On the Solution of Multi-Term Nonlinear Partial Derivative Delay
Differential Equations
WASAN AJEEL AHMOOD
Department of Al-Quran Science,
Al-Iraqia University,
Faculty of Education for Women, Baghdad,
IRAQ
MARWA MOHAMED ISMAEEL
Department of Arabic Language,
Al-Iraqia University,
Faculty of Education for Women, Baghdad,
IRAQ
Abstract: - In our paper, we used the Adomian decomposition method to solve multi-term nonlinear delay
differential equations of partial derivative order, these types of equations are studied. When we used this
method, the being of an exclusive solution will be provided, approximate analytics of this method applied to
these kinds of equations will be disputed, and the maximum absolute briefed error of the Adomians series
solution will be rated. A digital example is made ready to make clear the effectiveness of the offered method.
Key-Words:- Non-linear Delay Differential Equation, Integer Order, Adomian decomposition method, and
convergence analysis.
Received: October 25, 2021. Revised: September 20, 2022. Accepted: October 22, 2022. Published: November 25, 2022.
1 Introduction
Fractional differential equations are practiced to
sample expansive space of physical problems
including non-linear vacillation of earth shakes
[4], fluid-dynamic passing (in 1999), [5], and
hesitancy dependent on the waning behavior of
many relativistic materials. Differential equations of
integer orders and delay differential equations have
many applications in engineering and science,
including electrical networks, fluid flow, control
theory, fractals theory, electromagnetic theory,
viscoelasticity, potential theory, chemistry, biology,
visual and neural network systems with delayed
feedback, see [14] and [15].
The authors in [1], [2] and the author in [3]
examined the qualitative behavior of the delay
differential equation of the form:
Where k is a given function and q is greater than 0,
and investigated the solution of this equation by
expanding them into the Dirichlet series.
Delay differential equations were originally
introduced in the 18th century by Laplace and
Condorcet and the authors in [6] presented Lambert
functions to obtain the complete solution for
systems of delay differential equations.
The authors in [7] used the Legendre-Pseudospectra
method to find the exact and approximate solutions
of the fractional-order delay differential equations.
The author in [8] collected with the linear
interpolation method to devise Adams Bash forth-
Moulton method for non-linear fractional positive
real number differential equations with well-
established or time-changing delay, then employed
this method to approximate the delayed fractional-
order differential equations.
The authors in [9] converted the fractional delay
differential equation to the fractional non-delay
differential equation and then applied the Hermite
wavelet method by utilizing the method of steps on
the obtained fractional non-delay differential
equation to find the solution.
The authors in [10] used the Laguerre Wavelets
method and combined it with the stages method to
solve linear and nonlinear delay differential
equations of fractional order.
The authors in [11] gave the digital solution of A
range of fractional delay differential equations,
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS
DOI: 10.37394/232011.2022.17.21
Wasan Ajeel Ahmood,
Marwa Mohamed Ismaeel
E-ISSN: 2224-3429
166
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comparative between higher absolute errors for the
suggested method and the results obtained by Haar
wavelet.
The authors in [12] presented a new method
(Gegenbauer Wavelets steps method) for solving
non-linear fractional delay differential equations by
using two methods: Gegenbauer polynomials and
method of steps. They converted the fractional non-
linear fractional delay differential equation into a
fractional non-linear differential equation and
applied the Gegenbauer wavelet method at each
iteration of the fractional differential equation to
find the solution.
In 2019, the authors in [13] used the spiritual
collocation method for solving fractional order
delay-differential equations by Chebyshev
operational matrix.
2 Formulation of the Issue with the
Solution Algorithm
Let
and consider the nonlinear delay differential
equation:
(1)
Where [0,T] and f satisfies
Lipschitz condition with Lipschitz constant k such
as:
which implies that,
Operating with L-1 to both sides of equation (1),
where
we get
(4)
The solution algorithm of equation (4) using the
domain decomposition method is:
Where, Aq are Adomian polynomials of the non-
linear term
taken the form,
Now, the solution of the equations (1) and (2) will
be,
(6)
3 The Issue with the Partial Order
Solution Algorithm
Let
and consider the Partial delay differential equation:
(7)
(8)
Where f satisfies Lipschitz condition h and x(t) 2 [0;
T] such as:
(9)
which implies that for,
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By using L−1 to both sides of the above first
equation with the solution algorithm using the
domain decomposition method, we get:
(11)
Anywhere, Aq are Adomian polynomials of the
non-linear idiom f (t, x(t – α1), x(t – α2),…, x(t – αn))
taken the form:
)
(12)
Now, the solution of the above first and second
equations will be:
(13)
4 Convergence analysis
Existence and uniqueness theorem
Define the mapping M: E →E where M is the
Banach space
The space of all continuous functions on J with the
norm N˃ 0.
