# APPLICATIONS OF GENERALIZED SUMUDU TRANSFORM

## Keywords:

Sumudu Transform, Generalized Sumudu Transform## Abstract

**Objective: **To solve Cauchy Problem for the Wave Equation in One Dimensional Space using the generalized Sumudu transform.

**Materials and Methods**: We use the new integral transform called the Sumudu transform for the extension to a generalized space and we use the

method for extension followed by Zemanian[29].

**Results and Conclusion**: In this paper, I have established the application of generalized Sumudu transform. To illustrate the applications of

generalized Sumudu transform, I have applied it to solve the cauchy problem for the wave equation in one dimensional space where the initial

conditions are generalized functions.

## References

[1] Asiru M.A.;Sumudu transform and solution of integral equations of convolution

type,Internat. J. Math.Ed.Sci.Tech.,32(2011),906–910.

[2] Asiru M.A.; Further Properties of the Sumudu Transform and its Applications, Internat.

J.Math.Ed.Sci.Tech.,33(2) (2002),441–449.

[3] Asiru M. A.; Applications of the Sumudu Transform to Discrete Dynamic system. Internat.

J. Math. Ed. Sci. Tech.,34(6),(2003),944–949.

[4] Belgacem F.B.M.;Applications of the Sumudu Transformation Indefinite Periodic Parabolic

Equations,Proceedings of the International Conference on Nonlinear Problems and

Aerospace Applications,Budapest,Hungary, (2007),1–10.

[5] Belgacem F.B.M.;Introducing and Analyzing Deeper Sumudu Properties,Nonlinear Studies

Journal,13(4),(2006),23–41.

[6] Belgacem F.B.M.,Karaballi A.A.,Kalla L.S.; nalytical investigations of the Sumudu transform

and applications to integral production equations, A,Math.Probl. Eng.,3(4),(2003),

103-118.

[7] Belgacem F.B.M.,Karaballi A.A.,Kalla L.S.; Sumudu transform fundamental properties

investigations and applications, Journal of Applied Mathematics and Stochastic Analysis,

3(4),(2006),1–23.

[8] Chindhe A.D.,Kiwne S.B.; Application of Sumudu Transform and Homotopy Perturbation

Method to Solve Nonlinear Differential Equations,MJIS,Chitkara University Publications,

6(1),(2017),1–9.

[9] Chindhe A.D.,Kiwne S.B.;On The Generalized Natural Transform,Mathematical Journal of

Interdisciplinary Sciences,5(1),(2018),1–14.

[10] Deshna Loonker,Banerji PK.;Natural transform for distribution and Boehmian

spaces,Mathematics in Engineering,Science and Aerospace(MESA),Vol. 4, No.

1,(2013),69-76 .

[11] Deshna Loonker,Banerji PK.;On the solution of distributional Abel integral equation by

distributional Sumudu transform,Internat. J. Math. Math. Sci.,Article ID 480528,(2011),1-

8.

[12] Deshna Loonker,Banerji PK.;Fractional Integrals and Derivatives for Sumudu transform

on distribution spaces,Appl. Appl. Math. Internat. J.,7 (1),(2012),188-200.

[13] Deshna Loonker,Banerji PK.;Hartley Transform for Integrable Boehmians,Int.Trans.

Spl.Funct.,21(6),(2010),459-464.

[14] Hassan Eltayeb ,Adem Kilicman, Brian Fisher ;New integral transform and associated

distributions,A Integral Transforms Special functions,21,(2010),367-379.

[15] Khan ZH,Khan WA.;N-transform properties and applications,NUST Journal of Engineering

Sciences,1(1),(2008),127-133.

[16] Pathak R.S.;Integral Transforms of Generalized Functions and their Applications,Gordon

and Breach Science Publishers,Australia, Canada, India,Japan,(1997).

[17] Sadikali Latif Shaikh;A Study of some Integral Transforms and their Operational Calculus,

Ph.D.Thesis,Shivaji University,Kolhapur,(2011).

[18] Sadikali Latif Shaikh ;Some Abelian Theorems for the Classical Sumudu Transform,

American International Journal of Research in Science, Technology, Engineering

and Mathematics,18(1),(2017),95-97.

[19] Shiraishi Risai;On the value of a distribution at a point and multiplicative products,J.

Sci.Hiroshima Univ. Ser. A-I 31,(1967),89-104.

[20] S.K.Q.Al-Omari;Generalized functions for double Sumudu transformation,International

Journal of Algebra, 6(3),(2011),139-146.

[21] S.K.Q.Al-Omari;On the Generalized Sumudu Transformation,journal of applied functional

analysis ,9(1-2),(2014),42-53.

[22] Thange T.G. and Chudhary M.S.;Some Integral Transform on Distribution Spaces,Ph. D.

Thesis, Shivaji University, Kolhapur,(2013).

[23] Watugala G. K.;Sumudu transform: a new integral transform to solve differential

equations and control engineering problems,Model Assisted Statistics and Applications,

24(1),(1993),35–43.

[24] Watugala G. K.;Sumudu Transform A New Integral Transform to solve Differential

Equations and Control Engineering Problems,Mathematical Engineering in Industry,

6(4),(1998),319-329.

[25] Watugala G. K.;The Sumudu transform for functions of two variables,Mathematical Engineering

in Industry,8(4),(2002),293-302.

[26] Weerakoon S.;Applications of Sumudu Transform to Partial Differential Equations,Int. J.

Math. Educ. Sci. Tech.,25(2),(1994),1277-283.

[27] Weerakoon S.;Complex Inversion formula for Sumudu Transform,Int. J. Math. Educ. Sci.

Tech., 29(4),(1998),618-621.

