IJRAR Research Journal



COMPARATIVE STUDY OF LAPLACE, SUMUDU, ABOODH, ELZAKI AND MAHGOUB TRANSFORMS AND APPLICATIONS IN BOUNDARY VALUE PROBLEMS

D. P. Patil

Associate Professor

Department of Mathematics

K. K. W. Arts, Science and Commerce College, Pimpalgaon B, Tal Niphad Dist Nashik, India

Abstract: In this paper we discuss some relationship between above five Integral Transforms. We apply these transforms to solve the differential equations of first and second order and system of differential equations formed in physical sciences.

IndexTerms - Differential Equations, System of differential equations, Laplace Transform, Sumudu Transform, Aboodh Transform, Mahgoub Transform, Physical Sciences.

Introduction

Integral transforms including Laplace Transform and Fourier Transform have been successfully used since 1780’s. Its origin. Can be traced back to celebrated work of P. S, Laplace and Joseph Fourier. They play an important role in many fields of science. There are many widely used integral transformations including Mellin Transform, Hankel Transform, the Stleitjes Transform and the Hilbert Transform to solve initial and boundary value problems involving ordinay differential equations and partial differential equations.

Hilbert Transform and its properties are studied by G. H. Hardy, but was named after one of the greatest mathematician of twentieth century. , David Hilbert.

Basically, Integral Transform of a function f(x) is defined in is denoted by and is defined as

[pic]

where K(x , k) is kernel of the transform , F(K) is referred as the image of the given object function f(x) and k is called as transform variable. The idea of integral transform operator is somewhat similar to that of linear differential operator.[pic]. Different integral transform can be defined by taking different kernels and different values for a and b.

We have further more integral transforms [LB 2006] like Laplace Transform, Fourier Transform, Sumudu Transform [EK2010], Elzaki Transform [E 2011][EE2011], Aboodh Transform [A2013], Mahgoub Transform [M 2016] for solving differential equations anf system of differential equations. These transforms are widely used in mathematical physics, optics, Engineering and few other fields. Recently, in 2013, Aboodh has introduced new, Aboodh Transform, which has deeper connection with Laplace and Elzaki Transform..It is useful for solving ordinary and partial differential equations in the time domain. The main aim of the paper is to introduce a comparative study to solve boundary value problem by using different integral transforms. The plan of the paper is as follows.

In section II we introduce Laplace Transform , Elzaki transform in section III. Fourth section deals with Aboodh Transform , while fifth section is devoted to Mahgoub Transform. Sumudu transform is in sixth section Application of transform is discussed in seventh section.

LAPLACE TRANSFORM

2.1 Definition : Let f(t) be defined for [pic]. The Laplace transform of f(t) denoted by f(s) or L[f(t)] is an integral transform given by

[pic]

Provided that this improper integral exists i.e. the improper integral is convergent.

2.2 Laplace transforms of some functions:

[1] [pic] [ 2] [pic]

[3] [pic] [4] [pic]

[5] [pic] [6] [pic]

[7] [pic] [8] [pic]

[9][pic] [10] [pic]

[11] [pic] [12] [pic]

[13] [pic] [14] [pic]

2.3 Properties of Laplace Transform

[1] Linear Property

[pic]

[2] Change of scale property

[pic]

[3] Shifting Property

[pic]

[pic]

2.4 Laplace Transform of derivative

If f(t) is continuous function for all [pic] and is exponential order a as [pic] and if [pic] is of class A then

[pic]

Further

[pic]

and

[pic]

Elzaki transform

Elzaki transform is derived from Fourier integral and was introduced by Tarig Elzaki .

Definition

For any function f(t) the sufficient condition for the existence of Elzaki transform are that f(t) for nonnegative t be piecewise continuous and of exponential order

[pic]

Elzaki transform is modified Laplace and Sumudu transform and it has been used to solve effectively, easily and accurately a large class of linear differential equation, ordinary differential equations with variable coefficients and system of all these equations.

Elzaki transforms of some functions

[1] [pic] [2] [pic] [3] [pic]

[4] [pic] [5] [pic] [6] [pic]

Elzaki transform of Derivatives

[pic]

and

[pic]

Shifting Property

[pic]

ABOODH TRANSFORM

Definition:

Aboodh Transform is defined for functions of exponential order. Function is defined in set A as follows

[pic]

For a function as defined above , M must be finite and [pic] may be finite or infinite. Aboodh Transform is defined by the equation

[pic]

Aboodh transform for some functions

[1] [pic] [2] [pic] [3] [pic]

[4] [pic] [5] [pic]

Aboodh transform of derivatives

If Aboodh transform of the function f(t) is given by [pic] , then

[pic]

[pic]

Simillarly

[pic]

MAHGOUB TRANSFORM

Mahand Mahgoub has introduced integral transform called as Mahgoub transform. It can be used

Mahgoub transform for some functions f(t) is defined as

[pic]

where M is Mahgoub transform operator. Sufficient conditions for the existence of Mahgoub transform of the function f(t) are f(t) is piecewise continuous and of exponential order. Fadil [F 2017] study the convolution for Mahgoub transforms. Taha et al. [TNAH] obtain the dualities Kamal and Mahgoub integral transforms and some famous transforms. Aggarwal et al. [ ASCGK 2018] proved the new application of Mahgoub transform for solving linear ordinary differential equations eith variable coefficients. A new application of Mahgoub transform for solving Linear Volterra integral equations was given by Aggarwal et al. [ASC 2018].

