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Numerical Solution of Singularly Perturbed Convection Delay Problems Using Self-adaptive Differential Evolution Algorithm

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Intelligent Computing Methodologies (ICIC 2018)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10956))

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Abstract

In this paper, a new numerical technique is constructed to solve singularly perturbed convection delay problems. First of all, based on Taylor’s series expansion, the given problem is transformed into a singularly perturbed convection-diffusion problem without delay term, which is discretized by using the rational spectral collocation method with a sinh transformation. It should be pointed out that the width of boundary layer, which is chosen as a parameter in the sinh transformation, can be determined. Then, a nonlinear unconstrained optimization problem is designed to determine the width of boundary layer. Finally, the numerical solution of the singularly perturbed problem is converted into minimizing the nonlinear unconstrained optimization problem, which is solved by using a self-adaptive differential evolution (SADE). The numerical results show that the proposed algorithm is a robust and accurate procedure for solving singularly perturbed convection delay problems. Furthermore, the obtained accuracy for the solutions using SADE is much better than results obtained using some others algorithms.

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References

  1. Elsgolts, L.E., El-Sgol-Ts, L.E., El-Sgol-C, L.E.: Qualitative Methods in Mathematical Analysis. American Mathematical Society, Providence (1964)

    Google Scholar 

  2. Mackey, M.C., Glass, L.: Oscillation and chaos in physiological control systems. Science 197, 287–289 (1977)

    Article  Google Scholar 

  3. Lasota, A., Wazewska, M.: Mathematical models of the red blood cell systems. Mat. Stos 6, 25–40 (1976)

    Google Scholar 

  4. Lange, C.G., Miura, R.M.: Singular perturbation analysis of boundary value problems for differential- difference equations. V. small shifts with layer behavior. SIAM J. Appl. Math. 54, 249–272 (1994)

    Article  MathSciNet  Google Scholar 

  5. Kadalbajoo, M.K., Sharma, K.K.: Numerical analysis of singularly perturbed delay differential equations with layer behavior. Appl. Math. Comput. 157, 11–28 (2004)

    MathSciNet  MATH  Google Scholar 

  6. Kadalbajoo, M.K., Sharma, K.K.: A numerical method based on finite difference for boundary value problems for singularly perturbed delay differential equations. Appl. Math. Comput. 197, 692–707 (2008)

    MathSciNet  MATH  Google Scholar 

  7. Kadalbajoo, M.K., Ramesh, V.P.: Hybrid method for numerical solution of singularly perturbed delay differential equations. Appl. Math. Comput. 187, 797–814 (2007)

    MathSciNet  MATH  Google Scholar 

  8. Kadalbajoo, M.K., Kumar, D.: Fitted mesh B-spline collocation method for singularly perturbed differential-difference equations with small delay. Appl. Math. Comput. 204, 90–98 (2008)

    MathSciNet  MATH  Google Scholar 

  9. Rao, R.N., Chakravarthy, P.P.: A finite difference method for singularly perturbed differential-difference equations with layer and oscillatory behavior. Appl. Math. Model. 37, 5743–5755 (2013)

    Article  MathSciNet  Google Scholar 

  10. Mohapatra, J., Natesan, S.: Uniform convergence analysis of finite difference scheme for singularly perturbed delay differential equation on an adaptively generated grid. Numer. Math. Theor. Meth. Appl. 3, 1–22 (2010)

    MathSciNet  MATH  Google Scholar 

  11. Liu, L.-B., Chen, Y.: Maximum norm a posterior error estimations for a singularly perturbed differential-difference equation with small delay. Appl. Math. Comput. 227, 1–10 (2014)

    Article  MathSciNet  Google Scholar 

  12. Baltensperger, R.J., Berrut, P., Noël, B.: Exponential convergence of a linear rational interpolant between transformed Chebyshev points. Math. Comp 68, 1109–1120 (1999)

