New Two Parameter Integral Transform “MAHA Transform”
and its Applications in Real Life
MAHA ALSAOUDI1, GHARIB GHARIB2, EMAD KUFFI3, AHLAM GUIATNI4*
1 Applied Science Private University, JORDAN.
2 Zarqa University, JORDAN.
3Al-Qadisiyah University, IRAQ.
4*The University of Jijel, ALGERIA.
Abstract: - In this paper, an integral transform with two parameters was proposed, and this transform was
applied to obtain the exact solutions for the linear ordinary differential equations with constant coefficients of
higher orders. This integral transform was also applied and we called it Maha transform in nuclear physics and
for medical applications.
Key-Words: - MAHA transform, Inverse MAHA transform, Integral transform, Ordinary differential equations,
coefficients of higher orders, Integral transform for two parameters.
Received: June 5, 2024. Revised: July 31, 2024. Accepted: August 17, 2024. Published: September 2, 2024.
1. Introduction
Emad A., Ku_ and Maha Al-Soudi are gotten
from the old-style Laplace transform in dispensable.
In light of the numerical effortlessness of the
MAHA transform and its principal properties.
MAHA transform was introduced by Emad A.,
Kuffi, Maha to work with the way toward
addressing common and halfway differential
equations in the time area. Regularly, Fourier,
Laplace, Elzaki, Aboodh, Mohanad, Al-Zughair,
Kamal, Mahgoub and SEE transforms are the
helpful numerical instruments for addressing
differential equations, likewise MAHA transform
and a portion of its crucial properties are utilized to
tackle differential equations.
A new integral transform said to be MAHA
transform characterized for capacity of out-standing
request we think about capacities in the set A
characterized by:
A
=
{
f
(
t
)
:
ther
e
exist
M
,
l
1
,
l
2
>
0
.
|
f
(
t
)
|
<
M
e
l
i
|
t
|
,
if
t
∈
(
−
1)
i
x
[0
,
∞
)
}
(1)
For a given capacity in the arrangement of A,
the constant M should be limited number, l1, l2
might be limited or boundless. MAHA fundamental
change signified by the administrator M(.)
characterized by the vital condition
(2)
The parameters u and v in this vital transform is
utilized to figure the variable t the contention of the
capacity f. This necessary transforms has further
association with the Laplace, Aboodh, and Mohanad
transforms. The reason for this examination is to
show the pertinence of this intriguing new transform
and it productivity in tackling the direct differential
conditions.
Through studying and researching integral
transformations and their applications in life,we
found that there are many integral transformations
with two parameters, the most important of which,
according to the year of publication, are ZZ
transformations. Group Ramadan, Shehu, ZMA,
Formable, Khalouta, KKAT, Rishi, SEA, Abaoub-
Shkheam Quideen, Honaiber.
When = 0 the Maha transform becomes Shehu
transform.
When = -1 then Maha transform becomes
KKAT transform and Quideen transform.
In addition, the form of our transformation is the
Maha integral transformation, which is a more
general transformation than the previous
transformations with two parameters.
Through this , which belongs to the set of
integers, we can choose the appropriate number for
the application of physics, engineering, or any life
application to convert the ordinary differential
equation into an easy and simple algebraic equation,
then by taking the inverse of this transform to obtain
the exact solution.
And that is the one who gave this
transformation the generality and the preference by
using it in important applications.
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.56
Maha Alsaoudi, Gharib Gharib, Emad Kuffi, Ahlam Guiatni
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And that this can be referred to as, for
example, the number of samples taken in the
application. There are many mathematical models in
the linear modeling that depend on this , for
example, drug concentration problems, chemical
problems, and others. Through
our study and our knowledge of many integral
transforms with one parameter or two parameters or
more, there is no preference between one transform
and another, but there is an important matter, which
is that each transform has a use for some
applications through after converting differential
equations, integral equation or systems into
algebraic equations
or algebraic system. Conversion is made easier by
simplifying that application or example
([1], [4], [5], [10], [11], [13], [15], [16])
2. Methodology
2.1 MAHA Integral Transform of Some
Functions
For any capacity f(t), we accept that the
fundamental condition (2) exists. The adequate
conditions for the presence of MAHA integral
transform are that f(t) for t ≥ 0 be piecewise
ceaseless and of remarkable request, in any case
MAHA integral transform could conceivably
exists.
In this segment we find MAHA integral
transform of basic capacities:
1) If f(t) = k, where k is a constant function,
then by the definition we have:
2) If f(t) = t then:
Integration by parts, we get:
Also:
(i)
(ii)
(iii) In general case if n is a positive integer
number, then , and if
n > -1 then ,
where Gamma function.
3) If f(t) = , where a is a constant
number, then:
,
also
4) If f(t) = sin(at), where a is a constant number,
then:
5)
If f(t)
=
cos(at), where a is a constant
number, then:
After simple computations, we get:
6)
If f(t)
=
sinh(at), where a is a constant
number, then:
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After simple computations, we get:
7)
If f(t)
=
cosh(at), where a is a constant
number, then:
8)
Shifting property of MAHA integral
transform
If MAHA integral transform of f(t) is F(u,v),
then MAHA transform of function
is given by
.