[1] [pic] [2] [pic]

[3] [pic] [4] [pic]

[5] [pic] [6] [pic]

[7] [pic]

Convolution theorem for Mahgoub Transform [ F 2017]

If [pic] and [pic] then [pic]

sumudu transform

Definition [BK2006]

The Sumudu transform over the set of functions

[pic] is defined as

[pic]

Sumudu transforms for some basic functions

[1] [pic] [2] [pic]

[3] [pic] [4] [pic]

[5] [pic] [6] [pic]

[7] [pic]

Properties of Sumudu Transform

I) Linearity Property

[pic]

II) Duality with Laplace transform

[pic]

III) Sumudu transform of integral of a function

[pic]

IV) Sumudu transform of function derivatives

[pic]

[pic]

Generally

[pic]

V) Shifting theorems

[pic]

[pic]

APPLICATION:

Application of Transform in system of differential equation arises in various physical phenomenons. We can also see the applications of integral equations in Engineering, Chemical and biological sciences. Applications are also found in the field of Economics, Commerce, Banking, Pharmaceutical Sciences and the fields where the ordinary differential equations, Partial differential equations and systems of these equations formed. Solutions can be obtained easily and accurately by using these integral equations.

Consider the boundary value problem

[pic]

[pic]

We solve above boundary value problem by using Laplace Transform.

Consider [pic]

Applying Laplace on both sides of equations (1) and (2)

[pic]

[pic]

Applying equation (3) and rearranging the terms,

[pic]

[pic]

[pic]

[pic]

[pic]

Solving equations (4) and (5) simultaneously we get

[pic]

and [pic]

Using partial fractions to equations ( 6) and (7)

[pic]

[pic]

Taking inverse Laplace of equations (8) and (9) we get

[pic]

[pic]

Conclusion

The main aim of this paper is to compare between different five integral transforms for solving the system of ordinary differential equations. By using any one of the integral transform we can obtain the similar results as obtained by using Laplace Transform.These integral transforms are powerful and efficient

References

[1] Abdebagy A Alshikh, Mohand M. Abdelrahim Mahgoub [AM 2016], A comparative study between Laplace Transform and Two

New integrals “Elzaki” Teansform and Aboodh Transform; Pure and Applied Mathematics Journal (2016), 5(5) pp. 145 – 150.

[2] Aboodh K.S[A 2013] . The new Integral Transform “ Aboodh Transform” Global Journal of Pure and Applied Mathematics, 9 (1) ,

pp 35 – 43.

[3] Aggarwal S. Sharma N., Chauhan R. Gupta A. and Khandelwal A [ASCGK 2018] A new application of Mahgoub transform for

solving linear ordinary differential equations with variable coefficients , Journal of computer and Mathematical Sciences. , 9(6) ,

2018 pp. 520 – 525.

[4]Aggarwal S. Sharma N. and Chauhan R. [ASC 2018] A new application of Mahgoub transform for solving linear Volterra Integral

equations , Asian Resonance, 7(2) , 2018 , pp. 46 – 48.

[5] Belgacem F. and Karaballi A. [ BK 2006] Sumudu transform fundamental properties , investigations and applications, International

Journal of Stochastic Analysis 2006.

[6] Loknath Debnath and D. Bhatta [DB 2006] Integral transforms and their applications , second Edition, Chapman and Hall / CRC

(2006).

[7] Hassan Eltayeb and Adem Kilicman [EK 2010] A note on the Sumudu transform and differential equations, Applied Mathematical

Sciences, Vol.4 (2010) No.22 pp. 1089 -1098.

[8] Tarig M. Elzaki [E2011] The new Integral Transform “Elzaki Transform” Global Journal of Pure and Applied Mathematics No.1 pp.

57 – 64.

[9] Tarig M. Elzaki and Salih M. Elzaki [EE 2011] “On the Elzaki Transform and Ordinary Differential Equations with variable

coefficients”, Advances in Theoretical and Applied Mathematics, Vol. 6 No. 1 pp. 13 – 18.

[10] Eadil R. A. [F2017] Convolution for Kamal and Mahgoub transform , Bulletin of mathematics and statistics research, 5 (4) , 2017,

pp 11 -17.

[11] S. R. Kushare, D. P. Patil and A. M. Takate [KPT 2018] , Comparison between Laplace, Sumudu and Mahgoub transforms for

solving system of first order first degree differential equations.; Journal of Emerging Technology and Innovative Research (Nov.

2018) Vol. 5 Issue 11 pp. 612 – 617.

[12] Mohand M. Abdelrahim Mahgoub. [M 2016] “The new integral transforms Mahgoub Transform” Advances in Theoretical and

Applied Mathematics, (2016)Vol. 11 No. 4 pp 391 – 398.

[13] Mohand M. Abdelrahim Mahgoub, Khalid Suliman Aboodh and Abdelbagy A Alshikh [MAA 2016], On the solution of Ordinary

Differential Equation with variable coefficients using Aboodh Transform; Advances in Theoretical and Applied Mathematics, 2016

Vol. 11 Number4 4 , pp. 383 – 389.

[14] Sunil Shrivastava,[S] Introduction of Laplace Transform and Elzaki Transform with Applications (Electrical Circuits).

[15] Taha N. E. H. , Nuruddeen R.I., Abdelilah K. and Hassan S. [ TNAH 2017] Dualities between ‘Kamal and Mahgoub integral

transforms” and ‘some famous integral transforms “ , British Journal of Applied Science and Technology, 20(3) , 2017, pp. 1 – 8.

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