A New Method for Solving a Neutral Functional-Differential Equation
with Proportional Delays
OSAMA ALA’YED
Department of Mathematics,
Jadara University,
JORDAN
Abstract: - This study presents and implements a new hybrid technique that combines the Sawi transform (ST)
and Homotopy perturbation method (HPM) to solve neutral functional-differential equations with proportional
delays. Some of the important properties of the method are established and validated. We start the method by
first applying ST to obtain the recurrence relation. We, next, implement HPM to find convergent series
solutions of the recurrence relation. The series is free of assumptions and restrictions, highlighting its
adaptability and robustness. Moreover, the convergence of the method is established through convincing proof.
To demonstrate its effectiveness and applicability, we provide five examples. The method yields accurate
approximate solutions, or in some cases exact solutions, with a few number of iterations, reinforcing its
reliability and validity. Moreover, the performance of the method is compared with some available methods
and demonstrates its superiority and efficiency.
Key-Words: - Sawi transform, Integral transform, Homotopy perturbation method, Proportional delay,
Pantograph equations, Neutral functional-differential equations.
Received: March 16, 2023. Revised: October 15, 2023. Accepted: November 17, 2023. Published: February 6, 2024.
1 Introduction
Functional–differential equations (FDEs) with
proportional delays are usually indicated as
pantograph equations. The term "pantograph" was
first introduced by Ockendon and Tayler in their
study, [1]. These equations frequently appear in
industry and studies based on economy, biology,
electrodynamics, and control theory among others,
[2]. One noteworthy characteristic of such equations
is the presence of compactly supported solutions,
[3]. Pantograph equations play a significant role in
describing various phenomena and are distinguished
by the presence of a linear functional argument.
They become especially essential when the ODEs-
based models fail. Recently, numerous methods
have been established for solving pantograph
differential equations including the Taylor operation
method, perturbation iteration algorithms, and the
Adomian decomposition method, [4], [5], [6], [7],
[8], [9], [10], [11], [12], [13], [14], [15], [16], [17],
[18], [19], [20], [21], [22], [23], [24], [25], [26],
[27].
HPM, [28], [29], is an efficient and reliable
technique for solving enormous classes of
differential problems. HPM does not require any
discretization, perturbation, or linearization. This
strategy yields the solution in the form of a
polynomial. Lately, HPM has been utilized to obtain
approximate solutions for numerous classes of
differential problems, [30], [31], [32]. It also has
been demonstrated that HPM is a fast and
trustworthy method compared to other methods.
More details regarding HPM can be found in, [33],
and references within.
In 2019, the work of, [34] proposed a new
integral transform, known as the Sawi transform
(ST). This transform is very simple to implement,
requiring no assumptions in its procedure, [35].
Recently, the Sawi transform has been widely
utilized for solving various integral and differential
equations, [35], [36], [37], [38], [39], [40], [41].
In this study, we combined the ST and HPM to
create the strategy of the Sawi homotopy
perturbation transform method (SHPTM) and find
the analytic results of FDEs with proportional
delays, in convergent series form. HPM is utilized to
handle delay components. ST is used to minimize
the computational work and to provide more
accurate results. We observe that HPM is an
efficient method for addressing various phenomena.
Outcomes demonstrate that the approach is simple
and easy to employ.
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2 Sawi Transform
In this section, we outline the fundamental concepts
and properties of the Sawi transform, [42], [43].
Definition 1. The Sawi transform for a function
is given by
where S is designated as Sawi transform. If is
the Sawi transform of a function , then is
the inverse of so that,
where is said to be inverse
Sawi transform.
Definition 2. If and
, then
where and are arbitrary constants.
Table 1. Sawi Transforms of Some Fundamental
Functions
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Definition 3. If , we can consider
the following differential properties as
i.
ii.
iii.
Table 1 provides Sawi transforms of various
essential functions that are useful in solving
significant issues in the fields of science and
engineering.
3 Method of Solution
In this section, we introduce the new SHPTM to
solve the following FDEs with proportional delays,
[23],
(1)
with the initial conditions
where and ( denote given
analytic functions, and designate given
constants, with for
Notably, this approach does not rely on integration
or any assumptions in its formulation. We express
(1) as:
(2)
Applying ST to (2) yields:
(3)
Using ST properties, we have:
(4)
Thus, using the initial conditions in (4), we obtain:
(5)
Operating inverse ST on (5), we get:
(6)
where
.
