Solution of Multi-Dimensional Non-linear Fractional Differential
Equations of Higher Orders
MARWA MOHAMED ISMAEEL1, WASAN AJEEL AHMOOD2
1Department of Arabic Language,
Al-Iraqia University,
Faculty of Education for Women, Baghdad,
IRAQ
2Department of Al-Quran Science,
Al-Iraqia University,
Faculty of Education for Women, Baghdad,
IRAQ
Abstract: - In our paper, we are used here two methods to solve non-linear differential equations from a higher
order: the first-one is domain decomposition method is used to estimate the Maxi. Abso. Trunc. Error of
Adomain series and the second-one proposed numerical (PNM), these types of equations are studied. When we
use these methods, an exclusive solution will be provided, and the approximate analyses of this method applied
to these types of equations will be overlooked, and the maximum error that has been informed to solve the
ADOMIANS series will be classified. A digital example is prepared clarify the impact method provided and
significant following of these equations in our paper is Bagley-Torvik equation.
Key-Words: - Non-linear fractional differential equation, higher order, adomian decomposition method (ADM),
and proposed numerical method.
Received: December 19, 2022. Revised: August 26, 2023. Accepted: October 2, 2023. Published: October 27, 2023.
1 Introduction
Fractional differential equations are practiced to
sample expansive space of physical problems
including non-linear vacillation of earth shakes,
[1], fluid-dynamic passing (in 1999), [2], and
hesitancy dependent on the waning behavior of
many relativistic materials.
Fractional differential equations that contain
only one fracture derivative are a good
understandable tool, they are often emploied for the
sporty depiction of plenty material procedures, but
they are not eternity enough to reverse all suitable
phenomena
The authors in, [3], investigated strategies for
the numerical solution of the initial value problem
with initial conditions where 0<α1<α2<<αν.
Here denotes the derivative of order
(not necessarily α j ℕ) in the sense of Caputo.
The authors in, [4], returned and expanded
multi-term homogeneous differential equations with
caputo-type derivatives and fixed transactions
through the necessary stability, instability conditions
and stability and caffeine.
The authors in, [5], reviewed two methods of
the most action groups of digital methods of
fractional arrangement problems and discussed
some major mathematical issues such as the
effective treatment of the term continuous memory
and the solution of nonlinear systems participating
in implicit ways.
The use of fractional differential operators in
mathematical models has become increasingly
widespread in recent years. Several forms of
fractional differential equations have been proposed
in standard models, and there has been significant
interest in developing numerical schemes for their
solution the authors in, [6], show how the numerical
approximation of the multi-wheel -term differential
differential formula solution can be calculated by
reducing the problem to a system of regular
differential equations and fracture in each unit in
most unit.
The author described in, [7], two sports ways to
use equations of multi-term and multi-arranged
systems have shown the relationships between these
two concepts. Then examine its most important
analytical characteristics. Finally, he is considerd
the digital methods of its approach solution.
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The author was collected in, [8], with the linear
way to devise the Adams Fource Molton method for
real, non-linear, non-linear equations with a firm
delay or change of time, then use this method to
estimate the late fractures- arrange differential
equations.
The authors in, [9], introduced a new method of
analytical and digital solution of a non-Dynican N-
Range N-Range by the well-known vibration
engineers, that is, the average consensual balance
method.
The authors have suggested at, [10], the
definition of Mittag-Lfler's stability and entered the
direct Lyapunov method. The principle of broken
comparison is presented and the Riemann -Liouville
Fairville system is extended using the Order Caputo
systems.
The authors were presented in, [11], and used
the results of modern stability of broken equations
and methods of analytical types include linear, non-
linear and chronological delay differential
equations. Some of the inferences of regularity are
similar to the inferences of differential equations for
the classic order.
The authors were obtained in, [12], for multi-
term homogeneous differential equations with three
Caputo derivatives and fixed laboratories through
the necessary and sufficient stability of instability
conditions.
In 2013, the authors in, [13], were informed of
the theories of stable point, the presence and
peerless of solutions for non -linear non -non-linear
equations, and presented two examples to clarify the
results.
The authors in, [14], built two new schemes to
solve a numerical solution to the non-linear
differential equation of fractional kinds in one-
dimensions and two-dimension. The scheme-I one
and two-dimension mythical (SLP) uses
fundamental functions while the chart-II uses 1D
and 2D (IBF) basis functions as main functions.
The authors in, [15], presented a simple and
effective analytical algorithm of two steps from two
steps to solve multidimensional times.
2 Formulation of Issue with the
Solution Algorithm
2.1 First method: Adomian decomposition
method
Let
Let nonlinear fractional differential equation:
(1)
Where
and the fractional derivative is,
=
Let f confirmed Lipschitz condition with constant L
such as
which implies that,
The solution algorithm by using the domain
decomposition method is:
(4)
(5)
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Where, Aj are Adomian polynomials of non-linear
which take
the form,
And the solution of the equations (1) and (2) will be,
(7)
Finally,
(8)
3 Convergence Analysis
Consider F is mapping with Banach space E,
All continual functions on J with
N˃ 0.
