On double Natural transform and its applications

Volume 10, Issue 4, pp 1744--1754 http://dx.doi.org/10.22436/jnsa.010.04.36

Publication Date: April 20, 2017 Submission Date: December 28, 2016

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Authors

Adem Kılıçman - Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, 43400 Serdang, Selangor, Malaysia. Maryam Omran - Institute for Mathematical Research, University Putra Malaysia , 43400 Serdang, Selangor, Malaysia.

Abstract

In this work, we generalize the concept of one dimensional Natural transform to two dimensional Natural transform namely, double Natural transform and some of its properties are given. We also set a relation between double Natural transform and double Laplace, double Sumudu transforms. Further the convolution theorem with a proof is investigated with some details. Double Natural transform is applied to get the solutions of some general linear telegraphs, wave and partial integro-differential equations.

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Keywords

• Double Natural transform
• single Natural transform
• partial differential and integro-differential equations
• convolution theorem.

MSC

•  44A30
•  44A35
•  35Axx
•  45K05

References

• [1] S. K. Q. Al-Omari, On the application of natural transforms, Inter. J. Pure Appl. Math., 85 (2013), 729–744.
  • Google Scholar


• [2] M. A. Asiru, Sumudu transform and the solution of integral equations of convolution type, Internat. J. Math. Ed. Sci. Tech., 32 (2001), 906–910.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] F. B. M. Belgacem, Introducing and analysing deeper Sumudu properties, Nonlinear Stud., 13 (2006), 23–41.
  • MathSciNet
  • Google Scholar
  • MATH


• [4] F. B. M. Belgacem, Sumudu transform applications to Bessel functions and equations, Appl. Math. Sci. (Ruse), 4 (2010), 3665–3686.
  • MathSciNet
  • Google Scholar
  • MATH


• [5] F. B. M. Belgacem, A. A. Karaballi, Sumudu transform fundamental properties investigations and applications, J. Appl. Math. Stoch. Anal., 2006 (2006), 23 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] F. B. M. Belgacem, A. A. Karaballi, S. L. Kalla, Analytical investigations of the Sumudu transform and applications to integral production equations, Math. Probl. Eng., 2003 (2003), 103–118.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] F. B. M. Belgacem, R. Silambarasan, Advances in the natural transform, AIP Conf. Proc., 1493 (2012), 106–110.
  • View Article
  • Google Scholar


• [8] F. B. M. Belgacem, R. Silambarasan, Maxwell’s equations solutions by means of the natural transform, Int. J. Math. Eng. Sci. Aerosp., 3 (2012), 313–323.
  • Google Scholar
  • MATH


• [9] F. B. M. Belgacem, R. Silambarasan, Theory of natural transform, Int. J. Math. Eng. Sci. Aerosp., 3 (2012), 99–124.
  • Google Scholar


• [10] Y. A. Brychkov, H.-J. Glaeske, A. P. Prudnikov, Vu Kim Tuan, Multidimensional integral transformations, Gordon and Breach Science Publishers, Philadelphia, PA (1992)
  • MathSciNet
  • Google Scholar


• [11] V. A. Ditkin, A. P. Prudnikov, Operational calculus in two variables and its applications, Translated by D. M. G. Wishart, International Series of Monographs on Pure and Applied Mathematics, Pergamon Press, New York- Oxford-London-Paris (1962)
  • MathSciNet
  • Google Scholar


• [12] D. G. Duffy, Transform methods for solving partial differential equations, Second edition, Chapman & Hall/CRC, Boca Raton, FL (2004)
  • Google Scholar


• [13] H. Eltayeb, A. Kılıçman, A note on solutions of wave, Laplace’s and heat equations with convolution terms by using a double Laplace transform, Appl. Math. Lett., 21 (2008), 1324–1329.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] H. Eltayeb, A. Kılıçman, A note on the Sumudu transforms and differential equations, Appl. Math. Sci. (Ruse), 4 (2010), 1089–1098.
  • MathSciNet
  • Google Scholar
  • MATH


• [15] H. Eltayeb, A. Kılıçman, On double Sumudu transform and double Laplace transform, Malays. J. Math. Sci., 4 (2010), 17–30.
  • Google Scholar


• [16] H. Eltayeb, A. Kılıçman, A note on double Laplace transform and telegraphic equations, Abstr. Appl. Anal., 2013 (2013), 6 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] T. A. Estrin, T. J. Higgins, The solution of boundary value problems by multiple Laplace transformations, J. Franklin Inst., 252 (1951), 153–167.
  • View Article
  • MathSciNet
  • Google Scholar


• [18] Z. H. Khan, W. A. Khan, N-transform properties and applications, NUST J. Eng. Sci. (NJES), 1 (2008), 127–133.
  • Google Scholar


• [19] A. Kılıçman, H. Eltayeb, A note on integral transforms and partial differential equations, Appl. Math. Sci. (Ruse), 4 (2010), 109–118.
  • MathSciNet
  • Google Scholar


• [20] A. Kılıçman, H. Eltayeb, On a new integral transform and differential equations, Math. Probl. Eng., 2010 (2010), 13 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [21] A. Kılıçman, H. Eltayeb, Some remarks on the Sumudu and Laplace transforms and applications to differential equations, ISRN Appl. Math., 2012 (2012), 13 pages.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] A. Kılıçman, H. Eltayeb, R. P. Agarwal, On Sumudu transform and system of differential equations, Abstr. Appl. Anal., 2010 (2010), 11 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [23] A. Kılıçman, H. E. Gadain, An application of double Laplace transform and double Sumudu transform, Lobachevskii J. Math., 30 (2009), 214–223.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [24] A. Kılıçman, H. E. Gadain, On the applications of Laplace and Sumudu transforms, J. Franklin Inst., 347 (2010), 848– 862.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [25] M. Omran, A. Kılıçman, Natural transform of fractional order and some properties, Cogent Math., 3 (2016), 1–8.
  • View Article
  • MathSciNet
  • Google Scholar


• [26] R. Silambarasan, F. B. M. Belgacem, Applications of the natural transform to Maxwells equations, Prog. Electromagnetic Research Symposium Proc. Suzhou, China, (2011), 899–902.
  • Google Scholar


• [27] I. N. Sneddon, The use of integral transforms, McGraw-Hill, New York (1972)
  • Google Scholar


• [28] M. R. Spiegel, Schaum’s outline of theory and problems of Laplace transforms, McGraw-Hill, New York (1965)
  • Google Scholar


• [29] J. M. Tchuenche, N. S. Mbare, An application of the double Sumudu transform, Appl. Math. Sci. (Ruse), 1 (2007), 31–39.
  • MathSciNet
  • Google Scholar
  • MATH


• [30] G. K. Watugala , Sumudu transform: a new integral transform to solve differential equations and control engineering problems, Internat. J. Math. Ed. Sci. Tech., 24 (1993), 35–43.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [31] G. K. Watugala, The Sumudu transform for functions of two variables, Math. Engrg. Indust., 8 (2002), 293–302.
  • MathSciNet
  • Google Scholar
  • MATH