Dejdumrong Collocation Approach and Operational Matrix for a Class
of Second-Order Delay IVPs: Error Analysis and Applications
NAWAL SHIRAWIA1, AHMED KHERD2, SALIM BAMSAOUD3,
MOHAMMAD A. TASHTOUSH4,1, ALI F. JASSAR1, EMAD A. AZ-ZO’BI5
1Department of Mathematics Education, Faculty of Education & Arts, Sohar University,
Sohar, OMAN
2Faculty of Computer Science & Engineering, Al-Ahgaff University, Mukalla, YEMEN
3Department of Physics, Faculty of Sciences, Hadhramout University, Mukalla, YEMEN
4Department of Basic Sciences, AL-Huson University College, AL-Balqa Applied University,
Al-Salt, JORDAN
5Department of Mathematics, Faculty of Science, Mutah University, Karak, JORDAN
Abstract: In this paper, a collocation method based on the Dejdumrong polynomial matrix approach was used
to estimate the solution of higher-order pantograph-type linear functional differential equations. The equations
are considered with hybrid proportional and variable delays. The proposed method transforms the functional-
type differential equations into matrix form. The matrices were converted into a system of algebraic equations
containing the Dejdumrong polynomial. The coefficients of the Dejdumrong polynomial were obtained by solving
the system of algebraic equations. Moreover, the error analysis is performed, and the residual improvement
technique is presented. The presented methods are applied to three examples. Finally, the obtained results are
compared with the results of other methods in the literature and were found to be better compared. All results in
this study have been calculated using Matlab R2021a.
Key-Words: - Dejdumrong polynomial; Functional differential equations; Numerical solutions; Proportional and
variable delays; Residual error analysis.
Received: September 24, 2023. Revised: April 23, 2024. Accepted: June 24, 2024. Published: July 19, 2024.
1 Introduction
A multitude of physical phenomena cannot be suffi-
ciently accounted for ordinary differential equations
(ODEs) when the employed model constructed bases
on a particular previous state in addition to its present
state. As a consequence, DEs with time delays are
utilized to model real-world scenarios for instance
chemical engineering, electric circuits, fluid mechan-
ics, human body control systems, multibody control
systems, stage-structured populations, spread of bac-
teriophage infection, the dynamic diseases model in
physiology, and the epidemic model in biology [1],
[2], [3], [4], [5], [6], [7], [8].
Numerous numerical and analytical methodolo-
gies were developed in order to address nonlinear
differential equations (NDEs) which involve propor-
tional and unchanged delays. Some notable methods
include the Aboodh transformation method [9],
spectral method [10], residual power series method
[11], Adomian decomposition method [12], [13],
polynomial least squares approach [14], reproduc-
ing kernal Hilbert space [15], variable multistep
techniques [16], differential transform method and
its reduction [17], [18], the cuckoo optimization
algorithm [19], Hermite wavelet-based approach
[20], variational iteration approach [21], [22],
[23], quasilinearization technique [24], expanded
ansatz method [25], generalized Riccati equation
mapping method [26], generic algorithm [27], and
Kudryashov modified simplest equation method [28].
In contrast, NDEs involving variable delays have
been the subject of relatively few studies. [29]
have conducted research on the presence of positive
solutions that repeat every ω-periodic. Asymptotic
stability was evaluated using new criteria that were
proposed in [30]. [31] and [32] have investigated
the asymptotic behavior of solutions. The analysis
of fixed points and stability is conducted in [33],
[34], [35], [36] and included bibliography. There
are only a few numerical strategies that have been
used in order to solve equations of this kind. These
techniques include a novel multi-step strategy [37],
the Legendre-Gauss collocation approach [38], and
the Runge-Kutta method employing Hermite inter-
polation [39].
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DOI: 10.37394/23206.2024.23.49
Nawal Shirawia, Ahmed Kherd,
Salim Bamsaoud, Mohammad A. Tashtoush,
Ali F. Jassar, Emad A. Az-Zo’Bi
E-ISSN: 2224-2880
467
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There is significant interest in finding numerical
solutions for ODEs, fractional differential equations
(FDEs), and integro-differential equations. Recently,
researchers have proposed methods for solving FDEs
that are based on Pell-Lucas, Fibonacci, and Fermat
polynomials [40], [41], [42], [43], [44], [45], [46],
[47], [48]. Researchers were able to develop the
operational matrix of fractional derivatives via these
investigations. They also made the observation that
the numerical solutions had less errors compared
to the ones that were acquired through the use of
orthogonal polynomials.