Theorem 3.1.(Existence and uniqueness): Let f
satisfies the Lipschitz condition
Then, the nonlinear fractional differential equation
has a unique solution
Proof:
Let F: E →E is defined as
(10)
Let
then,
This implies that:
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(14)
Now, we choosen N large enough s.t
,
then, we get:
Therefore, the mapping F is constriction.
Theorem 3.2. (Proof of convergence):
the series solution
Then, by using Adomain decomposition method
converges if , where c is positive no.
Proof:
Let seq. , s.t.
of partial sums
from the series
since,
So, we can write
From equations (4) and (5), we have:
Let Sp and Sq be partial sums and p greater than q,
one can have:
And
Now, the Cauchy sequence {Sp} in this E,
=
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Let p=q+1 then,
Since,
, and
Consequently,
But, , and as then,
and hence, is a Cauchy sequence in Banach
space E so,
convergence.
Theorem 3.3. (Error analysis):
The max. absolute trunc. error of solution equation
(6) to the problem (1) is estimated to be,
if n is odd
And n is even
Proof:
By theorem (3.2), we have
but,
then
So,
So,
(17)
From equation (1), we get
(18)
By using equation (1) and the above equation, we
can get:
from the equation (17), we get:
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If n is odd
(19)
and
If n is even
(20)
This completes the proof.
4 Numerical Examples
Example (4.1), [16]:
The Bagley-Torvik equation
(21)
x(t)=(t+1),
is the valid solution
By using Adomian decomposition method, we will
solve it:
X(t)=x(t)-t-1,
equation (21):
(22)
Applying the Adomian decomposition method gets:
(23)
(24)
From the equation (23) and (24), the solution:
(25)
The solution of equation (22) is:
Finally, the above equation (21) is:
.
is the valid solution.
Example (4.2):
Let non-linear FDE,
(26)
which has x(t)=.
By using the equations (23) and (24), the solution is:
Applying the Adomian decomposition method gets:
Where are Adomain polynomials of non-linear
. Finally, the solution (26):
(27)
Table 1. illustrates the absolute error of the ADM
solution, while Table 2. illustrates the max. absolute
sectioned. The figure shows ADM and exact
solutions (when m=15).
Table 1. Absolute Error
m
max. error (N=5)
5
0.00418546
10
0.000325204
15
0.0000366795
Table 2. Maximum absolute error
m
max. error (N=5)
5
0.0093013
10
0.00050822
15
0.0000277689
4.1 Second Method: Proposed Numerical
Method
This method is a disadvantage to numerail method.
It solve only fractional differential equations with
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initial condition, we get past this shortcoming by
using PNM.
The solution of steps:
Step 1: Use the transform the initial conditions to
homogenous.
By the substitution for eqs. (1) and (2),
(28)
Step 2: Acquire solution algorithm.
By using the next equations:
〗
obtain values of
Step 3: coming back to the main new conditions.
By relevance (28), we receive:
5 Numerical Examples
Example (5.1):
Let non-linear FDE:
(29)
Set
X(t)=x(t)-c,
The equation (27) will be:
(30)
The solution algorithm of equation (30) is:
The solution of equation (29) is
Figure 1, shows PNM and ADM solutions (when
m=5, h=0.01).
Fig. 1: PNM and ADM solutions.
Example (5.2):
Let non-linear FDE,
(31)
The exact solution x(t)=.
X(t)=x(t)-t
Equation (31):
(32)
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By algorithm of equation (32) is:
Finally, the solution of equation (31) is
Table 3. illustrates the results passed from PNM and
ADM solutions.
Table 3. Absolute Error at (t=1)
PNM
ADM
h
N
0.1
2
0.01
4
0.001
6
We used two methods to solve FDEs, each
method has an advantage over the other. If the
solution is needed in a narrow interval, ADM is
preferred to be used, as it gives a more accurate
solution but if the solution is needed in a wide
interval, PNM is preferred to be used (see the result
in Table 3). We see that after we overcome the
disadvantage of the numerical method it gives a
more accurate solution than the numerical methods.
6 Conclusion
We used the ADM for solving the non-linear
fractional differential equations, we introduced
some new theorems are give the existence,
uniqueness, convergence, and maximum absolute
truncation error to the Adomian decomposition
method series solution when applied to these
equations. Some numerical examples are discussed
and solved by using the Adomian decomposition
method.
We see from the results that the exact error
coincides with the approximate error obtained from
using the theorems, see for example.
We use a numerical method for comparison, we
see that after we overcome the disadvantage of this
method. In the two methods that we used to solve
fractional differential equations (ADM with
numerical method), each method has an advantage
over the other.
The method is still open for investigation,
especially in fractional differential equations with
higher 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
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 authors have no conflict of interest to declare.
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(Attribution 4.0 International, CC BY 4.0)
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