A Novel Hybrid Method for Singularly Perturbed Delay Differential Equations

A Novel Hybrid Method for Singularly Perturbed Delay Differential Equations

The aim of the present study is to solve singularly perturbed second order linear delay differentialequations by combining the flexibility of differential transform method and the efficiency ofTaylor series expansion method. For this purpose, we use two-term Taylor series expansionmethod for delayed parameter linearization and then apply the differential transform method. Twoexamples are presented to demonstrate the efficiency, rapidity and reliability of the proposedhybrid method.

___

  • Zhou, J.K., Differential Transform and Its Application for Electrical Circuits, Huazhong University Press, China, (1986).
  • Ertürk, V.S., “Differential transform method for solving differential equations of Lane-Emden type”, Mathematical and Computational Applications, 12(3): 135-139, (2007).
  • Ayaz, F., “Solution of the system of differential equations by differential transform method”, Applied Mathematics and Computation, (147): 547-567, (2004).
  • Ayaz, F., “Applications of differential transform method to differential algebraic equations”, Applied Mathematics and Computation, (152): 649-657, (2004).
  • Nayfeh, A.H., Perturbation Methods, John Wiley&Sons, (2008).
  • Gemechis, F., Reddy, Y.N., “Terminal boundary-value technique for solving singularly perturbed delay differential equations”, Journal of Taibah University for Sciences, 8(3): 289–300, (2014).
  • Pinney, E., Ordinary Difference-Differential Equations, University of California Press, (1958).
  • Bellman, R.E., Cooke, K.L., Differential-Difference Equations, Academic Press, New York, USA, (1963).
  • Kadalbajoo, M.K., Sharma, K.K., “A numerical method based on finite difference for boundary value problems for singularly perturbed delay differential equations,” Applied Mathematics and Computation, 197(2): 692–707, (2008).
  • Nageshwar Rao, R., Pramod Chakravarthy, P., “A finite difference method for singularly perturbed differential-difference equations with layer and oscillatory behavior,” Applied Mathematical Modelling, 37(8): 5743–5755, (2013).
  • Nicaise, S., Xenophontos, C., “Robust approximation of singularly perturbed delay differential equations by the hp finite element method”, Computational Methods in Applied Mathematics, 13(1): 21–37, (2013).
  • Zarin, H., “On discontinuous Galerkin finite element method for singularly perturbed delay differential equations”, Applied Mathematics Letters. An International Journal of Rapid Publication, (38): 27–32, (2014).
  • Shakeri, F., Dehghan, M., “Solution of delay differential equations via a homotopy perturbation method ”, Mathematical and Computer Modelling, 48(3-4): 486–498, (2008).
  • Akgül, A., Kiliman, A., “Solving delay differential equations by an accurate method with interpolation”, Abstract and Applied Analysis, 2015: 7 pages, (2015).
  • Geng, F.Z., Qian, S.P., Cui, M.G., “Improved reproducing kernel method for singularly perturbed differential-difference equations with boundary layer behavior”, Applied Mathematics and Computation, (252): 58–63, (2015).
  • El- Hawary, H.M., Mahmoud, S.M., “Spline collocation methods for solving delay-differential equations”, Applied Mathematics and Computation, 146(2-3): 359–372, (2003).
  • Driver, R.D., Ordinary and Delay Differential Equations, Applied Mathematical Sciences, vol. 20, Springer, New York, (1977).
  • Cengizci, S., “An asymptotic-numerical hybrid method for solving singularly perturbed linear delay differential equations”, International Journal of Differential Equations, vol. 2017, 8 pages, (2017).
  • Reddy, Y.N., Soujanya, GBSL., Phaneendra, K., “Numerical integration method for singularly perturbed delay differential equations”, International Journal of Applied Science and Engineering, 10, 3:249-261, (2012).