Let us introduce HPM as
(7)
Substituting (7) in (6) and matching terms with the
same power of , we obtain:
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Following this process, we can calculate
and so on. These functions can
be combined to derive a solution for (1) as:
(8)
To prove the convergence of the solution in (8),
we illustrate the following theorem.
Theorem. Assume that and are Banach spaces
and : is a contraction function, that is
Then, according to Banach’s fixed point theorem,
the existence of a unique fixed point γ is guaranteed.
Moreover, suppose that the sequence produced by
the HPM can be written as
and suppose that where
, then we have
a.
b.
c.
Proof.
a. By the mathematical induction method on for
we have
Assume that for some
as an induction assumption, then
Hence
b. Using (a), we have
Therefor,
c. Since and
(as
we have
that is,
4 Numerical Examples
In this section, we provide several examples to
demonstrate the effectiveness of the method
introduced in Section 3. All examples are
implemented using MATHEMATICA 12.
Example 1. Consider the following first-order
neutral FDEs with proportional delays:
(9)
Applying ST to (9), we get:
(10)
Using the properties of ST, we have:
(11)
Thus, yields:
(12)
Operating inverse ST on (12), we obtain:
(13)
Substituting (7) into (13), we get:
(14)
Equating the coefficients of that have the same
exponent leads to:
which is the analytical solution of (9).
Example 2. Consider the following first-order
neutral FDEs with proportional delays:
(15)
Applying ST on (15), we have:
(16)
Using the properties of ST, we get:
(17)
Thus, yields:
(18)
Operating inverse ST on (18), we get:
(19)
Substituting (7) into (19), we get:
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(20)
Matching terms with the same power of , we
obtain:
In Table 2 we compare the absolute errors of
the 5 and 6-term solutions with those of the
VIM, [23], HPM, [24], two-stage order-one
Runge-Kutta method (RK method), [25], and the
one-leg-method (leg-method), [26], [27]. In
Figure 1 we show the comparison of the 3, 4,
and 5-term solutions with the exact solution
.
Example 3. Consider the following second-
order neutral FDEs with proportional delays:
(21)
Applying ST on (21), we have:
(22)
Utilizing the properties of ST, we get:
Table 2. Comparison of the absolute errors for Example 2.
leg-method
RK method
VIM
HPM
SHPTM
5-term solution
6-term solution
0.6
0.7
0.8
0.9
1.0
Table 3. Comparison of the absolute errors for Example 4.
leg-method
RK method
VIM
HPM
SHPTM
5-term solution
6-term solution
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Fig. 1: Comparison of the exact solution with the
approximate solutions for Example 2.
(23)
Therefore, yields:
(24)
Upon applying inverse ST to (24), we obtain:
(25)
Substituting (7) into (25) and equating the
coefficients of with the same exponent gives:
which coincides with the analytical solution of (21).
Example 4. Consider the following second-order
neutral FDEs with proportional delays:
(26)
Using ST on (26), and utilizing the properties of ST,
we have:
(27)
Thus, yields:
(28)
Applying inverse ST to (28), we obtain:
(29)
Substituting (7) into (29) and matching the
coefficients of with the same exponent results in:
Fig. 2: Comparison of the exact solution with the
approximate solutions for Example 4.
In Table 3 we compare the absolute errors of the
5 and 6-term solutions with the VIM, [23], HPM,
[24], RK method, [25], and the leg-method, [26],
[27]. In Figure 2 we show the comparison of the 3,
4, and 5-term solutions with the exact solution
.
Example 5. Consider the following third-order
neutral FDEs with proportional delays:
. (30)
Applying ST to (30), and using the properties of ST,
we have:
(31)
Thus, yields:
(32)
Operating inverse ST on (32), we obtain:
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(33)
Substituting (7) into (33) and equating the
coefficients of that have the same exponent leads
to:
which is the analytical solution of (30).
Moreover, we demonstrate that the proposed
method is quite simple and efficient in solving such
problems. The graphical illustrations expose that the
obtained results are extremely close, and in some
cases are identical, to the exact results.
5 Conclusion
In this study, we confirm the capability of the
SHPTM for solving neutral FDEs with proportional
delays. This approach does not rely on integration or
any assumptions in its formulation. The approach
starts by first applying ST to the considered problem
and using HPM to generate a series solution. This
series yields accurate approximations, or in some
cases exact solutions, with a few number of
iterations. All the computations and the graphical
illustration are made using MATHEMATICA 12.
The proposed examples demonstrate that the results
of the SHPTM agree excellently with the exact
solution and with those of some other methods.
