Solving delay differential equations via Sumudu transform

Document Type : Research Paper

Authors

1 Institute for Systems Science & KZN e-Skills CoLab, Durban University of Technology, Durban 4000, South Africa

2 DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), Johannesburg, South Africa

3 KZN e-Skills CoLab, Durban University of Technology, Durban 4000, South Africa

4 Institute for Systems Science \& Office of the DVC Research, Innovation \& Engagement, Milena Court, Durban University of Technology, Durban 4000, South Africa

Abstract

A new technique which is known as Sumudu Transform Method (STM) is instituted for solving delay differential equations. STM is used to obtain the solutions of general nonlinear systems. The strength of STM is illustrated in reducing the complex computational work as compared to the well-known methods. This paper shows how to succinctly identify the Lagrange multipliers for nonlinear delay differential equations with variable coefficients, using the STM. The potency and suitability of the STM are exhibited by giving expository examples. The method is used to obtain the exact and approximate solutions of pantograph type equations with variable coefficients and nonlinear Volterra integro-differential equations of pantograph type.

Keywords

[1] A.T. Ademola, S. Moyo, B.S. Ogundare, M.O. Ogundiran and O.A. Adesina, New conditions on the solutions of
a certain third order delay differential equations with multiple deviating arguments, Diff. Equ. Control Process.
2019 (2019), no. 1, 33–69.
[2] W.G. Ajello, H.I. Freedman and J. Wu, A model of stage structured population growth with density depended time
delay, SIAM J. Appl. Math. 52 (1992), 855–869.
[3] W.R.A. AL-Hussein and S.N. Al-Azzawi, Approximate solutions for fractional delay differential equations by
using Sumudu transform method AIP Conf. Proc. 2096 (2019), no. 1, 020007.
[4] A.K. Alomari, M.I. Syam, N. R. Anakira and A.F. Jameel, Homotopy Sumudu transform method for solving
applications in physics, Results Phys.18 (2020), 103265.[5] A. Arikoglu and I. Ozkol, Solution of fractional integro-differential equations by using fractional differential transform method, Chaos Solitons Fractals, 34, (2007), 1473-1481.
[6] M.M. Bashi and M. Cevik, Numerical solution of pantograph-type delay differential equations using perturbationiteration algorithms, J. Appl. Math. 2015 (2015), Article ID: 139821, 10 pages.
[7] M.D. Buhmann and A. Iserles, Stability of the discretized pantograph differential equation, Math. Comp. 60 (1993),
575–589.
[8] A.K. Golmankhaneh and C. Tunς, Sumudu transform in fractal calculus, Appl. Math. Comput. 350 (2019), no.
1, 386–401.
[9] J.R. Graef and C. Tunς, Global asymptotic stability and boundedness of certain multi-delay functional differential
equations of third order, Math. Methods Appl. Sci. 38 (2015), no. 17, 3747–3752.
[10] J. He, Variational iteration methoda kind of non-linear analytical technique: Some examples, Int. J. Non-Linear
Mech. 34 (1999), no. 4, 699–708.
[11] J. He, Variational iteration method-some recent results and new interpretations, J. Comput. Appl. Math. 207
(2007),, no. 1, 3–17.
[12] J.H. He and X.H. Wu, Variational iteration method: new development and applications, Comput. Math. Appl.
54 (2007), no. 7-8, 881–894.
[13] J.H. He, G.C. Wu andF. Austin, The variational iteration method which should be followed, Nonlinear Sci. Lett.
A 1 (2010), no. 1, 1–30.
[14] N. Herisanu, V. Marinca, A modifed variational iteration method for strongly nonlinear problems, Nonlinear Sci.
Lett. A: Math. Phys. Mech. 1 (2010), no. 2, 183–192.
[15] X.H. Ma and C.M. Huang, Numerical solution of fractional integro-differential equations by a hybrid collocation
method, Appl. Math. Comput. 219 (2013), 6750–6760.
[16] Z. Meng, L. Wang, H. Li and W. Zhang, Legendre wavelets method for solving fractional integro-differential
equations, Int. J. Comput. Math. 92 (2015), 1275–1291.
[17] S.V. Meleshko, S. Moyo and G.F. Oguis, On the group classification of systems of two linear second-order ordinary
differential equations with constant coefficients, J. Math. Anal. Appl. 410 (2014), no. 1, 341–347.
