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An operational Haar wavelet collocation method for solving singularly perturbed boundary-value problems

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Abstract

The basic aim of this study is to introduce and describe a numerical scheme for the approximate solutions of the one-dimensional singularly perturbed boundary-value problems. The method is based on Haar wavelets and its main characteristic is that, it converts the given problem into a system of algebraic equations that can be solved easily with any of the usual methods. Another distinguishing feature of the this method is that unlike several other numerical methods, it does not require conversion of a boundary value problem into initial-value problem and hence eliminates the possibility of unstable solutions. To show the accuracy and the efficiency of the method, several benchmark problems are implemented and the comparisons are given with other methods existing in the recent literature. The results of numerical tests confirm that the Haar wavelet collocation method is superior to other existing ones and is highly accurate.

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References

  1. Aziz, T., Khan, A.: Quintic spline approach to the solution of a singularly perturbed boundary value problem. J. Optim. Theory Appl. 112(3), 517–527 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aziz, T., Khan, A.: A spline method for second-order singularly perturbed boundary-value problems. J. Comput. Appl. Math. 147, 445–452 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, C.F., Hsiao, C.H.: Haar wavelet method for solving lumped and distributed-parameter systems. IEE Proc. Control Theory Appl. 144, 87–94 (1997)

    Article  MATH  Google Scholar 

  4. Debnath, L., Shah, F.A.: Wavelet transforms and their applications. Birkhäuser, New York (2015)

    Book  MATH  Google Scholar 

  5. Geng, F.Z., Cui, M.G.: A novel method for solving a class of singularly perturbed boundary value problems based on reproducing kernel method. Appl. Math. Comput. 218, 4211–4215 (2011)

    MathSciNet  MATH  Google Scholar 

  6. Kadalbajoo, M.K., Awasthi, A.: A parameter uniform difference scheme for singularly perturbed parabolic problem in one space dimension. Appl. Math. Comput. 183, 42–60 (2006)

    MathSciNet  MATH  Google Scholar 

  7. Kadalbajoo, M.K., Bawa, R.K.: Variable mesh difference scheme for singularly perturbed boundary-value problems using splines. J. Optim. Theory Appl. 9, 405–416 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kadalbajoo, M.K., Patidar, K.C.: Spline techniques for the numerical solution of singular perturbation problem. J. Optim. Theory Appl. 112(3), 575–594 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kadalbajoo, M.K., Aggarwal, V.K.: Fitted mesh \(B\)-spline collocation method for solving self adjoint singularly perturbed boundary value problems. Appl. Math. Comput. 161(3), 973–987 (2005)

    MathSciNet  MATH  Google Scholar 

  10. Kadalbajoo, M.K., Gupta, V.: Numerical solution of singularly perturbed convection-diffusion problem using parameter uniform \(B\)-spline collocation method. J. Math. Anal. Appl. 355, 439–452 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kadalbajoo, M.K., Gupta, V., Awasthi, A.: A uniformly convergent \(B\)-spline collocation method on a nonuniform mesh for singularly perturbed one-dimensional time-dependent linear convectiondiffusion problem. J. Comput. Appl. Math. 220, 271–289 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Khan, A., Khan, I., Aziz, T.: Sextic spline solution of singularly perturbed boundary-value problems. Appl. Math. Comput. 181, 432–439 (2006)

    MathSciNet  MATH  Google Scholar 

  13. Khan, A., Khandelwala, P.: Non-polynomial sextic spline solution of singularly perturbed boundary-value problems. Int. J. Comput. Math. 91(5), 1122–1135 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kumar, V., Mehra, M.: Cubic spline adaptive wavelet scheme to solve singularly perturbed reaction diffusion problems. Int. J. Wavelets Multiresoult. Inf. Process 5(2), 317–331 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kumar, M., Singh, P., Mishra, H.K.: An initial-value technique for singularly perturbed boundary value problems via cubic spline. Int. J. Comput. Methods Eng. Sci. Mech. 8:419–427 (2007)

  16. Lenferink, W.: A second order scheme for a time-dependent, singularly perturbed convection-diffusion equation. J. Comput. Appl. Math. 143, 49–68 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lepik, U., Hein, H.: Haar wavelets with applications. Springer, New York (2014)

    Book  MATH  Google Scholar 

  18. Lepik, U.: Solving integral and differential equations by the aid of nonuniform Haar wavelets. Appl. Math. Comput. 198, 326–332 (2008)

    MathSciNet  MATH  Google Scholar 

  19. Lubuma, J.S., Patidar, K.C.: Uniformly convergent non-standard finite difference methods for self-adjoint singular perturbation problems. J. Comput. Appl. Math. 191, 228–238 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Mallat, S.G.: Multiresolution approximations and wavelet orthonormal bases of \(L^2(\mathbb{R})\). Trans. Am. Math. Soc. 315, 69–87 (1989)

    MathSciNet  MATH  Google Scholar 

  21. Miller, J.H., ORiordan, E., Shishkin, G.I.: Fitted numerical methods for singular perturbation problems. World Scientific, Singapore (1996)

  22. 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 

  23. Phongthanapanich, S., Dechaumphai, P.: Combined finite volume element method for singularly perturbed reaction-diffusion problems. Appl. Math. Comput. 209, 177–185 (2009)

    MathSciNet  MATH  Google Scholar 

  24. Rashidinia, J., Ghasemi, M., Mahmoodi, Z.: Spline approach to the solution of a singularly-perturbed boundary-value problems. Appl. Math. Comput. 189, 72–78 (2007)

    MathSciNet  MATH  Google Scholar 

  25. Roos, H.G., Stynes, M., Tobiska, L.: Numerical methods for singularly perturbed differential equations. Springer, New York (1996)

    Book  MATH  Google Scholar 

  26. Shah, F.A., Abass, R., Iqbal, J.: Numerical solution of singularly perturbed problems using Haar wavelet collocation method. Cogent Math. 3, 1202504 (2016)

    Article  MathSciNet  Google Scholar 

  27. Surla, K., Herceg, D., Cvetkovic, L.: A family of exponential spline difference schemes. Rev. Res. Fac. Sci. Math. Ser. Univ. Novi. Sad. 20, 17–26 (1990)

    MathSciNet  MATH  Google Scholar 

  28. Surla, K., Stojanovic, M.: Solving singularly-perturbed boundary-value problems by splines in tension. J. Comput. Appl. Math. 24, 355–363 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  29. Surla, K., Vukoslavcevic, V.: A spline difference scheme for boundary-value problems with a small parameter. Rev. Res. Fac. Sci. Math. Ser. Univ. Novi. Sad. 25, 159–166 (1995)

    MathSciNet  MATH  Google Scholar 

  30. Wazwaz, A.M.: A new method for solving singular initial value problems in the second-order ordinary differential equations. Appl. Math. Comput. 128, 45–57 (2002)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Kashmir, South Campus, Anantnag, 192101, Jammu and Kashmir, India

    Firdous A. Shah

  2. Department of Mathematical Sciences, BGSB University, Rajouri, 185234, Jammu and Kashmir, India

    R. Abass

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  1. Firdous A. Shah
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  2. R. Abass
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Correspondence to Firdous A. Shah.

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Shah, F.A., Abass, R. An operational Haar wavelet collocation method for solving singularly perturbed boundary-value problems. SeMA 74, 457–474 (2017). https://doi.org/10.1007/s40324-016-0094-9

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  • DOI: https://doi.org/10.1007/s40324-016-0094-9

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