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A new numerical method for singularly perturbed turning point problems with two boundary layers based on reproducing kernel method

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Abstract

In this paper, a simple numerical method is proposed for solving singularly perturbed boundary layers problems exhibiting twin boundary layers. The method avoids the choice of fitted meshes. Firstly the original problem is transformed into a new boundary value problem whose solution does not change rapidly by a proper variable transformation; then the transformed problem is solved by using the reproducing kernel method. Two numerical examples are given to show the effectiveness of the present method.

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References

  1. Natesan, S., Jayakumar, J., Vigo-Aguiar, J.: Parameter uniform numerical method for singularly perturbed turning point problems exhibiting boundary layers. J. Comput. Appl. Math. 158, 121–134 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Kadalbajoo, M.K., Arora, P., Gupta, V.: Collocation method using artificial viscosity for solving stiff singularly perturbed turning point problem having twin boundary layers. Comput. Math. Appl. 61, 1595–1607 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Rai, P., Sharma, K.K.: Nnmerical method for singularly perturbed differential-difference equation with turning point. Int. J. Pure Appl. Math. 73, 451–470 (2011)

    MathSciNet  MATH  Google Scholar 

  4. Rai, P., Sharma, K.K.: Numerical study of singularly perturbed differential-difference equation arising in the modeling of neuronal variability. Comput. Math. Appl. 63, 118–132 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Rai, P., Sharma, K.K.: Numerical analysis of singularly perturbed delay differential turning point problem. Appl. Math. Comput. 218, 3483–3498 (2011)

    MathSciNet  MATH  Google Scholar 

  6. Phaneendra, K., Rakmaiah, S., Chenna Krishna Reddy, M.: Numerical treatment of singular perturbation problems exhibiting dual boundary layers. Ain Shams Eng. J. (2015). doi:10.1016/j.asej.2015.02.012

  7. Becher, S., Roos, H.-G.: Richardson extrapolation for a singularly perturbed turning point problem with exponential boundary layers. J. Comput. Appl. Math. 290, 334–351 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Geng, F.Z., Qian, S.P.: Reproducing kernel method for singularly perturbed turning point problems having twin boundary layers. Appl. Math. Lett. 26, 998–1004 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337–404 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  10. Geng, F.Z., Cui, M.G.: Solving a nonlinear system of second order boundary value problems. J. Math. Anal. Appl. 327, 1167–1181 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cui, M.G., Geng, F.Z.: Solving singular two-point boundary value problem in reproducing kernel space. J. Comput. Appl. Math. 205, 6–15 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cui, M.G., Lin, Y.Z., et al.: Nonlinear Numerical Analysis in Reproducing Kernel Space. Nova Science Publishers, Inc., Commack (2009)

    MATH  Google Scholar 

  13. Li, X.Y., Wu, B.Y.: Error estimation for the reproducing kernel method to solve linear boundary value problems. J. Comput. Appl. Math. 243, 10–15 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, X.Y., Wu, B.Y.: A continuous method for nonlocal functional differential equations with delayed or advanced arguments. J. Math. Anal. Appl. 409, 485–493 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Wang, W.Y., Yamamoto, M., Han, B.: Numerical method in reproducing kernel space for an inverse source problem for the fractional diffusion equation. Inverse Probl. 29, 1–15 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Wang, W.Y., Han, B., Yamamoto, M.: Inverse heat problem of determining time-dependent source parameter in reproducing kernel space. Nonlinear Anal. Real World Appl. 14, 875–887 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Wang, Y., Su, L., Cao, X., Li, X.: Using reproducing kernel for solving a class of singularly perturbed problems. Comput. Math. Appl. 61, 421–430 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jiang, W., Lin, Y.Z.: Representation of exact solution for the time-fractional telegraph equation in the reproducing kernel space. Commun. Nonlinear Sci. Numer. Simulat. 16, 3639–3645 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Inc, M., Akgül, A.: Numerical solution of seventh-order boundary value problems by a novel method. Abstr. Appl. Anal. 2014, 1–9 (2014)

    Article  MathSciNet  Google Scholar 

  20. Mohammadi, M., Mokhtari, R.: Solving the generalized regularized long wave equation on the basis of a reproducing kernel space. J. Comput. Appl. Math. 235, 4003–4014 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  21. Akram, G., Ur Rehman, H.: Numerical solution of eighth order boundary value problems in reproducing Kernel space. Numer. Algorithms 62, 527–540 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Arqub, O.A., Al-Smadi, M., Shawagfeh, N.: Solving Fredholm integro-differentialequations using reproducing kernel Hilbert space method. Appl. Math. Comput. 219, 8938–8948 (2013)

    MathSciNet  MATH  Google Scholar 

  23. Abu, O. A.: Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm–Volterra integrodifferential equations. Neural Comput. Appl. (2015). doi:10.1007/s00521-0152110-x

  24. Arqub, O.A., Mohammed, A.S., Momani, S., Hayat, T.: Numerical solutions of fuzzy differential equations using reproducing kernel Hilbert space method. Soft Comput. (2015). doi:10.1007/s00500-0151707-4

  25. Momani, S., Arqub, O.A., Hayat, T., Al-Sulami, H.: A computational method for solving periodic boundary value problems for integro-differential equations of Fredholm-Volterra type. Appl. Math. Comput. 240, 229–239 (2014)

    MathSciNet  MATH  Google Scholar 

  26. Abbasbandy, S., Azarnavid, B., Alhuthali, M.S.: A shooting reproducing kernel Hilbert space method for multiple solutions of nonlinear boundary value problems. J. Comput. Appl. Math. 279, 293–305 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ketabchi, R., Mokhtari, R., Babolian, E.: Some error estimates for solving Volterra integral equations by using the reproducing kernel method. J. Comput. Appl. Math. 273, 245–250 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ghasemi, M., Fardi, M., Ghaziani, R.K.: Numerical solution of nonlinear delay differential equations of fractional order in reproducing kernel Hilbert space. Appl. Math. Comput. 268, 815–831 (2015)

    MathSciNet  Google Scholar 

  29. Geng, F.Z., Qian, S.P.: Modified reproducing kernel method for singularly perturbed boundary value problems with a delay. Appl. Math. Model. 39, 5592–5597 (2015)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

The work was supported by the National Natural Science Foundation of China (Nos. 11201041, 11026200), the Special Funds of the National Natural Science Foundation of China (No. 11141003) and Qing Lan Project of Jiangsu Province.

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Authors and Affiliations

  1. Department of Mathematics, Changshu Institute of Technology, Changshu, 215500, Jiangsu, People’s Republic of China

    F. Z. Geng & S. P. Qian

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  1. F. Z. Geng
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  2. S. P. Qian
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Correspondence to F. Z. Geng.

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Geng, F.Z., Qian, S.P. A new numerical method for singularly perturbed turning point problems with two boundary layers based on reproducing kernel method. Calcolo 54, 515–526 (2017). https://doi.org/10.1007/s10092-016-0196-x

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  • DOI: https://doi.org/10.1007/s10092-016-0196-x

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