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Year 2019, Volume: 11 Issue: 4, 476 - 493, 05.12.2019
https://doi.org/10.24107/ijeas.647640

Abstract

References

  • Derstine M. W., Gibbs F.A.H.H.M. and Kaplan D. L., Bifurcation gap in a hybrid optical system, Phys. Rev. A, 26, 3720-3722, 1982
  • Longtin A. and J. Milton J., Complex oscillations in the human pupil light reflex with mixed and delayed feedback, Math. Biosci, 90, 183-199, 1988.
  • Mackey M.C. and Glass L., Oscillation and chaos in physiological control systems, Science. 197, 287-289, 1977.
  • Mallet-Paret J. and Nussbaum R. D., A differential-delay equations arising in optics and physiology, SIAMJ. Math. Anal. 20, 249-292, 1989.
  • Lange C.G. and Miura R.M., Singular perturbation analysis of boundary value problems for differential difference equations, SIAM J. Appl. Math. Vol. 42(3), 502-531, 1982.
  • Kadalbajoo M.K and Sharma K.K., An exponentially fitted finite difference scheme for solving boundary value problems for singularly-perturbed differential-difference equations: small shifts of mixed type with layer behaviour, J. Comput. Anal. Appl. Vol. 8(2), 151-171, 2006.
  • Patidar K. C. and Sharma K.K., ε-uniformly convergent non-standard finite difference methods for singularly perturbed differential difference equations with small delay, Appl. Math. Comput. Vol. 175(1), 864-890, 2006.
  • Kadalbajoo M.K. and Sharma K.K., An ε-uniform convergent method for a general boundary-value problem for singularly perturbed differential-difference equations: Small shifts of mixed type with layer behaviour, J. Comput. Methods Sci. Eng, 6, 39-55, 2006.
  • Kadalbajoo M.K. and Sharma K.K., Numerical treatment of boundary value problems for second order singularly perturbed delay differential equations, Comput. Appl. Math, vol. 24(2), 151-172, 2005.
  • Kadalbajoo M.K. and Sharma K.K., Numerical treatment for singularly perturbed nonlinear differential difference equations with negative shift, Nonlinear Anal. Theory Methods Appl, vol. 63(5), e1909-e1909, 2005.
  • Kadalbajoo M.K. and Sharma K.K., Numerical analysis of singularly perturbed delay differential equations with layer behaviour, Appl. Math. Comput, vol. 157(1), 11-28, 2004.
  • Kadalbajoo M.K. and Sharma K.K., Numerical analysis of boundary-value problems for singularly perturbed differential-difference equations with small shifts of mixed type, J. Optim. Theory Appl, vol.115(1), 145-163, 2002.
  • Hongijiong T., Numerical methods for singularly perturbed delay differential equations, In proceeding of the International conference on Boundary and Interior Layers, Computational and Asymptotic Methods-BAIL 2004, ONERA, Toulouse.
  • Amiraliyev G. M and Cimen E., Numerical method for a singularly perturbed convection-diffusion problem with delay, Appl. Math. Comput. 216, 2351-2359, 2010.
  • Subburayan V. and Ramanujam N., An initial value technique for singularly perturbed convection-diffusion problems with a negative shift, J. Optim Theory Appl. 158, 234-250, 1992.
  • Sakar E. and Tamilselvan A., Singularly perturbed delay differential equations of convection–diffusion type with integral boundary condition, J. Appl. Math. Comput, 2018, https://doi.org/10.1007/s12190-018-1198-4.
  • Clavero C., Gracia J.L. and Jorge J.C., High-order numerical methods for one dimensional parabolic singularly perturbed problems with regular layers, Numer. Methods Partial differential equations, vol. 21(1), 149-169, 2005.
  • OMalley R.E., Singular Perturbation Methods for Ordinary Differential Equations. Springer-Verlag,New York, (1991).
  • Woldaregay M.M. and Duressa G.F., parameter uniform numerical method for singularly perturbed differential difference equations, J. Nigerian Math. Soc. Vol.38(2), 223-245, 2009.

Exponentially fitted finite difference method for singularly perturbed delay differential equations with integral boundary condition

Year 2019, Volume: 11 Issue: 4, 476 - 493, 05.12.2019
https://doi.org/10.24107/ijeas.647640

Abstract

In this paper, exponentially fitted finite difference method for
solving singularly perturbed delay differential equation with integral boundary
condition is considered. To treat the integral boundary condition, Simpson’s
rule is applied. The stability and parameter uniform convergence of the
proposed method are proved. To validate
the
applicability of the scheme, two model problems are considered for
numerical experimentation and solved for different values of the perturbation
parameter,  and mesh size,  The numerical
results are tabulated in terms of maximum absolute errors and rate of
convergence and it is observed that the present method is more accurate and -uniformly convergent for  where the
classical numerical methods fails to give good result and it also improves the
results of the methods existing in the literature.