Investitigang the analytical solutions of nonlinear
differential equations with delay variables is more
challenging in comparison to linear differential equa-
tions. Another difference between linear and nonlin-
ear differential equations is that the former often do
not have analytical solutions. Consequently, numer-
ical methods assume a critical role in the solution of
these nonlinear differential equations.
Lately, there has been a significant emphasis
among researchers on finding numerical solutions
for nonlinear differential equations. We provide our
precise numerical solutions, which exhibit increasing
accuracy. It is worth mentioning that we may antici-
pate that a small number of Dejdamrong polynomial
basis functions will be enough to get an approximate
solution that closely matches the precise solution
with a precision of up to 10 digits. Dejdamrong
polynomial basis functions are utilized for the first
time, to the best of our knowledge.
With regard to the current investigation, we take
into consideration the NDE having variable delays of
this kind
2
P
m=0
1
P
n=0
Pmn(ν)χ(m)(ν−¯hmn(ν))+
2
P
r=0
r
P
s=0
Qrs(ν)χ(r)(ν)χ(s)(ν) = h(ν).
(1)
subject to the given initial conditions (ICs)
χ(a) = η1and χ0(b) = η2,(2)
where χ(ν)is the unknown function, Pmn(ν), Qrs(ν)
and h(ν)are assumed to be continuous on the given
domain 0≤a≤ν≤b. Also, the variable de-
lays ¯hmn(ν)are assumed to be so. Determining ap-
proximate solution to the considered problem Eq.(1)-
Eq.(2) using the Dejdumrong collocation method is
the objective of this analysis.
2 Dejdumrong Polynomial
Representation
When dealing with polynomials of degree m, one may
express them directly as [49], [50], [51], [52]:
Dm
i(ν) =
(3ν)i(1 −ν)i+3,
for 0≤i < 2−1m−1,
(3ν)i(1 −ν)m−i,
if i=2−1m−1,
2·3i−1(1 −ν)iνi,
if iis even and i= 2−1m,
Dm
m−i(1 −ν),
for 2−1m+ 1 ≤i≤m.
(3)
Definition 1. The monomial matrix of Dejdumrong is
denoted by [52]
N=
n00 n01 · · · · · · n0m
n10 n11 · · · · · · n1m
.
.
..
.
.....
.
.
.
.
..
.
........
.
.
nm0nm1· · · · · · nmm
(m+1)×(m+1)
(4)
where nrs is given as:
nrs =
(−1)(s−r)3rr+ 3
s−r,
for 0≤r≤2−1m−1,
(−1)(s−r)3rm−r
s−r,
for r =2−1m−1,
(−1)(s−r)2(3r−1)r
s−r,
for r = 2−1mand mis even,
(−1)(s−r)3m−rm−r
s−r,
for r =2−1m+ 1,
(−1)(s−m+r−1)3m−km−r
s−m+r−3,
for 2−1m+ 1 ≤r≤m.
(5)
with bν
2crepresents GI ≤νand dν
2erepresents LI ≥
ν, where GI is the greatest integer and LI is smallest
integer.
The following properties are satisfied by the Dej-
dumrong basis function, [53], [54], [55]:
1. The basis function of Dejdumrong is non-
negative, which might be interpreted as,
Dm
i(ν)≥0,∀i= 0,1,· · · , m.
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2. The partitioning of unity, which is,
m
X
i=0
Dm
i(ν) = 1.
3 Basic Matrix Relations
In this subsection, we use the Dejdumrong poly-
nomial to depict the matrix representations of the
problems Eq.(1)-Eq.(2).
Lemma 1. One possible representation of the vector
DN(ν)is as follows:
DN(ν) = T(ν)NT
N,(6)
where T(ν) = 1ν ν2· · · νNand NT
Nis given
in Eq.(4).
Proof. By multiplying the vector T(ν)by the matrix
NT
Nfrom the right side, the vector
DN(ν) = T(ν)NT
Nis acquired.
Lemma 2. The form that may be used to represent
the approximated solution based on the Dejdumrong
polynomial in Eq. (1) is as follows:
χ(ν)∼
=χN(ν) = T(ν)NT
NAN,(7)
where AN= [ a0a1· · · aN]T.
Proof. By multiplying the vector DN(ν) = T(ν)NT
N
by the vector ANfrom the right, the relation Eq.(7) is
found.
Lemma 3. The matrix relations for the kth deriva-
tives of the solution form Eq.(7) are respectively as
follows
χ(k)(ν)∼
=χ(k)
N(ν) = T(ν)ΛkNT
NAN,(8)
where,
Λ =
0 1 0 · · · 0
0 0 2 · · · 0
.
