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  31. R. Rach, “A convenient computational form for the Adomian polynomials,” Journal of Mathematical Analysis and Applications, vol. 102, no. 2, pp. 415–419, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  32. G. Adomian and R. Rach, “Generalization of Adomian polynomials to functions of several variables,” Computers & Mathematics with Applications, vol. 24, no. 5-6, pp. 11–24, 1992. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. A.-M. Wazwaz, “A new algorithm for calculating Adomian polynomials for nonlinear operators,” Applied Mathematics and Computation, vol. 111, no. 1, pp. 53–69, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  34. J. Biazar, E. Babolian, G. Kember, A. Nouri, and R. Islam, “An alternate algorithm for computing Adomian polynomials in special cases,” Applied Mathematics and Computation, vol. 138, no. 2-3, pp. 523–529, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  35. J.-S. Duan, “Recurrence triangle for Adomian polynomials,” Applied Mathematics and Computation, vol. 216, no. 4, pp. 1235–1241, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  36. J.-S. Duan, “An efficient algorithm for the multivariable Adomian polynomials,” Applied Mathematics and Computation, vol. 217, no. 6, pp. 2456–2467, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  37. J.-S. Duan, “Convenient analytic recurrence algorithms for the Adomian polynomials,” Applied Mathematics and Computation, vol. 217, no. 13, pp. 6337–6348, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  38. Y. Cherruault and G. Adomian, “Decomposition methods: a new proof of convergence,” Mathematical and Computer Modelling, vol. 18, no. 12, pp. 103–106, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  39. K. Abbaoui and Y. Cherruault, “Convergence of Adomian's method applied to differential equations,” Computers & Mathematics with Applications, vol. 28, no. 5, pp. 103–109, 1994. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  40. K. Abbaoui and Y. Cherruault, “New ideas for proving convergence of decomposition methods,” Computers & Mathematics with Applications, vol. 29, no. 7, pp. 103–108, 1995. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  41. A. Abdelrazec and D. Pelinovsky, “Convergence of the Adomian decomposition method for initial-value problems,” Numerical Methods for Partial Differential Equations, vol. 27, no. 4, pp. 749–766, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus