Numerical Approach for Differential-Difference Equations with Layer Behaviour

G. Sangeetha, G. Mahesh, K. Phaneendra


A numerical scheme is proposed using a non polynomial spline to solve the differential-difference equations having layer behaviour, with delay as well advanced terms. The retarded terms are handled by using Taylor’s series, subsequently the given problem is substituted by an equivalent second order singular perturbation problem. A finite difference scheme using non polynomial spline is derived and it is applied to the singular perturbation problem using non standard differences of the first derivatives. Tridiagonal algorithm is used to solve the resulting system. The method is exemplified on numerical examples with various values of perturbation, delay and advance parameters. Maximum absolute errors are computed and tabulated to support the method. Numerical solutions are pictured in graphs and the effects of small shifts on the boundary layer region has been investigated. Also, the convergence of the proposed method has also been established.


Differential-difference equations; Boundary Layer; Non polynomial spline; Maximum absolute error

Full Text:



R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York (1963), DOI: 10.1002/zamm.19650450612.

M. Bestehornand and E. V. Grigorieva, Formation and propagation of localized states in extended systems, Ann. Phys. 13 (2004), 423 – 431, DOI: 10.1002/andp.200410085.

M. W. Derstine, H. M. Gibbs, F. A. Hopf and D. L. Kaplan, Bifurcation gap in a hybrid optical system, Phys. Rev. A 26 (1982), 3720 – 3722, DOI: 10.1103/PhysRevA.26.3720.

D. K. Swamy, K. Phaneendra and Y. N. Reddy, Accurate numerical method for singularly perturbed differential-difference equations with mixed shifts, Khayyam J. Math. 4 (2018), 110 – 122, DOI: 10.22034/kjm.2018.57949.

E. P. Doolan, J. J. H. Miller and W. H. A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, Dublin (1980).

R. D. Driver, Ordinary and Delay Differential Equations, Springer, New York (1977), DOI: 10.1007/978-1-4684-9467-9.

L. E. El’sgol’ts and S. B. Norkin, Introduction to the Theory and Application of Differential Equations With Deviating Arguments, Mathematics in Science and Engineering, Academic Press (1973).

M. K. Kadalbajoo and K. K. Sharma, Numerical treatment of mathematical model arising from a model of neuronal variability, Journal of Mathematical Analysis and Applications 307 (2005), 606 – 627, DOI: 10.1016/j.jmaa.2005.02.014.

P. V. Kokotovic, H. K. Khalil and J. O’Reilly, Singular Perturbation Methods in Control Analysis and Design, Academic Press, New York (1986).

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York (1993).

C. G. Lange and R. M. Miura, Singular perturbation analysis of boundary-value problems for differential-difference equations V Small shifts with layer behaviour, SIAM. J. Appl. Math. 54 (1) (1994), 249 – 272, DOI: 10.1137/S0036139992228120.

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. J. Appl. Math. 54(1) (1994), 273 – 283, DOI: 10.1137/S0036139992228119.

A. Lasota and M. Wazewska, Mathematical models of the red blood cell system, Mat. Stos. 6 (1976), 25 – 40.

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science 197 (1977), 287 – 289, DOI: 10.1126/science.267326.

A. Martin and S. Raun, Predetor-prey models with delay and prey harvesting, J. Math. Bio. 43(3) (2001), 247 – 267, DOI: 10.1007/s002850100095.

J. J. H. Miller, R. E. O’Riordan and G.I. Shishkin, Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore (1996), DOI: 10.1142/8410.

H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, Berlin (2010), DOI: 10.1007/978-1-4419-7646-8.

R. B. Stein, Some models of neuronal variability, Biophys. J. 7(1) (1967), 37-68. DOI: 10.1016/S0006-3495(67)86574-3.



  • There are currently no refbacks.

eISSN 0975-8607; pISSN 0976-5905