## Abstract and Applied Analysis

### Numerical Solution of Singularly Perturbed Delay Differential Equations with Layer Behavior

#### Abstract

We present a numerical method to solve boundary value problems (BVPs) for singularly perturbed differential-difference equations with negative shift. In recent papers, the term negative shift has been used for delay. The Bezier curves method can solve boundary value problems for singularly perturbed differential-difference equations. The approximation process is done in two steps. First we divide the time interval, into $k$ subintervals; second we approximate the trajectory and control functions in each subinterval by Bezier curves. We have chosen the Bezier curves as piecewise polynomials of degree $n$ and determined Bezier curves on any subinterval by $n+1$ control points. The proposed method is simple and computationally advantageous. Several numerical examples are solved using the presented method; we compared the computed result with exact solution and plotted the graphs of the solution of the problems.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 731057, 4 pages.

Dates
First available in Project Euclid: 26 March 2014

https://projecteuclid.org/euclid.aaa/1395858527

Digital Object Identifier
doi:10.1155/2014/731057

Mathematical Reviews number (MathSciNet)
MR3166649

Zentralblatt MATH identifier
07022967

#### Citation

Ghomanjani, F.; Kılıçman, A.; Akhavan Ghassabzade, F. Numerical Solution of Singularly Perturbed Delay Differential Equations with Layer Behavior. Abstr. Appl. Anal. 2014 (2014), Article ID 731057, 4 pages. doi:10.1155/2014/731057. https://projecteuclid.org/euclid.aaa/1395858527

#### References

• K. C. Patidar and K. K. Sharma, “$\epsilon$-uniformly convergent non-standard finite difference methods for singularly perturbed differential difference equations with small delay,” Applied Mathematics and Computation, vol. 175, no. 1, pp. 864–890, 2006.
• M. K. Kadalbajoo and K. K. Sharma, “Numerical treatment for singularly perturbed nonlinear differential difference equations with negative shift,” Nonlinear Analysis: Theory, Methods and Applications, vol. 63, no. 5, pp. e1909–e1924, 2005.
• M. K. Kadalbajoo and K. K. Sharma, “Parameter-uniform fitted mesh method for singularly perturbed delay differential equations with layer behavior,” Electronic Transactions on Numerical Analysis, vol. 23, pp. 180–201, 2006.
• P. Rai and K. K. Sharma, “Numerical analysis of singularly perturbed delay differential turning point problem,” Applied Mathematics and Computation, vol. 218, no. 7, pp. 3483–3498, 2011.
• C. G. Lange and R. M. Miura, “Singular perturbation analysis of boundary value problems for differential-difference equations. V: small shifts with layer behavior,” SIAM Journal on Applied Mathematics, vol. 54, no. 1, pp. 249–272, 1994.
• M. C. Mackey and L. Glass, “Oscillations and chaos in physiological control system,” Science, vol. 197, pp. 287–289, 1997.
• A. Longtin and J. G. Milton, “Complex oscillations in the human pupil light reflex with “mixed” and delayed feedback,” Mathematical Biosciences, vol. 90, no. 1-2, pp. 183–199, 1988.
• V. Y. Glizer, “Asymptotic analysis and solution of a finite-horizon ${H}_{\infty }$ control problem for singularly-perturbed linear systems with small state delay,” Journal of Optimization Theory and Applications, vol. 117, no. 2, pp. 295–325, 2003.
• C. G. Lange and R. M. Miura, “Singular perturbation analysis of boundary value problems for differential-difference equations,” SIAM Journal on Applied Mathematics, vol. 42, no. 3, pp. 502–531, 1982.
• C. G. Lange and R. M. Miura, “Singular perturbation analysis of boundary value problems for differential-difference equations. VI: small shifts with rapid oscillations,” SIAM Journal on Applied Mathematics, vol. 54, no. 1, pp. 273–283, 1994.
• M. K. Kadalbajoo and K. K. Sharma, “Numerical analysis of singularly perturbed delay differential equations with layer behavior,” Applied Mathematics and Computation, vol. 157, no. 1, pp. 11–28, 2004.
• M. K. Kadalbajoo and K. K. Sharma, “Numerical treatment of a mathematical model arising from a model of neuronal variability,” Journal of Mathematical Analysis and Applications, vol. 307, no. 2, pp. 606–627, 2005.
• M. K. Kadalbajoo and K. K. Sharma, “A numerical method based on finite difference for boundary value problems for singularly perturbed delay differential equations,” Applied Mathematics and Computation, vol. 197, no. 2, pp. 692–707, 2008.
• P. Rai and K. K. Sharma, “Numerical study of singularly perturbed differential-difference equation arising in the modeling of neuronal variability,” Computers & Mathematics with Applications, vol. 63, no. 1, pp. 118–132, 2012.
• G. M. Amiraliyev and F. Erdogan, “Uniform numerical method for singularly perturbed delay differential equations,” Computers & Mathematics with Applications, vol. 53, no. 8, pp. 1251–1259, 2007.
• I. G. Amiraliyeva and G. M. Amiraliyev, “Uniform difference method for parameterized singularly perturbed delay differential equations,” Numerical Algorithms, vol. 52, no. 4, pp. 509–521, 2009.
• F. Ghomanjani, M. H. Farahi, and M. Gachpazan, “Bézier control points method to solve constrained quadratic optimal control of time varying linear systems,” Computational & Applied Mathematics, vol. 31, no. 3, pp. 433–456, 2012.
• F. Ghomanjani, A. K\il\içman, and S. Effati, “Numerical solution for IVP in Volterra type linear integro-differential equations system,” Abstract and Applied Analysis, vol. 2013, Article ID 490689, 4 pages, 2013. \endinput