    Article  MathSciNet  Google Scholar 

  13. Baltensperger, R., Berrut, J.P., Dubey, Y.: The linear rational pseudo-spectral method with preassigned poles. Numer. Alg. 33, 53–63 (2003)

    Article  Google Scholar 

  14. Luo, W.-H., Huang, T.-Z., Gu, X.-M., et al.: Barycentric rational collocation methods for a class of nonlinear parabolic partial differential equations. Appl. Math. Lett. 68, 13–19 (2017)

    Article  MathSciNet  Google Scholar 

  15. Chen, S., Wang, Y.: A rational spectral collocation method for third-order singularly perturbed problems. J. Comput. Appl. Math. 307, 93–105 (2016)

    Article  MathSciNet  Google Scholar 

  16. Wang, Y., Chen, S., Wu, X.: A rational spectral collocation method for solving a class of parameterized singular perturbation problems. J. Comput. Appl. Math. 233, 2652–2660 (2010)

    Article  MathSciNet  Google Scholar 

  17. Raja, M.A.Z.: Stochastic numerical techniques for solving Troesch’s problem. Inform. Sci. 279, 860–873 (2014)

    Article  MathSciNet  Google Scholar 

  18. Raja, M.A.Z.: Numerical treatment for boundary value problems of pantograph functional differential equation using computational intelligence algorithms. Appl. Soft Comput. 24, 806–821 (2014)

    Article  Google Scholar 

  19. Raja, M.A.Z.: Numerical treatment of nonlinear MHD Jeffery-Hamel problems using Stochastic algorithms. Comput. Fluids 91, 28–46 (2014)

    Article  MathSciNet  Google Scholar 

  20. Sobester, A., Nair, P.B., Keane, A.J.: Genetic programming approaches for solving elliptic partial differential equations. IEEE. Trans. Evolut. Comput. 12, 469–478 (2008)

    Article  Google Scholar 

  21. Fateh, M.F., Zameer, A., Mirza, N.M., et al.: Biologically inspired computing framework for solving two-point boundary value problems using differential evolution. Neural Comput. Appl. 28, 1–15 (2016)

    Article  Google Scholar 

  22. Luo, X.-Q., Liu, L.-B., Ouyang, A., et al.: B-spline collocation and self-adapting differential evolution (jDE) algorithm for a singularly perturbed convection-diffusion problem. Soft. Comput. 22, 2683–2693 (2018)

    Article  Google Scholar 

  23. Liu, L.-B., Long, G., Ouyang, A., et al.: Numerical solution of a singularly perturbed problem with Robin boundary conditions using particle swarm optimization algorithm. J. Intell. Fuzzy Syst. 33, 1785–1795 (2017)

    Article  Google Scholar 

  24. Liu, L.-B., Long, G., Huang, Z., et al.: Rational spectral collocation and differential evolution algorithms for singularly perturbed problems with an interior layer. J. Comput. Appl. Math. 335, 312–322 (2018)

    Article  MathSciNet  Google Scholar 

  25. Storn, R., Price, K.: Differential evolution–A simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim. 11, 341–359 (1997)

    Article  MathSciNet  Google Scholar 

  26. Huang, D.S., Du, J.-X.: A constructive hybrid structure optimization methodology for radial basis probabilistic neural networks. IEEE Trans. Neural Netw. 19, 2099–2115 (2008)

    Article  Google Scholar 

  27. Huang, D.S.: Systematic Theory of Networks for Pattern Recognition. Publishing House of Electronic Industry of China, Beijing (1996)

    Google Scholar 

  28. Du, J.-X., Huang, D.S., Zhang, G.-J., et al.: A novel full structure optimization algorithm for radial basis probabilistic neural networks. Neurocomputing 70, 592–596 (2006)

    Article  Google Scholar 

  29. Du, J.-X., Huang, D.S., Wang, X.-F., et al.: Shape recognition based on neural networks trained by differential evolution algorithm. Neurocomputing 70, 896–903 (2007)