.
..
.
..
.
.....
.
.
0
0
0
0
0
0
· · ·
· · ·
N
0
.
Proof. When the kth derivative of Eq.(7) are taken,
we have
χ(k)(ν)∼
=χ(k)N(ν) = T(k)(ν)NT
NAN,(9)
Next, the kth derivative of T(ν)are taken to obtain
T(k)(ν) = T(ν)Λk.(10)
Therefore, by substituting the corresponding values
of Eq.(9) with the matrix relations Eq.(10) for the kth
derivative of the solution form Eq.(7)we get the de-
sired results.
Lemma 4. The corresponding matrix relations for
the proportional delay of the derivatives of the solu-
tion form Eq.(7) are as follows:
χ(k)(ν−¯hmn(ν)) ∼
=χ(k)N(ν−¯hmn(ν))
=T(ν)ΩN(−¯hmn(ν))ΛkNT
NAN,
where
ΩN(η) =
0
0(−¯hmn(ν))01
0(−¯hmn(ν))1
0
.
.
.
01
1(−¯hmn(ν))0
.
.
.
0
· · · N
0(−¯hmn(ν))N
· · ·
...
· · · N
1(−¯hmn(ν))N−1
.
.
.
N
N(−¯hmn(ν))0
.
Proof. If ν−¯hmn(ν)is written instead of νin Eq.(8),
then it is achieved
χ(k)(ν−¯hmn(ν)) ∼
=χ(k)
N(ν−¯hmn(ν))
=T(ν)ΩN(−¯hmn(ν))ΛkNT
NAN,
(11)
By multiplying the vector T(ν−¯hmn(ν)) by the vec-
tor ΩN(−¯hmn(ν)) from the right side, on the other
hand, gives us the result of
TN(ν−¯hmn(ν)) = TN(ν)ΩN(−¯hmn(ν)).
In addition, by using Eq. (8) we can obtain the matrix
forms of χ(r)(ν)χ(s)(ν)as:
χ(r)(ν)χ(s)(ν) = T(ν)NT
NΛrANT(ν)ΛsNT
N.(12)
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where
T(ν) =
T(ν) 0 · · · 0
0T(ν)· · · .
.
.
.
.
..
.
.....
.
.
0 0 · · · T(ν)
,
NT
N=
NT
N0· · · 0
0NT
N· · · .
.
.
.
.
..
.
.....
.
.
0 0 · · · N T
N
,
Vr=
Vr0· · · 0
0Vr· · · .
.
.
.
.
..
.
.....
.
.
0 0 · · · Vr
,
AN=
AN0· · · 0
0AN· · · .
.
.
.
.
..
.
.....
.
.
0 0 · · · AN
.
By substituting Eq.(11) and Eq.(12) in Eq.(1) yields
2
P
m=0
1
P
n=0
Pmn(ν)T(ν)ΩN(−¯hmn(ν))ΛkNT
NAN+
2
P
r=0
r
P
s=0
Qrs(ν)T(ν)NT
NΛrANT(ν)ΛsNT
NAN=
H(ν)
(13)
As a result, the following set of adjustable collocation
points is necessary to solve Eq.(13)
νk=1
2−1
2cos kπ
N, k = 0,1,2, . . . , N. (14)
When these collocation points are substituted into the
Eq.(13) one may derive as
2
P
m=0
1
P
n=0
Pmn(νk)T(νk)ΩN(−¯hmn(νk))ΛkNT
NAN+
2
P
r=0
r
P
s=0
Qrs(νk)T(νk)NT
NΛrANT(νk)ΛsNT
NAN=
H(νk), k = 0,1, ..., N.
Or simply,
2
P
m=0
1
P
n=0
PmnTΩNΛkNT
NAN+
2
P
r=0
r
P
s=0
QrsT V rNT
NANT V sNT
NAN=
H, k = 0,1, ..., N.
(15)
where
Pmn =
Pmn(ν0) 0 · · · 0
0Pmn(ν1)· · · 0
.
.
..
.
.....
.
.
0 0 · · · Pmn(νN)
,
Qrs =
Qrs(ν0) 0 · · · 0
0Qrs(ν1)· · · 0
.
.
..
.
.....
.
.
0 0 · · · Qrs(νN)
,
H=
h(ν0)
h(ν1)
.
.
.
h(νN)
, AN=
a0
a1
.
.
.
aN
, AN=
AN
AN
.
.
.
AN
,
ΩN=
ΩN(−¯hmn(ν0))
0
.