[18] T.G. Mkhize, K. Govinder, S. Moyo and S.V. Meleshko, Linearization criteria for systems of two second-order
stochastic ordinary differential equations, Appl. Math. Comput. 301 (2017), 25–35.
[19] S.A. Mohammed and C. Tunς, Qualitative analysis of nonlinear retarded differential equations of second order,
Dynamic Syst. Appl. 29 (2020), 53–70. equations with constant coefficients, J. Math. Anal. Appl. 410 (2014),
341–347.
[20] S. Momani and M.A. Noor, Numerical methods for fourth-order fractional integro-differential equations, Appl.
Math. Comput. 182 (2006), 754–760.
[21] K.S. Nisar, A. Shaikh, G. Rahman and D. Kumar, Solution of fractional kinetic equations involving class of
functions and Sumudu transform, Adv. Differ. Equ. 2020 (2020), Article Number 39.
[22] A.H. Nayfeh, Introduction to Perturbation Techniques, John Wiley & Sons, New York, NY, USA, 1981.
[23] Y. Nawaz, Variational iteration method and homotopy perturbation method for fourth-order fractional integrodifferential equations, Comput. Math. Appl. 61 (2011), 2330–2341.
[24] J.R. Ockendon and A.B. Tayler, The dynamics of a current collection system for an electric locomotive, Proc. R.
Soc. Lond. Ser. A. 322 (1971), no. 1551, 447–468.
[25] Z.A. Odibat, A study on the convergence of variational iteration method, Math. Comput. Model. 51 (2010),
1181–1192.
[26] E. Rawashdeh, Numerical solution of fractional integro-differential equations by collocation method, Appl. Math.
Comput. 176 (2006), 1–6.
[27] A. Saadatmandi and M. Dehghan, Variational iteration method for solving a generalized pantograph equation,Comput. Math. Appl. 58 (2009), no. 11-12, 2190–2196.
[28] H. Saeedi and M. Mohseni Moghadam, Numerical solution of nonlinear Volterra integro-differential equations of
arbitrary order by CAS wavelets, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), 1216–1226.
[29] H. Saeedi, M. Mohseni Moghadam, N. Mollahasani and G.N. Chuev, A CAS wavelet method for solving nonlinear
Fredholm integro-differential equations of fractional order, Commun. Nonlinear Sci. Numer. Simul. 16 (2011),
1154–1163.
[30] K. Sayevand, Analytical treatment of Volterra integro-differential equations of fractional order, Appl. Math. Model.
39 (2015), 4330–4336.
[31] M. Sezer and A. Akyuz-Dascioglu, A Taylor method for numerical solution of generalized pantograph equations
with linear functional argument, J. Comput. Appl. Math. 200 (2007), 217–225.
[32] M. Sezer, S. Yalcinbas and N. Sahin, Approximate solution of multipantograph equation with variable coefficients,
J. Comput. Appl. Math. 214 (2008), 406–416.
[33] S. Vilu, R.R. Ahmad and U.K. Salma Din, Variational iteration method and Sumudu transform for solving delay
differential equation, Int. J. Differ. Equ. 2019 (2019), Article ID 6306120, 6 pages.
[34] Y.X. Wang and L. Zhu, SCW method for solving the fractional integro-differential equations with a weakly singular
kernel, Appl. Math. Comput. 275 (2016), 72–80.
[35] G.K. Watugala, Sumudu transforma new integral transform to solve differential equations and control engineering
problems, Math. Engin. Ind. 24 (1993), no. 1, 35–43.
[36] J. Wei and T. Tian, Numerical solution of nonlinear Volterra integro-differential equations of fractional order by
the reproducing kernel method, Appl. Math. Model. 39 (2015), 4871–4876.
[37] G. Wu, Challenge in the variational iteration methoda new approach to identification of the Lagrange multipliers,
J. King Saud Univ. Sci. 25 (2013), no. 2, 175–178.
[38] G. Wu and D. Baleanu, Variational iteration method for fractional calculusa universal approach by Laplace transform, Adv. Differ. Equ. 2013 (2013), no. 1, 1–9.
[39] Z.-H. Yu, Variational iteration method for solving the multi-pantograph delay equation, Phys. Lett. A 372 (2008),
no. 43, 6475–6479.
[40] L. Zhu and Q.B. Fan, Solving fractional nonlinear Fredholm integro-differential equations by the second kind
Chebyshev wavelet, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 2333–2341.
[41] L. Zhu and Q.B. Fan, Numerical solution of nonlinear fractional-order Volterra integro-differential equations by
SCW, Commun. Nonlinear Sci. Numer. Simul. 18 (2013), 1203–1213.
Volume 13, Issue 2
July 2022
Pages 563-575
  • Receive Date: 17 February 2021
  • Accept Date: 20 March 2021