On the use of RAHMOH integral transform for solving differential equations

Abstract

In this paper, we introduced a new Laplace-type integral transform called RAHMOH transform which is generalized of Laplace and Sumudu transforms for solving ordinary and partial differential equations. We presented its existence, inverse transform and some essential properties with some theorems and applications.

Full text

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International Journal of Physics and Mathematics 2024; 6(2): 01-08
E-ISSN: 2664-8644
P-ISSN: 2664-8636
IJPM 2024; 6(2): 01-08
© 2024 IJPM
www.physicsjournal.net
Received: 01-05-2024
Accepted: 05-06-2024
RA Farah
Department of MIS- Statistics
and Quantum Methods unit,
Faculty of Business and
Economics, University of
Qassim, Buraidah, KSA, Saudi
Arabia
MA Hamad
Department of Mathematics,
Sudan University of Science and
Technology, Saudi Arabia
Corresponding Author:
RA Farah
Department of MIS- Statistics
and Quantum Methods unit,
Faculty of Business and
Economics, University of
Qassim, Buraidah, KSA, Saudi
Arabia
On the use of RAHMOH integral transform for solving
differential equations
RA Farah and MA Hamad
DOI: https://doi.org/10.33545/26648636.2024.v6.i2a.83
Abstract
In this paper, we introduced a new Laplace-type integral transform called RAHMOH transform which is
generalized of Laplace and Sumudu transforms for solving ordinary and partial differential equations. We
presented its existence, inverse transform and some essential properties with some theorems and
applications.
Keywords: RAHMOH transform, Integral transform, Sumudu transform, laplace transform, differential
equations
1. Introduction
The origin of the integral transforms can be traced back to the work of P. S.
Laplace in 1780 and Joseph Fourier in 1822. In recent years, there are many transforms
obtained from Fourier and Laplace transforms.
One of these is RAHMOH transform.
RAHMOH transform is derived from the classical Fourier, Laplace, Sumudu and Natural
integral Transforms. In order to solve the differential equations, the integral transform was
extensively used.
We will describe the definitions for some of these transforms. Fourier integral transform
was defined as:
(1)
The Fourier transform have many applications in physics and engineering processes. The
Laplace integral transform is similar with the Fourier transform and is defined as:
(2)
The Laplace transform is highly efficient for solving some class of ordinary and partial
differential equations
.
In 1993, Watugala obtained new integral transform called Sumudu integral transform and it is
applied in different sciences like physics and engineering recently, also many new transforms
based on Sumudu integral transform because of its properties. The mathematical definition of
Sumudu transform is
(3)
In 2008, the Natural transform has been introduced. Recently it is applied to many applications
in physics and engineers. It is defined as
(4)

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Provided the integral exists for some variables and .
In 2019, a new transform has been introduced and it is called Shehu transform, it is derived from Fourier, Laplace, Sumudu and
Natural transform. Shehu transform applied to solve ODEs and PDEs and defined as
(5)
Historically, there are many other transforms also obtained from Fourier, Laplace and Sumudu integral transform. However, most
of the existing integral transforms have some limitations and cannot be used directly to solve nonlinear problems or many
complex mathematical models.
2. Main Results
2.1 Definition
A new transform called the RAHMOH Transform defined for the function of exponential order; we consider functions in the set
defined by:
Which is constant must be finite number, may be finite or infinite.
The RAHMOH transform denoted by the operator for the function by the following integral:
(6)
It converges if the limit of the integral exists, else, it diverges.
The inverse of RAHMOH transform is given by:
(7)
where are RAHMOH transform variables
.
Properties of
RAHMOH
transform
Property 1
Let be RAHMOH transform then the following is true:
(8)
(9)
(10)
Proof
Apply in eq (2.1) we get:
(i)
(ii) , integrating by parts to find that:

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(iii) , integrating by parts times and take the limit we
find:
Property 2
Let be RAHMOH transform, then the following is true:
(i) (11)
(ii) (12)
(iii) (13)
(iv) (14)
(v) (15)
Proof
Applying eq (2.1) we get:
(i) , integrating directly as same as (2.3) in
property 1 and take the limit yields:
(ii)
Using the fact that:
Here taking and apply it in eq (2.1), so
Apply and take the limit yields:

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Using the fact that:
Also taking and apply it in eq (2.1), so
Apply and take the limit yields:
2.2 Theorem: (Existence of RAHMOH)
Let the function belong to the set A in definition 2.1. Then its
RAHMOH
transform exists.
Proof
Since
We take
by taking thus
Then the RAHMOH exists.
2.3 Theorem: (
Linearity property of
RAHMOH
transform
)
Let the functions and be in the set ,
then , where α and β are
nonzero arbitrary constants, and

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Proof:
Using the Definition 2.1 of
RAHMOH
transform, we get
2.4 Theorem: (Change of scale property of RAHMOH transform)
Let the function be in set , where
is an arbitrary constant. Then
Proof
Using definition 2.1 of RAHMOH transform, we deduce
Substituting yields
2.5 Theorem
: (RAHMOH of Derivatives)
Let in definition 2.1 and its RAHMOH transform, then:
(i)
(ii)
(iii)
Proof: It is obtained by applying in definition 2.1 and using integration by parts.
The following properties are obtained using Leibniz's rule

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3. Some Applications
RAHMOH transform can be used to solve Linear and non-Linear ODEs and PDEs. The following simple examples show that:
3.1 Example: Consider the following first order ordinary differential equation:
Subject to initial condition
Solution:
By taking RAHMOH transform
Using eq (2.14) yields
Using initial condition, then
By taking RAHMOH inverse of eq (3.3) yields
3.2 Example
Consider the following second order non-homogeneous ODE:
Subject to the initial conditions
Solution
Applying the RAHMOH transform on both sides of eq (3.4), we get
Substituting the given initial conditions and simplify, we get
Take RAHMOH transform we obtain:
3.3 Example
Consider the following homogeneous PDE:

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Subject to the boundary and initial conditions:
Solution
Applying the RAHMOH transform on both sides of eq (3.6), we get
Substituting the given initial conditions and simplify, we get
The general solution of eq (3.8) can be written as:
Where is the solution of the homogeneous part which is given by
And
is the solution of the nonhomogeneous part which is given by
Applying the boundary conditions on eq (3.10) yields
Using the method of undetermined coefficients on the nonhomogeneous part, we get
Then eq (3.9) become
Taking the inverse RAHMOH transform for eq (3.13), we get
Conclusion
We introduced an efficient Laplace-type integral transform called the RAHMOH transform for solving both ordinary and partial
differential equations. We presented its existence, inverse transform and some essential properties. We conclude that RAHMOH is
highly efficient because of the following Pros:
It is generalized of the Laplace and Sumudu transforms.

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International Journal of Physics and Mathematics https://www.physicsjournal.net
RAHMOH transform become Laplace transform when the variable .
For advanced research in physical science and engineering, the proposed integral transform can be considered a stepping-stone to
the Sumudu transform, the natural transform, and the Laplace transform.
The relation between RAHMOH transform and Shehu transform is
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