Furthermore, the findings seem to indicate that the
SHPTM is an efficient and convenient approach to
approximate the solution of such problems. We
expect that this method can be easily used as a
viable alternative for various problems in science
and engineering that lack exact solutions.
References:
[1] J.R. Ockendon, A.B. Tayler, The dynamics
of a current collection system for an electric
locomotive, Proc. Roy. Soc. London Ser. A
322, 1971, 447–468.
[2] M.M. Khader, Numerical and theoretical
treatment for solving linear and nonlinear
delay differential equations using variational
iteration method, Arab Journal of
Mathematical Sciences, 19, 2013, 243-56.
[3] M.Z. Liu, D. Li, Properties of analytic
solution and numerical solution of multi-
pantograph equation, Appl. Math. Comput.,
155, 2004, 853–871.
[4] S. YÜZBAŞI, and N. Ismailov, A Taylor
operation method for solutions of generalized
pantograph type delay differential equations,
Turk. J. Math., 42, 2, 2018, 395-406.
[5] M. M. Bahsi, M. Cevik, Numerical solution
of pantograph-type delay differential
equations using perturbation-iteration
algorithms, J. Appl. Math., 2015, Art. ID
139821, 10 pages.
[6] S. Yalcinbas, H. H. Sorkun, M. Sezer, A
numerical method for solutions of
pantograph type differential equations with
variable coefficients using Bernstein
polynomials, New Trends Math. Sci., 3,
2015, 179-195.
[7] F. O. Ogunfiditimi, Numerical solution of
delay differential equations using the
Adomian decomposition method (ADM), Int.
J. Engg. Sci., 4, 2015, 18-23.
[8] S. Davaeifar, J. Rashidinia, Solution of a
system of delay differential equations of
multi-pantograph type, J. Taibah University
Sci., 11, 2017, 1141-1157.
[9] S. Y. Reutskiy, A new collocation method
for approximate solution of the pantograph
functional differential equations with
proportional delay, Appl. Math. Comp., 266,
2015, 1642-1655.
[10] S. Sedaghat, Y. Ordokhani, and M. Dehghan,
Numerical solution of the delay differential
equations of pantograph type via Chebyshev
polynomials, Commu. Nonl. Sci. Numer.
Simul., 17, 2012, 4815-4830.
[11] S. Yuzbasi, M. Sezer, Shifted Legendre
approximation with the residual correction to
solve pantograph-delay type differential
equations, Appl. Math. Model., 39, 2015,
6529-6542.
[12] M. Behroozifar, S. A. Youse, Numerical
solution of delay differential equations via
operational matrices of hybrid of block-pulse
functions and Bernstein polynomials,
Comput. Meth. Differ. Equ., 1, 2013, 78-95.
[13] S. C. Shiralashetti, S. Kumbinarasaiah, R. A.
Mundewadi, B. S. Hoogar, Series solution of
pantograph equations using wavelets, Open
J. Appl. Theo. Math., 2, 2016, 505-518.
[14] M. Ghasemi, M. T. Kajani, Numerical
solution of time-varying delay systems by
Chebyshev wavelets, Appl. Math. Model., 35,
2011, 5235-5244.
[15] A. Ali, M. A. Iqbal, S. T. Mohyud-Din,
Chebyshev wavelts method for delay
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.9
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differential equations, Int. J. Mod. Math. Sci.,
8, 2013, 102-110.
[16] S. C. Shiralashetti, B. C. Hoogar, S.
Kumbinarasaiah, Hermite wavelets based
method for the numerical solution of linear
and nonlinear delay differential equations,
Int. J. Engg. Sci. Math., 6, 2017, 71-79.
[17] S. Gümgüm, D. E. Özdek, G. Özaltun, and
N. Bildik, Legendre wavelet solution of
neutral differential equations with
proportional delays, J Appl Math Comput,
61, 2019, 389-404.
[18] M. G. Sakar, Numerical solution of neutral
functional-differential equations with
proportional delays. Int. J. Optim. Control
Theor. Appl., 7(2), 2017, 186–194.
[19] D.J. Evans, K.R. Raslan, The Adomian
decomposition method for solving delay
differential equation, Int. J. Comput. Math.,
82 (1), 2005, 49–54.
[20] G. Derfel, N. Dyn, D. Levin, Generalized
refinement equation and subdivision process,
J. Approx. Theory, 80, 1995, 272–297.
[21] M. Shadia, Numerical Solution of Delay
Differential and Neutral Differential
Equations Using Spline Methods, PhD thesis,
Assuit University, 1992.