.
.
0
0
ΩN(−¯hmn((ν1))
.
.
.
0
· · ·
· · ·
...
0
0
0
.
.
.
ΩN(−¯hmn(νN))
T=
T(ν0)
T(ν1)
.
.
.
T(νN)
,NT
N=
NT
N0· · · 0
0NT
N· · · 0
.
.
..
.
.....
.
.
0 0 · · · N T
N
and T=
T(ν0) 0 · · · 0
0T(ν1)· · · 0
.
.
..
.
.....
.
.
0 0 · · · T(νN)
.
By using Eq. (33) in Eq. (32) we obtain
W A =Hor [W;H],(16)
where
W=
2
P
m=0
1
P
n=0
PmnTΩNΛkNT
NAN+
2
P
r=0
r
P
s=0
QrsTΛrNT
NANTΛsNT
NAN.
The initial form of system Eq.(2) is provided using the
matrix for
U1=χ(a) = η1,orT(a)NNA=η1,
U2=χ0(b) = η2,orT(b)ΛNNA=η2.(17)
Finally, the ICs mentioned in equation Eq.(17) have
been substituted in the final three rows using the aug-
mented matrix structure given in equation Eq.(16).
This process will provide a new augmented form as:
f
W A =e
Hor
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hf
W;e
Hi=
w0,0
w1,0
.
.
.
wN−2,0
U1,0
U2,0
w0,1
w1,2
.
.
.
wN−2,1
U1,1
U2,1
· · ·
· · ·
...
· · ·
· · ·
· · ·
w0,N
w1,N
.
.
.
wN−2,N
U1,N
U2,N
;
;
;
;
;
;
h(τ0)
h(τ1)
.
.
.
h(τN−2)
η1
η2
.
If rank f
W=rank hf
W;e
Hi=N+ 1,then it could
be conclude that AN=f
W−1e
Hwhere the coeffi-
cient matrix of the Dejdumrong polynomial Eq.(7) is
denoted by AN.Consequently, the solution to Eq.(1)
has been found.
4 Errors Analysis
In this part, we will give the error analysis that was
performed on the method that was employed. The is-
sue will be given a residual correction process, which
will attempt to provide an estimate of the absolute in-
accuracy.
Let χN(ν)andχ(ν)represent, respectively, the ap-
proximate and exact solutions to Eq.(1). For the pur-
pose of estimating the error analysis, the subsequent
process known as residual correction might be used.
First, let’s do some addition and subtraction on the
term
<N=
2
P
m=0
1
P
n=0
Pmn(ν)χ(m)
N(ν−¯hmn(ν))+
2
P
r=0
r
P
s=0
Qrs(ν)χ(r)
N(ν)χ(s)
N(ν)−h(ν)
to Eq.(1) produce the subsequent differential equation
2
P
m=0
1
P
n=0
Pmn(ν)e(m)
N(ν−¯hmn(ν))+
2
P
r=0
r
P
s=0
Qrs(ν)e(r)
N(ν)e(s)
N(ν) = h(ν)− <N,
(18)
with the ICs
eN(0) = 0,deN(0)
dν = 0.
where eN=χ(ν)−χN(ν).For a given value Mlet
e•(ν)be the approximate solution of Eq.(18), where
is a polynomials degree of Eq.(18) and M≥N.
Theorem 1. Assume that χN(ν)represents the ap-
proximate solution of equation (1) whereas e•(ν)is
the approximate solution of equation (18). Further-
more, χN(ν) + e•(ν)may be considered an approxi-
mation solution to equation (1) with an error function
denoted as eN(ν)−e•(ν).
We term the approximate solution χN(ν) + e•(ν)
as the corrected approximate solution. Note that if
keN(ν)−e•
Mk< ε, thereafter, e•
M.might be used
to estimate the AE. Moreover, if keN(ν)−e•
Mk<
kχ(ν)−χN(ν)k,then χN(ν) + e•
Mis a more accu-
rate solution than χN(ν)in any given norm.
Theorem 2. Consider χ(ν)and χN(ν) =
T(ν)NT
NANrepresent the exact solution and
the Dejdumrong polynomial of the Eq.(1) with a
given degree N. Furthermore, we make the assump-
tion that χNM (ν) = T(ν)b
Ais the expansion of the
generalised Maclaurin series [56] of χN(ν)with
degree of N. Consequently, the AE of the polyno-
mial solution χNM (ν)for Dejdumrong polynomial
solution is constrained as
kχ(ν)−χN(ν)k∞≤χ(N+1)(0)
(N+1)!
