About this Journal Submit a Manuscript Table of Contents
Discrete Dynamics in Nature and Society
Volume 2014 (2014), Article ID 401696, 15 pages
http://dx.doi.org/10.1155/2014/401696
Research Article

Optimization Solution of Troesch’s and Bratu’s Problems of Ordinary Type Using Novel Continuous Genetic Algorithm

1Department of Mechatronics Engineering, Faculty of Engineering, The University of Jordan, Amman 11942, Jordan
2Department of Mathematics, Faculty of Science, Al-Balqa’ Applied University, Salt 19117, Jordan
3Department of Mathematics, Faculty of Science, The University of Jordan, Amman 11942, Jordan
4Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Received 23 August 2013; Accepted 16 December 2013; Published 9 February 2014

Academic Editor: Stepan Agop Tersian

Copyright © 2014 Zaer Abo-Hammour et al. 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. S. M. Roberts and J. S. Shipman, “On the closed form solution of Troesch's problem,” Journal of Computational Physics, vol. 21, no. 3, pp. 291–304, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. U. M. Ascher, R. M. M. Mattheij, and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  3. E. S. Weibel, The Plasma in Magnetic Field, Edited by Landshoff R. K. M, Stanford University Press, Stanford, Calif, USA, 1958.
  4. E. S. Weibel, “On the confinement of a plasma by magnetostatic fields,” Physics of Fluids, vol. 2, pp. 52–56, 1959.
  5. V. S. Markin, A. A. Chernenko, Y. A. Chizmadehev, and Y. G. Chirkov, “Aspects of the theory of gas porous electrodes,” in Fuel Cells: Their Electrochemical Kinetics, Consultants Bureau, V. S. Bagotskii and Y. B. Vasilev, Eds., pp. 21–33, New York, NY, USA, 1966.
  6. D. Gidaspow and B. S. Baker, “A model for discharge of storage batteries,” Journal of the Electrochemical Society, vol. 120, pp. 1005–1010, 1973.
  7. G. Bratu, “Sur certaines équations intégrales non linéares,” Comptes Rendus, vol. 150, pp. 896–899, 1910.
  8. R. Buckmire, “Application of a Mickens finite-difference scheme to the cylindrical Bratu-Gelfand problem,” Numerical Methods for Partial Differential Equations, vol. 20, no. 3, pp. 327–337, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. J. S. McGough, “Numerical continuation and the Gelfand problem,” Applied Mathematics and Computation, vol. 89, no. 1–3, pp. 225–239, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. A. S. Mounim and B. M. de Dormale, “From the fitting techniques to accurate schemes for the Liouville-Bratu-Gelfand problem,” Numerical Methods for Partial Differential Equations, vol. 22, no. 4, pp. 761–775, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. S. A. Khuri, “A numerical algorithm for solving Troesch's problem,” International Journal of Computer Mathematics, vol. 80, no. 4, pp. 493–498, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. S. A. Khuri, “A new approach to Bratu's problem,” Applied Mathematics and Computation, vol. 147, no. 1, pp. 131–136, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. E. Deeba, S. A. Khuri, and S. Xie, “An algorithm for solving boundary value problems,” Journal of Computational Physics, vol. 159, no. 2, pp. 125–138, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. X. Feng, L. Mei, and G. He, “An efficient algorithm for solving Troesch's problem,” Applied Mathematics and Computation, vol. 189, no. 1, pp. 500–507, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. S. Momani, S. Abuasad, and Z. Odibat, “Variational iteration method for solving nonlinear boundary value problems,” Applied Mathematics and Computation, vol. 183, no. 2, pp. 1351–1358, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. H. Caglar, N. Caglar, M. Özer, A. Valarıstos, and A. N. Anagnostopoulos, “B-spline method for solving Bratu's problem,” International Journal of Computer Mathematics, vol. 87, no. 8, pp. 1885–1891, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. S. Abbasbandy, M. S. Hashemi, and C.-S. Liu, “The Lie-group shooting method for solving the Bratu equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 11, pp. 4238–4249, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. S. T. Cheng, “Topological optimization of a reliable communication network,” IEEE Transactions on Reliability, vol. 47, pp. 225–233, 1998.
