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APPLICATIONS OF A SHEHU TRANSFORM TO THE HEAT AND TRANSPORT EQUATIONS

Authors:

Abstract

The Shehu transform principle properties were showed by Maitama [13]. Atheros in 2019 used Shehu transform to solve differential equations. In this paper, we introduce Shehu transform, which used in solution of ordinary and partial differential equations, moreover, we extended Shehu transform application for solving transport and heat equations that satisfied known or unknown initial conditions.
International Journal of Psychosocial Rehabilitation, Vol. 24, Issue 05, 2020
ISSN: 1475-7192
DOI: 10.37200/IJPR/V24I5/PR2020141 Received: 08 Mar 2020 | Revised: 26 Mar 2020 | Accepted: 04 Apr 2020 4254
Abstract--- The Shehu transform principle properties were showed by Maitama [13]. Atheros in 2019 used
Shehu transform to solve differential equations. In this paper, we introduce Shehu transform, which used in solution
of ordinary and partial differential equations, moreover, we extended Shehu transform application for solving
transport and heat equations that satisfied known or unknown initial conditions.
Keywords--- Transport Equations, Applications of A Shehu, Uniqueness.
I. INTRODUCTION
Transport and Heat equations are partial differential equations which the applications wide in physics and
engineering [6].
Moreover, integral transformations are mathematical methods that extensively used in solution of differential
equations, therefore, there are several integral transforms such as the Laplace, Sumudu, Mellin, Elzaki and Temimi
[3,4,5,10], to name but a few.
Laplace transform is widely used to solve differential equations subjected to the initial or boundary conditions.
The result of solutions for initial value problems that used Laplace transform represent particular solutions.
Sumudu transform is similar to Laplace transform, but the first used to solve differential equations with variable
coefficient [5].
In 2013, Atangana and Kilicman introduced a new integral transform named the Abdon - Kilicman integral
transform for solution some differential equations with some kind of Uniqueness [1]. The new integral transform is
defined as:
 

The Atangana - Kilicman integral happened Laplace transform when n=0.
A new integral transform named the - transform, which is Moreover similar to normal transform is introduced
by Srivastava et al. [9] in 2015. Mathematically talking - transform is Closely related with the known Laplace
transform and the Sumudu integral transform. - transform was successfully used to first order initial-boundary
value problem. The - transform is defined as:
Athraa Neamah ALbukhuttar*, Department of Mathematics, Faculty of Education for Girls, University of Kufa, Najaf, Iraq.
E-mail: athraan.kadhim@uokufa.edu.iq
Zainab Dheyaa Ridha, Department of Banking & Financial, Faculty of Administration and Economics, University of Kufa, Najaf, Iraq.
Hayder Neamah Kadhim, Department of Banking & Financial, Faculty of Administration and Economics, University of Kufa, Najaf, Iraq.
E-mail: hayder.najaf1976@gmail.com
Applications of A Shehu Transform to the Heat and
Transport Equations
Athraa Neamah ALbukhuttar*
Zainab Dheyaa Ridha and Hayder Neamah Kadhim
International Journal of Psychosocial Rehabilitation, Vol. 24, Issue 05, 2020
ISSN: 1475-7192
DOI: 10.37200/IJPR/V24I5/PR2020141 Received: 08 Mar 2020 | Revised: 26 Mar 2020 | Accepted: 04 Apr 2020 4255



Recently, a Laplace - kind integral transform named the Yang transform for solution fixed heat transfer problems
was submitted in 2016[15]. The integral transform is defined as:
 

Under condition the integral convergent.
Because the quick development in the physical science and engineering models, there are a lot of another
integral transforms in the literature. Anyway, most of the present integral transforms have some determinants and
cannot be applied immediately to solved nonlinear problems or many composite mathematical models. As a result,
many authors exists very interested to reach with the replacement approach for solution a lot of genuine - life
problems.
In 2016, [2] Atangana and Alkaltani introduced a Novel double integral transform and their properties according
to on the Laplace transform and decomposition method. The double integral transform was successfully used to
second order partial differential equation with uniqueness named the two dimensional Mboctara equation. Recently,
Eltayeb used double Laplace decomposition method to Mon linear partial differential equations [8]. In 2017, Bel-
gacem el at, extended the applications of the normal and the Sumudu transforms to fractional diffusion and Stokes
fluid flow realms [7].
We benefited by the above - mentioned researches, in this paper we Recommended Laplace - type integral
transform named Shehu transform for solution both ordinary and partial differential equations. The Laplace - kind
integral transform converges to Laplace transform when k=1, and to Yang integral transform when ƅ=1 the
recommended integral transform is successfully used to solve many types of differential equations [11, 12, 14]. All
the results it got in the applications section can easily be verification applying the Laplace or Fourier integral
transforms. In this paper, the Shehu transform is denoted by an operator .
In our research, we apply Shehu transform to solve transport and heat equations in one - dimension. Whereas,
they are homogeneous or non- homogeneous, that subjected to known or unknown initial conditions. Section 2
showed the properties and theorems for Shehu transform, and the most important formulas of Shehu transform for
some functions. In section 3, we established the general formula to the transport and heat equations, in one
dimension by using Shehu transform. Finally, section 4 applied the general formula for equations in some examples.
II. BASIC DEFINITIONS AND PROPERTIES
Definition: The Shehu transform of the function  of exponential order is over the set of functions
 

International Journal of Psychosocial Rehabilitation, Vol. 24, Issue 05, 2020
ISSN: 1475-7192
DOI: 10.37200/IJPR/V24I5/PR2020141 Received: 08 Mar 2020 | Revised: 26 Mar 2020 | Accepted: 04 Apr 2020 4256
By the following integral:
 


 


It converges if the limit of the integral exists, and diverges if not.
The inverse Shehu transform is given by:

Equivalently



 

Where and are the Shehu transform variables, and is a real constant and the integral in equation (2.3) is
taken along in the complex plane.
Property: Linearity property of Shehu transform.
Let the functions  and  be in set A,
thenarbitrary constant and

Proof. Using the definition of Shehu transform, we get:













Lemma: Derivative of Shehu transform.
If the function  is the derivative of the function with respect to then its Shehu
transform is defined by:



 
When n=1, 2, and 3 in equation (2.5) above, we obtain the following derivatives with respect to
International Journal of Psychosocial Rehabilitation, Vol. 24, Issue 05, 2020
ISSN: 1475-7192
DOI: 10.37200/IJPR/V24I5/PR2020141 Received: 08 Mar 2020 | Revised: 26 Mar 2020 | Accepted: 04 Apr 2020 4257







Proof. Now suppose equation (2.5) is true for then using equation (2.8) and the induction hypothesis, we
deduce





 


 
Which implies that equation (2.8) holds for, by induction hypothesis the proof is complete.
The following important properties are obtain using the Leibniz's rule

  

 





 






.
.
.

 






Table 1: The following table showed the Shehu Transform for some functions such as:
S.No.


1
1
2
International Journal of Psychosocial Rehabilitation, Vol. 24, Issue 05, 2020
ISSN: 1475-7192
DOI: 10.37200/IJPR/V24I5/PR2020141 Received: 08 Mar 2020 | Revised: 26 Mar 2020 | Accepted: 04 Apr 2020 4258
3


4


5



6


7


8



III. FORMULA OF GENERAL SOLUTIONS OF HEAT AND TRANSPORT EQUATIONS
Formula 1: Consider the homogeneous transport equations:
(3.1) 
We take the Shehu transformation to both sides, we get:


By substitute, initial value:





Which represent linear equations of order one and it has the solution



We take the inverse of both sides:





Equation (3.3) represent the general formula solution of equation (3.1)
Formula 2: Consider the non- homogeneous transport equations:
International Journal of Psychosocial Rehabilitation, Vol. 24, Issue 05, 2020
ISSN: 1475-7192
DOI: 10.37200/IJPR/V24I5/PR2020141 Received: 08 Mar 2020 | Revised: 26 Mar 2020 | Accepted: 04 Apr 2020 4259


After we take the Shehu transformation to both sides, we have:

 
By substitute, initial value:







The above equation has the solution






The general formula solution of equation (3.4) can be obtained by taking the inverse of both sides to equation
(3.5)








Formula 3: Consider the homogeneous heat equations:


We take the Shehu transformation to both sides and substitute the initial value:



The last equation represent linear equation of second order, which has the solution:











We take the inverse of both sides to equation (3.8), we get:
International Journal of Psychosocial Rehabilitation, Vol. 24, Issue 05, 2020
ISSN: 1475-7192
DOI: 10.37200/IJPR/V24I5/PR2020141 Received: 08 Mar 2020 | Revised: 26 Mar 2020 | Accepted: 04 Apr 2020 4260












Formula 4: Consider the non- homogeneous heat equations:


After taking the Shehu transformation and applying the initial conditions of equation (3.10), we obtained:



Which represent linear equation of second order and it has the solution











The solution of equation (3.10) can be found by taking the inverse of both sides to equation (3.11):












Equation (3.12) represent the general formula solution of equation (3.10).
IV. APPLICATIONS
In the following section, the usefulness and the effectiveness of Shehu transformation are showed by finding
exact solution of transport and heat equations.
Example 1: To solve the equation 
International Journal of Psychosocial Rehabilitation, Vol. 24, Issue 05, 2020
ISSN: 1475-7192
DOI: 10.37200/IJPR/V24I5/PR2020141 Received: 08 Mar 2020 | Revised: 26 Mar 2020 | Accepted: 04 Apr 2020 4261

Sol: We can obtained the solution by utilizing of formula 1 in equation (3.3), as show:










Example 2: For solving the equation 

Sol: By utilizing the formula 2 in equation (3.6), we obtained:













 


 



Example 3: To solve the equation =
u(x,0) =
Sol: We can use the formula 3 in equation (3.9), so as:


 



 






 



 



International Journal of Psychosocial Rehabilitation, Vol. 24, Issue 05, 2020
ISSN: 1475-7192
DOI: 10.37200/IJPR/V24I5/PR2020141 Received: 08 Mar 2020 | Revised: 26 Mar 2020 | Accepted: 04 Apr 2020 4262























Example 4: To solve the non-homogenous heat equation
+=

Sol.: Similarly, if we use the formula 4 in equation (3.12), then also the above equations has the solution:



 




 






 





 








 





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International Journal of Psychosocial Rehabilitation, Vol. 24, Issue 05, 2020
ISSN: 1475-7192
DOI: 10.37200/IJPR/V24I5/PR2020141 Received: 08 Mar 2020 | Revised: 26 Mar 2020 | Accepted: 04 Apr 2020 4263
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Article
Full-text available
In this paper, we introduce definition of new transformation which we call it Temimi transformation. Also, we introduce properties, theorems, proofs and transformations of the polynomials functions, logarithms functions and other functions. Also, we introduce how we can use this transformation and it's inverse to solve the Euler's equation
Article
Full-text available
We introduced a novel integral transform operator. We proved the existence and the uniqueness of the relatively new operator. We presented some useful properties of the new operator. We presented the application of this operator for solving some kind of fractional ordinary and partial differential equation containing some kind of singularity.
A novel double integral transform and its applications
  • A Atangana
  • Alkaltani
A Atangana and B Alkaltani, A novel double integral transform and its applications, Jour-nal of Nonlinear Science and Applications, Vol.9(2), 2016.