χ(N+1)(ζ)
+
κN
b
A
∞
+
NT
N
∞kAk∞,
ν∈[0, b].
Proof. To begin, for proving the above theorem, we
use the same procedure as in [56]. we may extract
the following formula from the Maclaurin expansion
χNM (ν)with degrees Nby adding and subtracting
from the triangle inequality:
kχ(ν)−χN(ν)k∞=kχ(ν)−χNM (ν)+
χNM (ν)−χN(ν)k∞
≤ kχ(ν)−χNM (ν)k∞+
kχNM (ν)−χN(ν)k∞.
(19)
From Eq.(6), the Dejdumrong polynomial solution
χN(ν) = DN(ν)Ais possible to be expressed us-
ing the matrix form χN(ν) = T(ν)NT
NANand
χNM (ν) = T(ν)b
Ais the truncated Maclaurin series
of χ(ν)having degree N, we can write
kχNM (ν)−χN(ν)k∞=
T(ν)b
A− N T
NA
∞
≤ kT(ν)k∞
b
A
∞
+
NT
N
∞kAk∞,
ν∈[0, b].
(20)
Since ν∈[0, b],, then the inequality will be given as
kT(ν)k∞≤max bN,1=κN.Therefore, we are
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able to arrange Equation Eq.(20) as
kχNM (ν)−χN(ν)k∞≤κN
b
A
∞
+
NT
N
∞kAk∞.
(21)
Conversely, it is understood that the residual term of
the Maclaurin polynomial χNM (ν),which has a de-
gree of Nis
∞
P
n=N+1
χ(n)(0)
n!νnSo, we can write
kχ(ν)−χNM (ν)k∞≤
∞
X
n=N+1
χ(n)(0)
n!νn,
ν∈[0, b].
(22)
Then, by using Eq.(19), Eq.(21) and Eq.(22), we ob-
tain
kχ(ν)−χN(ν)k∞≤
∞
X
n=N+1
χ(n)(0)
n!τn+
κN
b
A
∞
+
NT
N
∞kAk∞
, ν ∈[0, b].
(23)
Given the existence of ζ∈(0, b)such that
∞
P
n=N+1
χ(n)(0)
n!νn=νN+1
(N+1)! uN+1(ζ), ν ∈[0, b],in the
In the residual term of Taylor’s Theorem, the inequal-
ity (23) may be represented as
kχ(ν)−χN(ν)k∞≤χ(N+1)(0)
(N+ 1)!
χ(N+1)(ζ)
+
κN
b
A
∞
+
NT
N
∞kAk∞
, ν ∈[0, b].
(24)
Hence, it may be concluded that the proof of the the-
orem has been completed.
Theorem 3. The Eq.(25) provides the convergence
condition of the Dejdumrong polynomial solution
χN(ν) = T(ν)NT
NANunder the supposition that the
maximal error in the interval 0≤ν≤bis, in fact,
equal to the upper bound Eq.(25) which is defined in
Theorem 2
∆e
AN+
NT
N
∞∆AN<bN+1
kN(N+ 1)!χN+1(0),
(25)
where χN(τ)is the Dejdumrong polynomial solution,
and its coefficient matrix is represented by AN. The
coefficient matrix e
ANrepresents the coefficients in
the generalized Maclaurin polynomial of χ(τ)with
degrees N, κN=max 1, bN. The delta operator
is defined as: κN=max 1, bNdegree e
AN,while
the definition of the delta operator is
∆e
AN=
e
AN+1
∞
−
e
AN
∞
.
Proof. The same approach is used to establish the
above theorem as in [56].The hypothesis of the theo-
rem supposes that the maximum error is equivalent to
its upper bound, which is stated in Theorem 2 Based
on the results of Theorem 2, the maximum errors for
χN(ν)and χN+1(ν)may be stated as
Emax
N=
∞
X
n=N+1
χ(n)(0)
n!bn+
κN
b
AN
∞
+
NT
N
∞kANk∞
and
Emax
N+1 =
∞
X
n=N+2
χ(n)(0)
n!bn+
κN+1
b
AN+1
∞
+
NT
N+1
∞kAN+1k∞.
To ensure that the solution χN(ν)converges, we want
to identify the condition under which Emax
N+1 < Emax
N
holds. Subsequently, we have the ability to write
Emax
N+1 −Emax
N=−χ(N)(0)
N!bN+κN+1
b
AN+1
∞
+
NT
N+1
∞kAN+1k∞
NT
N+1
∞kAN+1k∞
−κN
b
AN
∞
+
NT
N
∞kANk∞<0.
