Sumudu Transformation or What Else Can Laplace Transformation Do
- 1 Max-Planck-Group “Nonclassical Radiation” at Physical Institute of Humboldt University Berlin, Berlin, Germany
Abstract
The transition from a known Taylor series of a known function f(x) to a new function primarily defined by the infinite power series with coefficients f<sup>(n)</sup>(0) from the Taylor series of the function f(x) can be made by an integral transformation which is a modified Laplace transformation and is called Sumudu transformation. It makes the transition from the Exponential series to the Geometric series and may help to evaluate new infinite power series from known Taylor series. The Sumud u transformation is demonstrated to be a limiting case of Fractional integration. Apart from the basic Sumudu integral transformation we discuss a modification where the coefficients from the Taylor series are not changed to f<sup>(n)</sup>(0) but only to . Beside simple examples our applications are mainly concerned to calculate new Generating functions for Hermite polynomials from the basic ones.
- Watugala, G.K. (1993) Sumudu Transform: A New Integral Transform to Solve Differential Equations and Control Engineering Problems. International Journal of Mathematical Education in Science and Technology, 24, 35-43. https://doi.org/10.1080/0020739930240105
- Belgacem, F.B.M. and Karabelli, A.A. (2006) Sumudu Transform Fundamental Properties Investigations and Applications. Journal of Applied Mathematics and Stochastic Analysis, 2006, Article ID: 91083. https://doi.org/10.1155/JAMSA/2006/91083
- Belgacem, F.B.M. (2009) Sumudu Applications to Maxwell’s Equations. PIERS ONLINE, 5, 355-360. https://doi.org/10.2529/PIERS090120050621
- Alenezi, A.M. and Belgacem, F.B.M. (2014) Sumudu Transform Based Treatment of Krawtchouk Polynomials and Their Integral Transforms. AIP Conference Proceedings, 1637, 1395. https://doi.org/10.1063/1.4907306
- Demiray, S.T., Bulut, H. and Belgacem, F.B.-M. (2015) Sumudu Transform Method for Analytical Solutions of Fractional Type Ordinary Differential Equations. Mathematical Problems in Engineering, 10, Article ID: 131690.
- Katatbeh, Qu. and Belgacem, F.B.M. (2017) Applications of the Sumudu Transform to Fractional Differential Equations. Nonlinear Studies, 18, 99-112. https://www.researchgate.net/publication/234114475
- Srivastava, H.M., Luo, M. and Raina, R.K. (2015) A New Integral Transform and Its Applications. Acta Mathematica Scientia, 35, 1386-1400. https://doi.org/10.1016/S0252-9602(15)30061-8
- Doetsch, G. (1967) Anleitung zum praktischen Gebrauch der Laplace-Transformation und der Z-Transformation. 3. Aufl, R. Oldenburg, München.
- Bateman, H. and Erdélyi, A. (1954) Tables of Integral Transforms, Vol. I. McGraw-Hill, New York.
- Ditkin, V.A. and Prudnikov, A.P. (1965) Spravochnik po operatsionnomy ischisleniyu, Vysshaya shkola, Moskva. [Integral Transforms and Operational Calculus.] Pergamon Press, Oxford.
- Brychkov, Yu.A. and Prudnikov, A.P. (1977) Integral Transformations of Generalized Functions. Nauka, Moskva. (In Russian)
- Zemanian, A.H. (1987) Generalized Integral Transformations. Dover, New York.
- Erdélyi, A. (1974) Higher Transcendental Functions, Vol. 2. Nauka, Moskva.
- Wünsche, A. (2015) Generating Functions for Products of Laguerre 2D and Hermite 2D Polynomials. Applied Mathematics, 6, 2142-2168. https://doi.org/10.4236/am.2015.612188