    Article  Google Scholar 

  30. Huang, D.S.: Radial basis probabilistic neural networks: model and application. Int. J. Pattern Recogn. 13, 1083–1101 (1999)

    Article  Google Scholar 

  31. Eiben, A.E., Hinterding, R., Michalewicz, Z.: Parameter control in evolutionary algorithms. IEEE. Tran. Evol. Comput. 3, 124–141 (1999)

    Article  Google Scholar 

  32. Eiben, A.E., Smith, J.E.: Introduction to Evolutionary Computing. Natural Computing Series. Springer, Berlin, Germany (2003). https://doi.org/10.1007/978-3-662-05094-1

    Book  MATH  Google Scholar 

  33. Brest, J., Greiner, S., Boskovic, B., et al.: Self-adapting control parameters in differential evolution: a comparative study on numerical benchmark problems. IEEE Trans. Evol. Comput. 10, 646–657 (2006)

    Article  Google Scholar 

  34. Qin, A.K., Huang, V.L., Suganthan, P.N.: Differential evolution algorithm with strategy adaptation for global numerical optimization. IEEE Trans. Evol. Comput. 13, 398–417 (2009)

    Article  Google Scholar 

  35. Ali, M.M., Torn, A.: Population set-based global optimization algorithms: Some modifications and numerical studies. Comput. Oper. Res. 31, 1703–1725 (2004)

    Article  MathSciNet  Google Scholar 

  36. Yang, X.-S.: Flower pollination algorithm for global optimization. In: Durand-Lose, J., Jonoska, N. (eds.) UCNC 2012. LNCS, vol. 7445, pp. 240–249. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32894-7_27

    Chapter  Google Scholar 

  37. Kennedy, J., Eberhart, R.: Particle swarm optimization. In: Proceeding of the IEEE international conference on neural networks, pp. 1942–1948. IEEE press, Piscataway, NJ, USA (1995)

    Google Scholar 

  38. Hansen, N., Muller, S.D., Koumoutsakos, P.: Reducing the time complexity of the derandomized evolution strategy with covariance matrix adaptation (CMA-ES). Evol. Comput. 11, 1–18 (2014)

    Article  Google Scholar 

  39. Nelder, J.A., Mead, R.: A simplex method for function minimization. Comput. J. 7, 308–313 (1965)

    Article  MathSciNet  Google Scholar 

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Acknowledgement

This work is supported by National Science Foundation of China (11761015, 11461011, 11561009), the Natural Science Foundation of Guangxi (2017GXNSFBA198183), the key project of Guangxi Natural Science Foundation (2017GXNSFDA198014), “BAGUI Scholar” Program of Guangxi Zhuang Autonomous Region of China, and Innovation Project of Guangxi Graduate Education (JGY2017086).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Guangxi Teachers Education University, Nanning, 530001, China

    Guangqing Long, Li-Bin Liu & Zaitang Huang

Authors
  1. Guangqing Long
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  2. Li-Bin Liu
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  3. Zaitang Huang
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Corresponding author

Correspondence to Li-Bin Liu .

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Editors and Affiliations

  1. Tongji University, Shanghai, China

    De-Shuang Huang

  2. Indian Institute of Technology Madras, Chennai, India

    M. Michael Gromiha

  3. Inha University, Incheon, Korea (Republic of)

    Kyungsook Han

  4. Liverpool John Moores University, Liverpool, United Kingdom

    Abir Hussain

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Long, G., Liu, LB., Huang, Z. (2018). Numerical Solution of Singularly Perturbed Convection Delay Problems Using Self-adaptive Differential Evolution Algorithm. In: Huang, DS., Gromiha, M., Han, K., Hussain, A. (eds) Intelligent Computing Methodologies. ICIC 2018. Lecture Notes in Computer Science(), vol 10956. Springer, Cham. https://doi.org/10.1007/978-3-319-95957-3_67

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  • DOI: https://doi.org/10.1007/978-3-319-95957-3_67

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-95956-6

  • Online ISBN: 978-3-319-95957-3

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