We are also aware that κN< κN+1.Then we get
κN
b
AN+1
∞
+
NT
N+1
∞kAN+1k∞−
b
AN
∞
+
NT
N
∞kANk∞
<
χ(N+1)(0)
(N+ 1)! bN+1.
Based on the fact that
NT
N
∞<
NT
N+1
∞,here is
an example of an inequality that we may establish:
κN
b
AN+1
∞
−
b
AN
∞
+
kNN+1k∞(kAN+1k∞− kANk∞))
<χ(N+1)(0)
(N+ 1)! bN+1.
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Nawal Shirawia, Ahmed Kherd,
Salim Bamsaoud, Mohammad A. Tashtoush,
Ali F. Jassar, Emad A. Az-Zo’Bi
E-ISSN: 2224-2880
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Volume 23, 2024
Here, by utilizing the operators ∆AN=kAN+1k∞−
kANk∞.and ∆e
AN=
e
AN+1
∞
−
e
AN
∞
,we ob-
tain
∆e
AN+
NT
N+1
∞∆AN<χ(N+1)(0)
κN(N+ 1)!bN+1.
Consequently, it could be noted that the proof has
been completed.
.
5 Application
All the approaches discussed in Section3 and
Section4 are now being evaluated using three dif-
ferent examples. Both tables and graphs are used to
display the findings that were obtained. Additionally,
comparisons are done with other methodologies that
have been published in the literature. All of the
results were computed with the help of MATLAB
R2021a.
In this study, the symbol χ(ν)denotes the exact
solution, χN(ν)corresponds to the Dejdumrong
polynomial solution, χN,M (ν)denotes the enhanced
estimated solution, |eN(ν)|symbolizes the function
of the real error analysis, |eN,M (ν)|stands for the
function of the estimated AE and |EN,M (ν)|symbol-
izes the function of the enhanced absolute error.
Example 1. We start by thinking about the nonlinear
differential equation of second order with variable
delays ν3, ν2and ν
2[53],[54]
χ00(ν−ν3)+χ(ν−ν2)−χ(ν+ν
2)−νχ0(ν)2=h(ν),
(26)
with the ICs
χ(0) = 1, χ0(0) = 1.(27)
where h(ν) = (ν−ν2)2
2−ν(1 + ν)2−ν
2−17
8ν2+1.The
exact solution of Eq.(26) under the conditions Eq.(27)
is χ(ν) = ν2
2+ν+ 1.
Now, let us investigate the solution of the equations
Eq.(26)-Eq.(27) for the case when N= 2 in the form
χ2(ν) =
2
X
r=0
aiDN(ν).(28)
For N= 2, the collocation points are ν0= 0, ν1=1
2
and ν= 1.According to the method in Section 3,
from Eq.(15) the fundamental matrix equation be-
comes
T1Λ2NT
N+T2NT
N−T3NT
N−N0TNT
NAT NT
N=H
where
T1="1 0 0
1 3/8 9/64
1 0 0 #, T2="1 0 0
1 1/4 1/16
1 0 0 #,
T3="1 0 0
1 3/4 9/16
1 3/2 9/4 #,NT="1 0 0
−220
1−2 1 #,
Λ2="002
000
000#, Q00 ="000
0 1/2 0
001#,
T="1000 0 0 000
00011/21/4000
0000 0 0 111#,
NT
N=
NT
N0 0
0NT
N0
0 0 NT
N
, AN="a0
a1
a2#,
A="AN0 0
0AN0
0 0 AN#, H ="1
−7/8
−45/8 #.
Therefore the augmented matrix is
[W;H] =
2
−a0
16 −a1
8−a2
16 −4
−5
2
−4 2 ; 1
5
2−a1
16 −a2
32 −a0
32
3
2−a1
16 −a2
32 −a0
32 ;−7
8
11
4−a2−1
4;−45
8
(29)
Also, the matrix representations of the conditions
Eq.(27)are as follows:
U0= [ 100], U1= [ −220].
Replace the first and the last rows of Eq.(29) we ob-
tain
W;H=
1 0
a2
2−a0
2+5
2−4
−2 2
0 ; 1
a0
2−a2
2+3
2;−7
8
0 ; 1
(30)
The Matlab program was used to solve the ob-
tained system and so the Dejdumrong coefficients
matrix have been calculated as A= [1,3/2,5/2]T.
By substituting this coefficients matrix Ain
Eq. (28), we get the approximate solution as
χ(ν) = 0.5ν2+ν+ 1.0,considered to be the exact
solution.
