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Singularly Perturbed Convection-Diffusion Turning Point Problem with Shifts

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Mathematical Analysis and its Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 143))

Abstract

In this paper, a class of singularly perturbed turning point problem with shifts (i.e., delay as well as advance) is considered. Presence of turning point results into twin boundary layers in the solution of the problem under consideration. For the numerical approximation of the problem, a finite difference scheme is proposed on a uniform mesh. Interpolation is used to tackle the terms containing shifts and to deal with the difficulty arising due to presence of the turning point a combination of backward and forward difference is used in the first derivative term. Convergence analysis is given for the proposed numerical scheme. Numerical results are presented which illustrate the theoretical results and depict the effect of shifts on the layer behavior of the solution.

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References

  1. Amiraliyev, G.M., Cimen, E.: Numerical method for a singularly perturbed convection-diffusion problem with delay. Appl. Math. Comput. 216, 2351–2359 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bellman, R., Cooke, K.L.: Differential Difference Equations. Academic Press, New York (1991)

    Google Scholar 

  3. Cryer, C.W.: Numerical methods for functional differential equations, in delay and functional differential equations. In: Schmitt, K. (ed.) Proceeding of Conference held at Park city, Utah, 1972

    Google Scholar 

  4. E’lsgolts, L.E.: Qualitative Methods in Mathematical Analysis, Translations of Mathematical Monographs, vol. 12. American Mathematical Society, Providence (1964)

    Google Scholar 

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

    Article  Google Scholar 

  6. Henderson, J.: Boundary Value Problems for Functional Differential Equations. World Scientific (1995)

    Google Scholar 

  7. Lange, C.G., Miura, R.M.: Singular perturbation analysis of boundary value problems for differential-difference equations. SIAM J. Appl. Math. 42, 502–531 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  8. Lange, C.G., Miura, R.M.: Singular perturbation analysis of boundary value problems for differential-difference equations II. Rapid oscillations and resonance. SIAM J. Appl. Math. 45, 687–707 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  9. 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  MATH  Google Scholar 

  10. Lee, J.W., O’Regan, D.: Existence results for differential delay equations-II. Non-linear Anal. 17, 683–702 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. Lee, J.W., O’Regan, D.: Existence results for differential delay equations-I. J. Diff. Eqn. 102, 342–359 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ntouyas, S.K., Sficas, Y.G., Tsamatos, P.: An existence principle for boundary value problems for second order functional differential equations. Nonlinear Anal. 20, 215–222 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  13. Norkin, S.B.: Differential Equations of the Second Order with Retarded Arguments, Translations of Mathematical Monographs, vol. 31. American Mathematical Society, Providence (1972)

    Google Scholar 

  14. Ramos, J.I.: Exponential methods for singularly perturbed ordinary differential-difference equations. Appl. Math. Comput. 182, 1528–1541 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. 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)

    Google Scholar 

  16. Kadalbajoo, M.K., Sharma, K.K.: Parameter- uniform fitted mesh method for singularly perturbed delay differential equations with layer behaviour. Elect. Trans. Numer. Anal. 23, 180–201 (2006)

    MathSciNet  MATH  Google Scholar 

  17. 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)

    Article  MathSciNet  MATH  Google Scholar 

  18. Stein, R.B.: A theoretical analysis of neuronal variability. Biophys. J. 5, 173–194 (1965)

    Article  Google Scholar 

  19. Tuckwell, H.C.: On the first exit time problem for temporally homogeneous Markov process. J. Appl. Probab. 13, 39–46 (1976)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The first author wish to acknowledge Delhi University, India, 110007 for providing financial support for the above work.

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Authors and Affiliations

  1. Department of Mathematics, University of Delhi, Delhi, 110007, India

    Pratima Rai

  2. Department of Mathematics, South Asian University (SAU), Akbar Bhawan, Chanakyapuri, 110021, New Delhi, India

    Kapil K. Sharma

Authors
  1. Pratima Rai
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  2. Kapil K. Sharma
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Corresponding author

Correspondence to Kapil K. Sharma .

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Editors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, India

    P. N. Agrawal

  2. Department of Mathematics, University of Central Florida, Orlando, Florida, USA

    R. N. Mohapatra

  3. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, India

    Uaday Singh

  4. Dept. of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada

    H. M. Srivastava

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Rai, P., Sharma, K.K. (2015). Singularly Perturbed Convection-Diffusion Turning Point Problem with Shifts. In: Agrawal, P., Mohapatra, R., Singh, U., Srivastava, H. (eds) Mathematical Analysis and its Applications. Springer Proceedings in Mathematics & Statistics, vol 143. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2485-3_31

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