References

  • Derstine M. W., Gibbs F.A.H.H.M. and Kaplan D. L., Bifurcation gap in a hybrid optical system, Phys. Rev. A, 26, 3720-3722, 1982
  • Longtin A. and J. Milton J., Complex oscillations in the human pupil light reflex with mixed and delayed feedback, Math. Biosci, 90, 183-199, 1988.
  • Mackey M.C. and Glass L., Oscillation and chaos in physiological control systems, Science. 197, 287-289, 1977.
  • Mallet-Paret J. and Nussbaum R. D., A differential-delay equations arising in optics and physiology, SIAMJ. Math. Anal. 20, 249-292, 1989.
  • Lange C.G. and Miura R.M., Singular perturbation analysis of boundary value problems for differential difference equations, SIAM J. Appl. Math. Vol. 42(3), 502-531, 1982.
  • Kadalbajoo M.K and Sharma K.K., An exponentially fitted finite difference scheme for solving boundary value problems for singularly-perturbed differential-difference equations: small shifts of mixed type with layer behaviour, J. Comput. Anal. Appl. Vol. 8(2), 151-171, 2006.
  • Patidar K. C. and Sharma K.K., ε-uniformly convergent non-standard finite difference methods for singularly perturbed differential difference equations with small delay, Appl. Math. Comput. Vol. 175(1), 864-890, 2006.
  • Kadalbajoo M.K. and Sharma K.K., An ε-uniform convergent method for a general boundary-value problem for singularly perturbed differential-difference equations: Small shifts of mixed type with layer behaviour, J. Comput. Methods Sci. Eng, 6, 39-55, 2006.
  • Kadalbajoo M.K. and Sharma K.K., Numerical treatment of boundary value problems for second order singularly perturbed delay differential equations, Comput. Appl. Math, vol. 24(2), 151-172, 2005.
  • Kadalbajoo M.K. and Sharma K.K., Numerical treatment for singularly perturbed nonlinear differential difference equations with negative shift, Nonlinear Anal. Theory Methods Appl, vol. 63(5), e1909-e1909, 2005.
  • Kadalbajoo M.K. and Sharma K.K., Numerical analysis of singularly perturbed delay differential equations with layer behaviour, Appl. Math. Comput, vol. 157(1), 11-28, 2004.
  • Kadalbajoo M.K. and Sharma K.K., Numerical analysis of boundary-value problems for singularly perturbed differential-difference equations with small shifts of mixed type, J. Optim. Theory Appl, vol.115(1), 145-163, 2002.
  • Hongijiong T., Numerical methods for singularly perturbed delay differential equations, In proceeding of the International conference on Boundary and Interior Layers, Computational and Asymptotic Methods-BAIL 2004, ONERA, Toulouse.
  • Amiraliyev G. M and Cimen E., Numerical method for a singularly perturbed convection-diffusion problem with delay, Appl. Math. Comput. 216, 2351-2359, 2010.
  • Subburayan V. and Ramanujam N., An initial value technique for singularly perturbed convection-diffusion problems with a negative shift, J. Optim Theory Appl. 158, 234-250, 1992.
  • Sakar E. and Tamilselvan A., Singularly perturbed delay differential equations of convection–diffusion type with integral boundary condition, J. Appl. Math. Comput, 2018, https://doi.org/10.1007/s12190-018-1198-4.
  • Clavero C., Gracia J.L. and Jorge J.C., High-order numerical methods for one dimensional parabolic singularly perturbed problems with regular layers, Numer. Methods Partial differential equations, vol. 21(1), 149-169, 2005.
  • OMalley R.E., Singular Perturbation Methods for Ordinary Differential Equations. Springer-Verlag,New York, (1991).
  • Woldaregay M.M. and Duressa G.F., parameter uniform numerical method for singularly perturbed differential difference equations, J. Nigerian Math. Soc. Vol.38(2), 223-245, 2009.
There are 19 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Habtamu Garoma Debela 0000-0003-0109-6860

Gemechis File Duressa 0000-0003-1889-4690

Publication Date December 5, 2019
Acceptance Date December 5, 2019
Published in Issue Year 2019 Volume: 11 Issue: 4

Cite

APA Debela, H. G., & Duressa, G. F. (2019). Exponentially fitted finite difference method for singularly perturbed delay differential equations with integral boundary condition. International Journal of Engineering and Applied Sciences, 11(4), 476-493. https://doi.org/10.24107/ijeas.647640
AMA Debela HG, Duressa GF. Exponentially fitted finite difference method for singularly perturbed delay differential equations with integral boundary condition. IJEAS. December 2019;11(4):476-493. doi:10.24107/ijeas.647640
Chicago Debela, Habtamu Garoma, and Gemechis File Duressa. “Exponentially Fitted Finite Difference Method for Singularly Perturbed Delay Differential Equations With Integral Boundary Condition”. International Journal of Engineering and Applied Sciences 11, no. 4 (December 2019): 476-93. https://doi.org/10.24107/ijeas.647640.
EndNote Debela HG, Duressa GF (December 1, 2019) Exponentially fitted finite difference method for singularly perturbed delay differential equations with integral boundary condition. International Journal of Engineering and Applied Sciences 11 4 476–493.
IEEE H. G. Debela and G. F. Duressa, “Exponentially fitted finite difference method for singularly perturbed delay differential equations with integral boundary condition”, IJEAS, vol. 11, no. 4, pp. 476–493, 2019, doi: 10.24107/ijeas.647640.
ISNAD Debela, Habtamu Garoma - Duressa, Gemechis File. “Exponentially Fitted Finite Difference Method for Singularly Perturbed Delay Differential Equations With Integral Boundary Condition”. International Journal of Engineering and Applied Sciences 11/4 (December 2019), 476-493. https://doi.org/10.24107/ijeas.647640.
JAMA Debela HG, Duressa GF. Exponentially fitted finite difference method for singularly perturbed delay differential equations with integral boundary condition. IJEAS. 2019;11:476–493.
MLA Debela, Habtamu Garoma and Gemechis File Duressa. “Exponentially Fitted Finite Difference Method for Singularly Perturbed Delay Differential Equations With Integral Boundary Condition”. International Journal of Engineering and Applied Sciences, vol. 11, no. 4, 2019, pp. 476-93, doi:10.24107/ijeas.647640.
Vancouver Debela HG, Duressa GF. Exponentially fitted finite difference method for singularly perturbed delay differential equations with integral boundary condition. IJEAS. 2019;11(4):476-93.

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