Proof
dt
Theorem (2.1)
(i) +
F(u,v)
(ii)
+
F(u,v)
(iii)
+
F(u,v)
(iv)
+
F(u,v)
(iiv)
+
F(u,v)
Proof
(i) By the definition we get:
Integrating by parts, we get:
(ii)
Also, by the definition, we have:
Also, integrating by parts, we get:
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Similarly, the proof of (iii) and (iv).
(iiv) can be confirmation by Mathematical
Induction.
2.2 The Inverse of MAHA Integral
Transform
In this section, we introduce the inverse of
MAHA integral transform of simple functions:
(1)
, where n > 0 integer
number.
, where a is a constant
number.
(4)
(5)
(6)
(7)
(8)
2.3 Application of MAHA Integral
Transform of Ordinary Differential
Equations (ODEs)
As expressed in the presentation of this work,
the MAHA necessary transform can be utilized
as a successful device.
For investigating the essential attributes of a
direct framework represented by the differential
condition in light of introductory conditions.
The accompanying models represent the
utilization of the MAHA necessary transform
in taking care of certain underlying worth issue
depicted by common differential conditions.
Consider the 1
st
order linear Ordinary
differential equation:
Example 1:
Consider the differential equation:
Solution: take MAHA integral transform to this
condition, we get:
So
Take inverse to both sides, we get:
The exact solution:
Example 2:
Find the solution of differential equation:
Solution: take MAHA transform to above
differential condition gives:
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So
+
The inverse MAHA transform of this equation is
simply obtained as:
Example 3:
Consider
the
2
nd
request
di
ff
eren
tial
equation:
Solution: Take MAHA transform to
above differential equation, we get:
So
Now
After simple computations, we get: A=3, B=-
2.
So
The general exact solution is:
Example 4:
Consider the second-order linear
nonhomogeneous request differential equation:
Solution: aince y’ (0) is unknown, let y’(0) = a.
Take MAHA transform of this condition and
utilizing beginning conditions, we have:
So
After simple computations, we get:
Take inverse MAHA transform, then the exact
solution is:
To find a note that
Then we find
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Then
Example 5:
T
ac
kle
the
di
ff
eren
tial
equation
Solution: Take MAHA integral transform of this
differential equation and using the initial
conditions, gives:
So
+
]
After simple computations, we have: A = 2, B
= 4, C = −9
Take inverse transform, we get:
3. Applications
3.1
Maha Integral Transform in
”Nuclear Physics”
The following problem is based on nuclear
physics fundamentals.
Consider the first order linear ordinary
differential equation:
The essential relation ship describing
radioactive decay is given in above equation,
where N(t) during time t denotes the number of
undecayed atoms left in a sample of
radioactive
isotope, and α is the decay
constant. We can apply the Maha integral
transform M , to set:
Therefore
So
Take the inverse to both sides, we get the exact
solution:
This is the proper type of radioactive decay.
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3.2
Blood Glucose Concentration
During continuous intravenous glucose
injection, the concentration of glucose in the
blood is G(t) exceeding the baseline value at
the start of the infusion. The function G(t)
satisfies the initial value problem (I, V, P).
The variables in this equation are k, α and γ,
which respectively represent the constant
velocity of elimination, the rate of infusion,
and the volume in which glucose is
distributed. The Maha integral transform
technique will be utilized to assess the
concentration of glucose present in the blood
stream.
Upon bilateral application of the Maha
integral transform to equation (.), the
resulting
expression is obtained :
(4)
Let M { G(t)}
= F (u, v). By utilizing the
initial value problem (I, V, P) and the
integral
transform outlined in section 2.1. The
equation (4) can be rearranged with the aid of
equation
(3) as:
So
After simple computation and using inverse of
Maha transform to this expression, we get the
concentration of glucose in the blood as:
3.3
Aorta Pressure
The heart’s contraction facilitates the
transportation of blood into the aorta. The
initial value problem is concerned with the aortic
pressure function P(t) as:
(5)
where p(0)
=
p
0
c, k, A and P
0
are constants. The Maha
integral transform technique is utilized to
derive
the pressure in the aorta.
With the
bilateral application of the Maha integral
transform to equation (5). The resulting
expression is:
(6)
By applying the initial value problem and
utilizing the transform outlined in section 2.1,
the rearrangement of equation (6) can be
expressed as:
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After preforming basic calculations and
applying partition fractions along with the
inverse of the Maha integral transform to the
given expression, the resultant value obtained
represents the amount of pressure in the aorta:
4 Conclusions
A two-parameters integral transform was
introduced through this transform. We found
that conversion of linear ordinary differential
equations with constant coefficients and higher
orders turns into simple algebraic equations that
are simpler and easier than the previous two-
parameter as mentioned. In this paper important
medical applications were presented, as well as
the application of nuclear physics.
Conceptualization, Alsaoudi and Kuffi;
methodology, Kuffi; validation Alsaoudi and
Guiatni; formal analysis, Alsaoudi; investigation,
Alsaoudi; resources, Gharib and Kuffi; data
curation, Kuffi; writing/original draft preparation,
Guiatni; writing/review and editing, Guiatni;
supervision, Gharib; project administration, Gharib.
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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 conflicts of interest to declare
that are relevant to the content of this article.
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(Attribution 4.0 International, CC BY 4.0)
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DOI: 10.37394/23206.2024.23.56
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