Analytical technique for neutral delay differential equations with proportional and constant delays
Volume 20, Issue 4, pp 334348
http://dx.doi.org/10.22436/jmcs.020.04.07
Publication Date: March 06, 2020
Submission Date: December 17, 2019
Revision Date: January 04, 2020
Accteptance Date: January 16, 2020
Authors
Normah Maan
 Department of Mathematical Sciences, Universiti Teknologi, 81310 Skudai, Johor, Malaysia.
Aminu Barde
 Department of Mathematical Sciences, Universiti Teknologi, 81310 Skudai, Johor, Malaysia.
 Department of Mathematical Sciences, Abubakar Tafawa Balewa University, Bauchi, Nigeria.
Abstract
Neutral delay differential equations (NDDEs) are a type of delay differential equations (DDEs) that arise in numerous areas of applied sciences and play a vital role in mathematical modelling of reallife phenomena. Some techniques have experienced difficulties in finding the approximate analytical solution which converges rapidly to the exact solution of these equations. In this paper, an analytical approach is proposed for solving linear and nonlinear NDDEs with proportional and constant delays based on the homotopy analysis method (HAM) and natural transform method where the nonlinear terms are simply calculated as a series of, He's polynomial. The proposed method produces solutions in the form of a rapidly convergent series which leads to the exact solution from only a few numbers of iterations. Some illustrative examples are solved, and the convergence analysis of the proposed techniques was also provided. The obtained results reveal that the approach is very effective and efficient in handling both linear and nonlinear NDDEs with proportional and constant delays and can also be used in various types of linear and nonlinear problems.
Share and Cite
ISRP Style
Normah Maan, Aminu Barde, Analytical technique for neutral delay differential equations with proportional and constant delays, Journal of Mathematics and Computer Science, 20 (2020), no. 4, 334348
AMA Style
Maan Normah, Barde Aminu, Analytical technique for neutral delay differential equations with proportional and constant delays. J Math Comput SCIJM. (2020); 20(4):334348
Chicago/Turabian Style
Maan, Normah, Barde, Aminu. "Analytical technique for neutral delay differential equations with proportional and constant delays." Journal of Mathematics and Computer Science, 20, no. 4 (2020): 334348
Keywords
 Neutral delay differential equations
 He's polynomial
 natural transform method
 homotopy analysis method
MSC
References

[1]
A. K. Alomari, M. S. M. Noorani, R. Nazar, Solution of delay differential equation by means of homotopy analysis method, Acta Appl. Math., 108 (2009), 395412

[2]
K. Barzinji, N. Maan, N. Aris, Fuzzy delay predatorprey system: existence theorem and oscillation property of solution, Int. J. Math. Anal. (Ruse), 8 (2014), 829847

[3]
J. J. Batzel, T. H. Tran, Stability of the human respiratory control system I. Analysis of a twodimensional delay statespace model, J. Math. Biol., 41 (2000), 4579

[4]
F. B. M. Belgacem, R. Silambarasan, Advances in the natural transform, AIP conference proceedings, 2012 (2012), 106110

[5]
F. B. M. Belgacem, R. Silambarasan, Maxwell's equations solutions by means of the natural transform, Math. Eng. Sci. Aerosp., 3 (2012), 313323

[6]
F. B. M. Belgacem, R. Silambarasan, Theory of Natural Transform, Math. Eng. Sci. Aerosp., 3 (2012), 99124

[7]
A. Bellen, M. Zennaro, Numerical methods for delay differential equations, Oxford University Press, New York (2003)

[8]
A. H. Bhrawy, M. A. Alghamdi, D. Baleanu, Numerical solution of a class of functionaldifferential equations using Jacobi pseudospectral method, Abstr. Appl. Anal., 2013 (2013), 9 pages

[9]
R. H. Fabiano, A semidiscrete approximation scheme for neutral delaydifferential equations, Int. J. Numer. Anal. Model., 10 (2013), 712726

[10]
Z. H. Khan, W. A. Khan, Ntransform properties and applications, NUST J. Eng. Sci., 1 (2008), 127133

[11]
S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Shanghai Jiao Tong University (Ph. D. Thesis), Shanghai (1992)

[12]
S.J. Liao, K. F. Cheung, Homotopy analysis of nonlinear progressive waves in deep water, J. Engrg. Math., 45 (2003), 105116

[13]
S.J. Liao, J. Su, A. T. Chwang, Series solutions for a nonlinear model of combined convective and radiative cooling of a spherical body, Int. J. Heat Mass Transfer, 49 (2006), 24372445

[14]
L. P. Liu, T. KalmárNagy, Highdimensional harmonic balance analysis for secondorder delaydifferential equations, J. Vib. Control, 16 (2010), 11891208

[15]
D. Loonker, P. K. Banerji, Natural transform and solution of integral equations for distribution spaces, American J. Math. Sci., 3 (2014), 6572

[16]
N. Maan, K. Barzinji, N. Aris, Fuzzy delay differential equation in predatorprey interaction: analysis on stability of steady state, Proceedings of the World Congress on Engineering, 2013 (2013), 35

[17]
S. Maitama, An efficient technique for solving linear and nonlinear fractional partial differential equations, Math. Eng. Sci. Aerosp., 8 (2017), 521534

[18]
S. Maitama, I. Abdullahi, A new analytical method for solving linear and nonlinear fractional partial differential equations, Prog. Fract. Differ. Appl., 2 (2016), 247256

[19]
S. Maitama, W. D. Zhao, New homotopy analysis transform method for solving multidimensional fractional diffusion equations, Arab J. Basic Appl. Sci., 27 (2020), 2744

[20]
L. Muhsen, N. Maan, Modeling of Human Postural Balance Using Neutral Delay Differential Equation to Solvable Lie Algebra Classification, Life Sci. J., 11 (2014), 11451152

[21]
L. Muhsen, N. Maan, Lie group analysis of secondorder nonlinear neutral delay differential equations, Malaysian J. Math. Sci., 10 (2016), 117129

[22]
Z. Odibat, S. Momani, H. Xu, A reliable algorithm of homotopy analysis method for solving nonlinear fractional differential equations, Appl. Math. Model., 34 (2010), 593600

[23]
M. S. Rawashdeh, S. Maitama, Solving nonlinear ordinary differential equations using the NDM, J. Appl. Anal. Comput., 5 (2015), 7788

[24]
F. A. Rihan, D. H. Abdel Rahman, S. Lakshmanan, A. S. Alkhajeh, A time delay model of tumourimmune system interactions: Global dynamics, parameter estimation, sensitivity analysis, Appl. Math. Comput., 232 (2014), 606623

[25]
F. A. Rihan, A. A. Azamov, H. J. AlSakaji, An Inverse problem for delay differential equations: parameter estimation, nonlinearity, sensitivity, Appl. Math. Inf. Sci., 12 (2018), 6374

[26]
M. G. Sakar, Numerical solution of neutral functionaldifferential equations with proportional delays, Int. J. Optim. Control. Theor. Appl. IJOCTA, 7 (2017), 186194

[27]
S. K. Vanani, A. Aminataei, On the numerical solution of neutral delay differential equations using multiquadric approximation scheme, Bull. Korean Math. Soc., 45 (2008), 663670

[28]
S. Yüzbaşı, A numerical approximation based on the Bessel functions of first kind for solutions of Riccati type differentialdifference equations, Comput. Math. Appl., 64 (2012), 16911705

[29]
J. J. Zhao, Y. Cao, Y. Xu, LegendreGauss collocation methods for nonlinear neutral delay differential equations, Adv. Difference Equ., 2015 (2015), 24 pages