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Ali F. Jassar, Emad A. Az-Zo’Bi
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Example 2 For the second example, let us take the
second-order nonlinear differential equation with
variable delay ν−ν3/8,[53],[54]
χ00(ν) + χ(ν3/8) + 2χ(ν)−χ2(ν) = h(ν), ν ∈[0,1]
(31)
and the ICs
χ(0) = 0, χ0(0) = 1.(32)
where h(ν) = sin(ν3/8) + sin(ν)−sin2(ν).
In Table 1 (Appendix), the actual absolute errors, the
estimated absolute errors and the improved absolute
errors are given. According to Table 1 (Appendix),
we can infer three important conclusions. The first
important result is that the errors decrease as the
value of increases. The second important conse-
quence is that the results of the estimated absolute
errors are quite close to the results of the actual
absolute errors. From this result, it can be said that
the error estimation method described in Section 4
is effective. The final important result is that the
improved absolute errors yield better results than
the actual absolute errors at most points in the given
range. From this result, it can be concluded that the
technique of improving approximate solutions based
on the residual function is effective. However, Table
2 (Appendix) displays Example 2’s absolute error
for several values of N. We observe that increasing
the value of Nyields an approximate solution that
approaches the exact solution..
Table provides a presentation of the expected ab-
solute errors, the actual absolute errors, and the
improved absolute errors. Based on the data shown
in Table 1 (Appendix), we may deduce three sig-
nificant conclusions. The first significant finding is
that the errors decrease as the value of N grows. It is
also crucial to note that the results of the estimated
absolute errors are pretty similar to the results of the
real absolute errors. This is the second significant
consequence. Based on this outcome, it can be con-
cluded that the error estimate approach outlined in
Section 4 is highly effective. The ultimate significant
outcome is that the enhanced absolute errors provide
better results compared to the current absolute errors
at the majority of points in the given domine [0,1].
Based on this outcome, it can be inferred that the
approach of enhancing approximation solutions using
the residual function is efficacious. Yet the AE of
example 2 for multiple values is presented in Table.
It is clear from exploring Table that as the value of N
is raised, an approximation solution is achieved that
is a pretty close approximation to the exact solution.
Example 3 The third instance pertains to a
second-order nonlinear differential equation that
incorporates variable delays ν2and −ν/2.
Fig.1: Approximate and exact solutions for Exam-
ple 3.
Fig.2: The AE for Example 3 with several values
of N.
χ00(ν) + χ0(ν−ν2)−ν2χ(ν+ν/2)−
χ0(ν)χ(ν)+(χ0(ν))2=h(ν),0≤ν≤1(33)
and the initial conditions
χ(0) = 1, χ0(0) = 1.(34)
where h(ν) = eν+eν−ν2−ν2e3ν/2.
The analytical solution of this problem is χ(ν) =
eν.We solve the problem for several values of NFor
the values of N= 4,5,7and 9,the absolute errors
are shown in Table 3 (Appendix). It is clear that even
with N= 4, one may get an accuracy of up to 4 dec-
imal places. As Nincreases, the AE lowers for ev-
ery single collecting point. Figure1 displays the an-
alytical and approximate solution for the case where
N= 6.The approximate solution is in close agree-
ment with the exact solution.
Figure 2shows a comparison of the actual absolute
error functions for N= 5, N = 7, and N= 11.Con-
sequently, choosing a greater numerical value of N
leads to a more precise outcome. The nonlinear dif-
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DOI: 10.37394/23206.2024.23.49
Nawal Shirawia, Ahmed Kherd,
Salim Bamsaoud, Mohammad A. Tashtoush,
Ali F. Jassar, Emad A. Az-Zo’Bi
E-ISSN: 2224-2880
474
Volume 23, 2024
ferential equations with variable delays were reported
to be solved numerically using the Pell-Lucase and
Lucase basis functions with the collection method.
[53] and [54], in comparison with the present method,
the Dejdamrong polynomial as a basis function, have
given better results for absolution errors (Table 1 (Ap-
pendix) and Table 2 (Appendix)).
6 Conclusions
This study introduces a matrix approach that utilizes
the Dejdumrong polynomial to solve functional dif-
ferential equations of the pantograph type. These
equations include hybrid delays that are both propor-
tional and variable. In order to figure out the AE, the
function of the residual errors are developed for these
sorts of equations. Furthermore, the text specifies the
execution of aforementioned technique and the pro-
cesses for error analysis on specific problems. Upon
examination of the difficulties, it becomes evident
that the Dejdumrong polynomial coefficients may be
readily obtained by the use of a computer program im-
plemented in Matlab R2021a. If the truncation limit
Nis raised, it is possible to see that approximate so-
lutions become more similar to the precise solutions.