[22] A. El-Safty, M.S. Salim, M.A. El-Khatib,
Convergence of the spline function for delay
dynamic system, Int. J. Comput. Math., 80
(4), 2003, pp.509-518.
[23] X. Chen, L. Wang, The variational iteration
method for solving a neutral functional-
differential equation with proportional
delays, Computers & Mathematics with
Applications, vol. 59, 2010, pp.2696-2702.
[24] J. Biazar, B. Ghanbari, The homotopy
perturbation method for solving neutral
functional–differential equations with
proportional delays, Journal of King Saud
University-Science, vol.24, 2012, pp.33-37.
[25] A. Bellen, M. Zennaro, Numerical methods
for delay differential equations, in Numerical
Mathematics and Scientific Computation,
The Clarendon Press Oxford University
Press, New York, 2003.
[26] W.S. Wang, S.F. Li, On the one-leg θ-
methods for solving nonlinear neutral
functional differential equations, Appl. Math.
Comput., vol.193, 2007, pp.285–301.
[27] W.S. Wang, T. Qin, S.F. Li, Stability of one-
leg θ-methods for nonlinear neutral
differential equations with proportional
delay, Appl. Math. Comput., 213, 2009,
pp.177–183.
[28] J.H. He, Homotopy perturbation technique,
Comput. Methods Appl. Mech. Eng., 178,
1999, 257–262.
[29] J.H. He, Homotopy perturbation method: A
new nonlinear analytical technique. Appl.
Math. Comput., 135, 2003, 73–79.
[30] J. H. He, and Y. O. El-Dib, and A. A. Mady,
Homotopy perturbation method for the
fractal toda oscillator, Fractal and
Fractional, 5, 3, 2021, 93.
[31] T. Liu, Porosity reconstruction based on Biot
elastic model of porous media by homotopy
perturbation method, Chaos, Solitons &
Fractals, 158, 2022, 112007.
[32] J. H. He, and Y. O. El-Dib, Homotopy
perturbation method for Fangzhu oscillator,
Journal of Mathematical Chemistry, 58,
2020, 2245-2253.
[33] M. I. Liaqat, and A. Ali, A novel approach
for solving linear and nonlinear time-
fractional Schrödinger equations, Chaos,
Solitons & Fractals, 162, 2022, 112487.
[34] M.M.A. Mahgoub, The new integral
transform ''Sawi Transform'', Advances in
Theoretical and Applied Mathematics, 14,
2019, 81-87.
[35] M. Nadeem, S. A. Edalatpanah, I. Mahariq,
W. H. Aly, Analytical view of nonlinear
delay differential equations using Sawi
iterative scheme, Symmetry, 14, 2022, 2430.
[36] M. Higazy and S. Aggarwal, Sawi
transformation for a system of ordinary
differential equations with the application,
Ain Shams Engineering Journal, 12, 2021,
3173-3182.
[37] M. Kapoor and S. Khosla, An iterative
approach using Sawi transform for fractional
telegraph equation in diversified dimensions,
Nonlinear Engineering, 2023, 12, 20220285.
[38] M. Nadeem, New strategy for the numerical
solution of multi-dimensional diffusion
equations, International Journal of
Numerical Methods for Heat & Fluid Flow,
2023, 33, 1939-54.
[39] M. Higazy, S. Aggarwal, and T. A. Nofal,
Sawi decomposition method for Volterra
integral equation with application, Journal of
Mathematics, 2020, 2020.
[40] S. R. Khirsariya and S. B. Rao, On the semi-
analytic technique to deal with nonlinear
fractional differential equations, Journal of
Applied Mathematics and Computational
Mechanics, 2023, 22, 13-26.
[41] M. Sahoo and S. Chakraverty, Sawi
Transform Based Homotopy Perturbation
WSEAS TRANSACTIONS on MATHEMATICS
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Method for Solving Shallow Water Wave
Equations in Fuzzy Environment,
Mathematics, 10, 2022, 2900.
[42] S. Aggarwal, A.R. Gupta, Dualities between
some useful integral transforms and Sawi
transform, International Journal of Recent
Technology and Engineering, 8, 2019, 5978-
5982.
[43] G.P. Singh, S. Aggarwal, Sawi transform for
population growth and decay problems,
International Journal of Latest Technology
in Engineering, Management and Applied
Science, 8, 2019, 157-162.
Contribution of Individual Authors to the
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Policy)
The author contributed in the present research, at all
stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author has no conflicts of interest to declare.
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