  19. I. S. Misra, A. Raychowdhury, K. K. Mallik, and M. N. Roy, “Design and optimization of a nonplanar multiple array using genetic algorithms for mobile communications,” Microwave and Optical Technology Letters, vol. 32, pp. 301–304, 2002.
  20. J. Burm, “Optimization of high-speed metal semiconductor metal photodetectors,” IEEE Photonics Technology Letters, vol. 6, pp. 722–724, 1994.
  21. A. Vossinis, “Shape optimization of aerodynamics using nonlinear generalized minimal residual algorithm,” Optimal Control Applications & Methods, vol. 16, pp. 229–249, 1995.
  22. R. Fondacci, “Combinatorial issues in air traffic optimization,” Transportation Science, vol. 32, pp. 256–267, 1998.
  23. E. de Klerk, C. Roos, T. Terlaky et al., “Optimization of nuclear reactor reloading patterns,” Annals of Operations Research, vol. 69, pp. 65–84, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. Y. Cherruault, “Global optimization in biology and medicine,” Mathematical and Computer Modelling, vol. 20, no. 6, pp. 119–132, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. J. G. Rowse, “On the solution of optimal tax models and other optimization models in economics,” Economics Letters, vol. 18, pp. 217–222, 1985.
  26. G. V. Reklaitis, A. Ravindran, and K. M. Ragsdell, Engineering Optimization, John Wiley & Sons, New York, NY, USA, 1983, Methods and applications. View at MathSciNet
  27. M. Sebag and A. Ducoulombier, “Extending population-based incremental learning to continuous search spaces,” in Parallel Problem Solving from Nature, pp. 418–427, 1998.
  28. P. Larrañaga, R. Etxeberria, J. A. Lozano, and J. M. Pena, “Optimization in continuous domains by learning and simulation of Gaussian networks,” in Proceedings of the Genetic and Evolutionary Computation Conference Workshop Program, pp. 201–204, Las Vegas, Nev, USA, 2000.
  29. P. A. N. Bosnian and D. Thierens, “Expanding from discrete to continuous estimation of distribution algorithms: the IDEA,” in Parallel Problem Solving from Nature, pp. 767–776, 2000.
  30. D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley, Reading, Mass, USA, 1989.
  31. R. L. Haupt and S. E. Haupt, Practical Genetic Algorithms, John Wiley & Sons, 2004. View at MathSciNet
  32. Z. S. Abo-Hammour, Advanced continuous genetic algorithms and their applications in the motion planning of robotic manipulators and the numerical solution of boundary value problems [Ph.D. thesis], Quiad-Azam University, Islamabad, Pakistan, 2002.
  33. Z. S. Abo-Hammour, “A novel continuous genetic algorithms for the solution of the cartesian path generation problem of robot manipulators,” in Robot Manipulators: New Research, J. X. Lui, Ed., pp. 133–190, Nova Science Publishers, New York, NY, USA, 2005.
  34. Z. S. Abo-Hammour, N. Mirza, S. Mirza, and M. Arif, “Cartesian path planning of robot manipulators using continuous genetic algorithms,” Robotics and Autonomous Systems, vol. 41, pp. 179–223, 2002.
  35. Z. S. Abo-Hammour, O. Alsmadi, S. I. Bataineh, M. A. Al-Omari, and N. Affach, “Continuous genetic algorithms for collision-free cartesian path planning of robot manipulators,” International Journal of Advanced Robotic Systems, vol. 8, pp. 14–36, 2011.