This is shown by the numerical results. Addition-
ally, the method may be adapted to work with vari-
ous kinds of equations and systems by making a few
adjustments to it.
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WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.49
Nawal Shirawia, Ahmed Kherd,
Salim Bamsaoud, Mohammad A. Tashtoush,
Ali F. Jassar, Emad A. Az-Zo’Bi
E-ISSN: 2224-2880
478
Volume 23, 2024
Table 1.Comparing the absolute errors, estimate, and absolute errors for the improved of the problem Eq.30 &
Eq.32 for (N, M)= (4, 5), (4, 6), (7, 8), (7, 9), (10, 11), and (10, 12).
νiAbsolute errors Estimated errors Absolute errors for the improved
|e5| |e5,6| |e5,7| |E5,6| |E5,7|
0.0 0.0000e+00 2.3798e-37 3.8968e-37 2.3798e-37 3.8968e-37
0.2 9.5303e-09 3.9484e-09 9.2624e-09 5.5820e-09 2.6789e-10
0.4 3.4432e-07 3.6505e-07 3.4282e-07 2.0725e-08 1.5033e-09
0.6 2.4805e-07 3.3208e-07 2.5695e-07 8.4037e-08 8.9049e-09
0.8 1.0733e-06 8.8959e-07 1.0756e-06 1.8370e-07 2.2670e-09
1.0 2.8254e-05 2.5584e-05 2.8333e-05 2.6699e-06 7.8980e-08
νi|e8| |e8,9| |e8,10| |E8,9| |e8,10|
0.0 0.0000e+00 1.0269e-40 0.0000e+00 1.0269e-40 0.0000e+00
0.2 1.0138e-11 1.0095e-11 1.0130e-11 4.3173e-14 8.9848e-15
0.4 2.3723e-11 2.3849e-11 2.3714e-11 1.2620e-13 8.3870e-15
0.6 6.0895e-11 6.1364e-11 6.0845e-11 4.6956e-13 5.0386e-14
0.8 7.8434e-11 8.0852e-11 7.9129e-11 2.4177e-12 6.9453e-13
1.0 3.1914e-09 3.2434e-09 3.1979e-09 5.1977e-11 6.4979e-12
Table 2.Comparison of the absolute errors for of Example 2 in [53]
νiRef. [53] PM
|e4| |e7| |e10| |e4| |e7| |e10|
0.0 0.0000e+00 6.3527e-22 1.2914e-22 0.0000e+00 0.0000e+00 0.0000e+00
0.2 1.3230e-05 7.2149e-10 1.4211e-13 4.4211e-06 1.1507e-10 1.5071e-15
0.4 3.8453e-05 1.4754e-09 2.8422e-13 1.3931e-05 2.6240e-10 1.7120e-14
0.6 4.2938e-05 2.1464e-09 2.2737e-13 7.2812e-05 4.6197e-10 2.3005e-14
0.8 2.1963e-04 2.7024e-09 4.5475e-13 8.9964e-05 9.8009e-10 4.1436e-14
1.0 1.2656e-03 1.07390e-07 1.5689e-11 1.2950e-03 4.6524e-08 2.2276e-12
Table 3.Comparison of the absolute errors for for N= 5,7and 9with Ref. [54]with respect to Example 3
νiRef. [54] PM
|e5| |e7| |e9| |e5| |e7| |e9|
0.0 — — — 0.0000e+00 6.5457e-26 0.0000e+00
0.2 2.05e-06 2.79e-09 1.86e-12 3.2417e-08 4.2243e-10 9.2773e-15
0.4 6.22e-06 5.53e-09 3.56e-12 1.5154e-06 1.0791e-09 8.4035e-14
0.6 1.37e-05 8.75e-09 4.95e-12 1.3661e-06 1.1409e-09 4.8594e-14
0.8 1.85e-05 4.84e-09 8.55e-11 2.4782e-06 4.7415e-09 1.6358e-11
1.0 9.09e-05 5.35e-07 4.78e-09 1.2852e-04 4.2872e-07 9.3254e-10
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.49
Nawal Shirawia, Ahmed Kherd,
Salim Bamsaoud, Mohammad A. Tashtoush,
Ali F. Jassar, Emad A. Az-Zo’Bi
E-ISSN: 2224-2880
479
Volume 23, 2024
APPENDIX