[28] Zemanian A.H.;Distribution Theory and Transform Analysis : An Introduction to Generalized

Functions with Applications,Mc-Graw Hill,New York,Republished by Dover Publication,

Inc., New York. 1965.

[29] Zemanian A.H.;Generalized Integral Transformations,Interscience Publishers, New

York,Republished by Dover Publications,Inc.,New York.1968.

type,Internat. J. Math.Ed.Sci.Tech.,32(2011),906–910.

[2] Asiru M.A.; Further Properties of the Sumudu Transform and its Applications, Internat.

J.Math.Ed.Sci.Tech.,33(2) (2002),441–449.

[3] Asiru M. A.; Applications of the Sumudu Transform to Discrete Dynamic system. Internat.

J. Math. Ed. Sci. Tech.,34(6),(2003),944–949.

[4] Belgacem F.B.M.;Applications of the Sumudu Transformation Indefinite Periodic Parabolic

Equations,Proceedings of the International Conference on Nonlinear Problems and

Aerospace Applications,Budapest,Hungary, (2007),1–10.

[5] Belgacem F.B.M.;Introducing and Analyzing Deeper Sumudu Properties,Nonlinear Studies

Journal,13(4),(2006),23–41.

[6] Belgacem F.B.M.,Karaballi A.A.,Kalla L.S.; nalytical investigations of the Sumudu transform

and applications to integral production equations, A,Math.Probl. Eng.,3(4),(2003),

103-118.

[7] Belgacem F.B.M.,Karaballi A.A.,Kalla L.S.; Sumudu transform fundamental properties

investigations and applications, Journal of Applied Mathematics and Stochastic Analysis,

3(4),(2006),1–23.

[8] Chindhe A.D.,Kiwne S.B.; Application of Sumudu Transform and Homotopy Perturbation

Method to Solve Nonlinear Differential Equations,MJIS,Chitkara University Publications,

6(1),(2017),1–9.

[9] Chindhe A.D.,Kiwne S.B.;On The Generalized Natural Transform,Mathematical Journal of

Interdisciplinary Sciences,5(1),(2018),1–14.

[10] Deshna Loonker,Banerji PK.;Natural transform for distribution and Boehmian

spaces,Mathematics in Engineering,Science and Aerospace(MESA),Vol. 4, No.

1,(2013),69-76 .

[11] Deshna Loonker,Banerji PK.;On the solution of distributional Abel integral equation by

distributional Sumudu transform,Internat. J. Math. Math. Sci.,Article ID 480528,(2011),1-

8.

[12] Deshna Loonker,Banerji PK.;Fractional Integrals and Derivatives for Sumudu transform

on distribution spaces,Appl. Appl. Math. Internat. J.,7 (1),(2012),188-200.

[13] Deshna Loonker,Banerji PK.;Hartley Transform for Integrable Boehmians,Int.Trans.

Spl.Funct.,21(6),(2010),459-464.

[14] Hassan Eltayeb ,Adem Kilicman, Brian Fisher ;New integral transform and associated

distributions,A Integral Transforms Special functions,21,(2010),367-379.

[15] Khan ZH,Khan WA.;N-transform properties and applications,NUST Journal of Engineering

Sciences,1(1),(2008),127-133.

[16] Pathak R.S.;Integral Transforms of Generalized Functions and their Applications,Gordon

and Breach Science Publishers,Australia, Canada, India,Japan,(1997).

[17] Sadikali Latif Shaikh;A Study of some Integral Transforms and their Operational Calculus,

Ph.D.Thesis,Shivaji University,Kolhapur,(2011).

[18] Sadikali Latif Shaikh ;Some Abelian Theorems for the Classical Sumudu Transform,

American International Journal of Research in Science, Technology, Engineering

and Mathematics,18(1),(2017),95-97.

[19] Shiraishi Risai;On the value of a distribution at a point and multiplicative products,J.

Sci.Hiroshima Univ. Ser. A-I 31,(1967),89-104.

[20] S.K.Q.Al-Omari;Generalized functions for double Sumudu transformation,International

Journal of Algebra, 6(3),(2011),139-146.

[21] S.K.Q.Al-Omari;On the Generalized Sumudu Transformation,journal of applied functional

analysis ,9(1-2),(2014),42-53.

[22] Thange T.G. and Chudhary M.S.;Some Integral Transform on Distribution Spaces,Ph. D.

Thesis, Shivaji University, Kolhapur,(2013).

[23] Watugala G. K.;Sumudu transform: a new integral transform to solve differential

equations and control engineering problems,Model Assisted Statistics and Applications,

24(1),(1993),35–43.

[24] Watugala G. K.;Sumudu Transform A New Integral Transform to solve Differential

Equations and Control Engineering Problems,Mathematical Engineering in Industry,

6(4),(1998),319-329.

[25] Watugala G. K.;The Sumudu transform for functions of two variables,Mathematical Engineering

in Industry,8(4),(2002),293-302.

[26] Weerakoon S.;Applications of Sumudu Transform to Partial Differential Equations,Int. J.

Math. Educ. Sci. Tech.,25(2),(1994),1277-283.

[27] Weerakoon S.;Complex Inversion formula for Sumudu Transform,Int. J. Math. Educ. Sci.

Tech., 29(4),(1998),618-621.

[28] Zemanian A.H.;Distribution Theory and Transform Analysis : An Introduction to Generalized

Functions with Applications,Mc-Graw Hill,New York,Republished by Dover Publication,

Inc., New York. 1965.

[29] Zemanian A.H.;Generalized Integral Transformations,Interscience Publishers, New

York,Republished by Dover Publications,Inc.,New York.1968.

## Published

31-05-2020

## How to Cite

*Innovare Journal of Sciences*,

*8*(7), 116–122. Retrieved from https://journals.innovareacademics.in/index.php/ijs/article/view/38554

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