  36. Z. S. Abo-Hammour, M. Yusuf, N. Mirza, S. Mirza, M. Arif, and J. Khurshid, “Numerical solution of second-order, two-point boundary value problems using continuous genetic algorithms,” International Journal for Numerical Methods in Engineering, vol. 61, pp. 1219–1242, 2004.
  37. Z. S. Abo-Hammour, A. Al-Asasfeh, A. Al-Smadi, and O. Alsmadi, “A novel continuous genetic algorithm for the solution of optimal control problems,” Optimal Control Applications and Methods, vol. 32, pp. 414–432, 2010.
  38. O. Abu Arqub, Z. Abo-Hammour, S. Momani, and N. Shawagfeh, “Solving singular two-point boundary value problems using continuous genetic algorithm,” Abstract and Applied Analysis, vol. 2012, Article ID 205391, 25 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  39. O. Abu Arqub, Z. S. Abo-Hammour, and S. Momani, “Application of continuous genetic algorithm for nonlinear system of second-order boundary value problems,” Applied Mathematics and Information Sciences, vol. 8, pp. 235–248, 2014.
  40. O. Abu Arqub, Numerical solution of fuzzy differential equation using continuous genetic algorithms [Ph.D. thesis], University of Jordan, Amman, Jordan, 2008.
  41. Z. Abo-Hammour, O. Alsmadi, S. Momani, and O. Abu Arqub, “A genetic algorithm approach for prediction of linear dynamical systems,” Mathematical Problems in Engineering, vol. 2013, Article ID 831657, 12 pages, 2012. View at Publisher · View at Google Scholar
  42. O. Abu Arqub, M. Al-Smadi, and S. Momani, “Application of reproducing kernel method for solving nonlinear Fredholm-Volterra integrodifferential equations,” Abstract and Applied Analysis, vol. 2012, Article ID 839836, 16 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  43. M. Al-Smadi, O. Abu Arqub, and S. Momani, “A computational method for two-point boundary value problems of fourth-order mixed integrodifferential equations,” Mathematical Problems in Engineering, vol. 2013, Article ID 832074, 10 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  44. O. Abu-Arqub, A. El-Ajou, S. Momani, and N. Shawagfeh, “Analytical solutions of fuzzy initial value problems by HAM,” Applied Mathematics & Information Sciences, vol. 7, no. 5, pp. 1903–1919, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  45. O. A. Arqub, M. Al-Smadi, and N. Shawagfeh, “Solving Fredholm integro-differential equations using reproducing kernel Hilbert space method,” Applied Mathematics and Computation, vol. 219, no. 17, pp. 8938–8948, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  46. O. Abu Arqub, A. El-Ajou, A. S. Bataineh, and I. Hashim, “A representation of the exact solution of generalized Lane-Emden equations using a new analytical method,” Abstract and Applied Analysis, vol. 2013, Article ID 378593, 10 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  47. O. Abu Arqub, Z. Abo-Hammour, R. Al-Badarneh, and S. Momani, “A reliable analytical method for solving higher-order initial value problems,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 673829, 12 pages, 2013. View at Publisher · View at Google Scholar
  48. N. Shawagfeh, O. Abu Arqub, and S. Momani, “Analytical solution of nonlinear second-order periodic boundary value problem using reproducing kernel method,” Journal of Computational Analysis and Applications, vol. 16, pp. 750–762, 2014.
  49. J. H. Jiang, J. H. Wang, X. Chu, and R. Q. Yu, “Clustering data using a modified integer genetic algorithm,” Analytica Chimica Acta, vol. 354, pp. 263–274, 1997.
  50. E. M. Rudnick, J. H. Patel, G. S. Greenstein, and T. M. Niermann, “A genetic algorithm framework for test generation,” IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, vol. 16, pp. 1034–1044, 1997.
  51. J. R. Vanzandt, “A Genetic Algorithm for Search Route Planning,” Tech. Rep. ESD-TR-92-262, United States Air Force, 1992.