# Laguerre Matrix-Collocation Method to Solve Systems of Pantograph Type Delay Differential Equations

## Abstract

In this study, an improved matrix method based on collocation points is developed to obtain the approximate solutions of systems of high-order pantograph type delay differential equations with variable coefficients. These kinds of systems described by the existence of linear functional argument play a critical role in defining many different phenomena and particularly, arise in industrial applications and in studies based on biology, economy, electrodynamics, physics and chemistry. The technique we have used reduces the mentioned delay system solution with the initial conditions to the solution of a matrix equation with the unknown Laguerre coefficients. Thereby, the approximate solution is obtained in terms of Laguerre polynomials. In addition, several examples along with error analysis are given to illustrate the efficiency of the method; the obtained results are scrutinized and interpreted.

## Keywords

Laguerre polynomials and series Matrix method Pantograph equations System of delay differential equations Collocation method## References

- 1.Murray, J.D.: Mathematical Biology. Interdisciplinary Applied Mathematics (2003)Google Scholar
- 2.Li, J., Kuang, Y., Mason, C.C.: Modeling the glucose-insulin regulatory system and ultradian insulin secretory oscillations with two explicit time delays. J. Theor. Biol.
**242**(3), 722–735 (2006)MathSciNetCrossRefGoogle Scholar - 3.Sturis, J., Polonsky, K.S., Mosekilde, E., Van Cauter, E.: Computer model for mechanisms underlying ultradian oscillations of insulin and glucose. Am. J. Physiol. Endocrinol. Metab.
**260**(5), E801–E809 (1991)CrossRefGoogle Scholar - 4.Tolić, I.M., Mosekilde, E., Sturis, J.: Modeling the insulin-glucose feedback system: the significance of pulsatile insulin secretion. J. Theor. Biol.
**207**(3), 361–375 (2000)CrossRefGoogle Scholar - 5.Abdel-Halim Hassan, I.H.: Application to differential transformation method for solving systems of differential equations. Appl. Math. Model.
**32**(12), 2552–2559 (2008)MathSciNetCrossRefGoogle Scholar - 6.Yüzbaşı, Ş.: On the solutions of a system of linear retarded and advanced differential equations by the Bessel collocation approximation. Comput. Math Appl.
**63**(10), 1442–1455 (2012)MathSciNetCrossRefGoogle Scholar - 7.Gökmen, E., Sezer, M.: Taylor collocation method for systems of high-order linear differential-difference equations with variable coefficients. Ain Shams Eng. J.
**4**, 117–125 (2013)CrossRefGoogle Scholar - 8.Akyüz, A., Sezer, M.: Chebyshev polynomial solutions of systems of high-order linear differential equations with variable coefficients. Appl. Math. Comput.
**144**(2–3), 237–247 (2003)MathSciNetzbMATHGoogle Scholar - 9.Oğuz, C., Sezer, M., Oğuz, A.D.: Chelyshkov collocation approach to solve the systems of linear functional differential equations. NTMSCI
**3**(4), 83–97 (2015)MathSciNetzbMATHGoogle Scholar - 10.Bhrawy, A.H., Doha, E.H., Baleanu, D., Hafez, R.M.: A highly accurate Jacobi collocation algorithm for systems of high-order linear differential-difference equations with mixed initial conditions. Math. Methods Appl. Sci.
**38**(14), 3022–3032 (2015)MathSciNetCrossRefGoogle Scholar - 11.Ali, I., Brunner, H., Tang, T.: A spectral method for pantograph-type delay differential equations and its convergence analysis. J. Comput. Math.
**27**, 254–265 (2009)MathSciNetzbMATHGoogle Scholar - 12.Biazar, J., Babolian, E., Islam, R.: Solution of the system of ordinary differential equations by Adomian decomposition method. Appl. Math. Comput.
**147**(3), 713–719 (2004)MathSciNetzbMATHGoogle Scholar - 13.Bennett, D.L., Gourley, S.A.: Asymptotic properties of a delay differential equation model for the interaction of glucose with plasma and interstitial insülin. Appl. Math. Comput.
**151**(1), 189–207 (2004)MathSciNetzbMATHGoogle Scholar - 14.Liu, L., Kalmár-Nagy, T.: High-dimensional harmonic balance analysis for second-order delay-differential equations. J. Vib. Control
**16**(7–8), 1189–1208 (2010)MathSciNetzbMATHGoogle Scholar - 15.Gu, K., Niculescu, S.-I.: Survey on recent results in the stability and control of time-delay systems. J. Dyn. Syst. Meas. Contr.
**125**(2), 158–165 (2003)CrossRefGoogle Scholar - 16.Gülsu, M., Gürbüz, B., Öztürk, Y., Sezer, M.: Laguerre polynomial approach for solving linear delay difference equations. Appl. Math. Comput.
**217**(15), 6765–6776 (2011)MathSciNetzbMATHGoogle Scholar - 17.Biazar, J.: Solution of the epidemic model by Adomian decomposition method. Appl. Math. Comput.
**173**, 1101–1106 (2006)MathSciNetzbMATHGoogle Scholar