Theorem 4.1. Let f satisfy the
Then, the nonlinear partial derivative delay
differential equation has a unique solution
for i=1,2,..,n. Where E is the Banach space
Proof:
Consider the mapping M: E →E is defined as
Let
then,
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This implies that:
(18)
Where nk/N is less than 1, we get:
Therefore, M there exists a unique solution and
contraction.
Theorem 4.2. the series solution
of the nonlinear delay differential equation
converges by using domain decomposition method
if │x1(t)│ is less than c, when c is any constant.
Proof
Define the sequence of partial sums from
(21)
So, we can write
(22)
Where Ai are a domain polynomials of the non-
linear idiom f(t,Sp(t − α1),…,Sp(t − αn))
taken the form,
(23)
The sequences Sp and Sq be to despotic partial sums
with p is greater than q, one can get
(24)
We are going to verify that, the Cauchy sequence
{Sp} in Banach space E,
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(25)
(26)
(28)
When p=q+1, J=[0,T], β=nk/N, one can get:
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Now, by triangle disparity, we can get:
Consequently,
(31)
When │x1│less than costant then,
And hence, {Sp} is a Cauchy sequence in Banach
space so,
converges.
Theorem 4.3.
The maximum absolute error of the series solution
to the nonlinear delay differential equation is
conjectured to be,
(32)
Proof
By the above theorem, we have
(33)
But,
(34)
When
So
(35)
so, the perfect error in J is
(36)
Theorem 4.4. The solution of the nonlinear partial
derivative delay differential equation
is uniformly stable
Proof
Let x(t1,t2,…,tn) be a solution of
And let
be a solution to the above problem such that,
Then
(38)
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(39)
Therefore, if
Less than , then
(40)
Which completes the proof.
5 Numerical Examples
Example 5.1
Consider the following nonlinear delay differential
equation,
(41)
x(t)=0.1, t
(42)
which has the exact solution (t+0.1).
Applying the Adomian decomposition method to
above (41) and (42), we have:
(43)
Where Ai are Adomian polynomials of the nonlinear
term x2(t-0.1). From the above equations, the
solution of problem x(t)=0.1, t≤0 is:
x(t)=
(44)
Table [1] shows the absolute error of Adomian
decomposition method series solution at different
values of m when t=1, while table [2] shows the
maximum absolute truncated error using theorem
4.3, when t=1, N=2. Fig [1]: shows ADM and exact
solutions (when m=20).
Table 1. Absolute Error
m
5
0.000238942
10
3.03189×10-6
15
3.96516×10-7
20
2.929×10-8
Table 2. Max. absolute error
m
max. error
5
0.0136826
10
0.000688625
15
0.0000346574
20
1.74425×10-6
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Fig. 1: ADM and exact solutions.
Example 5.2
Consider the logistic delay differential equation with
two different delays,
(45)
x(t)=x2, t ≤ 0.
The above equation can be written as
(46)
Applying ADM to the above equation, we have
(47)
(48)
Where Ai are Adomian polynomials of the
nonlinear term
From equations (47) and (48), the first six terms of
the series solution considering the following two
cases:
Case 1:
(When
x(t)=0.5+0.03164+0.00002565t2-
0.00004218t3+1.11262×10-7t4+6.35783×10-8t5. (49)
Table [3] shows the maximum absolute truncated
error at different values of m at t=10 and N=5,10.
Case 2:
(When
x(t)=0.25+0.00374962t+0.0000187547t2-
3.11281×10-8t3-7.80902×10-10t4+2.03125×10-12t5.
(50)
Table 3. Maximum absolute error
m
max. error (N=5)
max. error (N=10)
5
9.91686×10-7
2.97291×10-8
10
2.9009×10-12
2.71763×10-15
15
8.48579×10-18
2.48427×10-22
20
1.24114×10-23
1.13548×10-29
Fig. 2: ADM solutions.
6 Conclusion
In this paper, the Adomian decomposition method is
an interesting and powerful tool when applied to
different kinds of equations. Here we used it to
solve the non-linear multi-term Partial derivative
delay differential equation with new theorems
introduced which give the sufficient conditions of
existence, uniqueness, convergence, and estimations
of the maximum absolute truncation error to
Adomian decomposition method series solution
when applied to these equations. Some numerical
examples are discussed and solved by using the
Adomian decomposition method.
The method has given an analytical solution that is
still open for investigation, especially in Partial
derivative delay differential equations with arbitrary
orders.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
Wasan Ajeel: Theorems, examples, and
methodology
Marwa Mohamed: Investigation and writing
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