Classifications and duality relations for several integral transforms

Volume 10, Issue 12, pp 6324--6332 http://dx.doi.org/10.22436/jnsa.010.12.16

Publication Date: December 13, 2017 Submission Date: September 23, 2017

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 2098 Downloads
• 3808 Views

Authors

Xin Liang - State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou 221116, China. Feng Gao - State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou 221116, China. - School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, China. Shan-Jie Su - State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou 221116, China. Zhen Wang - College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China. Xiao-Jun Yang - State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou 221116, China. - School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, China. - College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China.

Abstract

In this paper, we classify several integral transforms into two categories according to the types of their kernel functions and two novel definitions of general integral transforms are suggested. Based on the general integral transforms, some of their basic properties are proved. In addition, the dualities between those two kinds of integral transforms are deducted and discussed in detail. The interesting coupling relations in symmetric form is illustrated graphically. The analysis shows that the classifications are reasonable and the dualities are significant.

Share and Cite

Keywords

• Integral transforms
• kernel functions
• dualities

MSC

•  44A99
•  83C15
•  76R50
•  35Q99

References

• [1] T. Abdeljawad, D. Baleanu, On fractional derivatives with exponential kernel and their discrete versions, Rep. Math. Phys., 80 (2017), 11–27.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] A. Atangana, Extension of the Sumudu homotopy perturbation method to an attractor for one-dimensional Keller-Segel equations, Appl. Math. Model., 39 (2015), 2909–2916.
  • View Article
  • MathSciNet
  • Google Scholar


• [3] A. Atangana, J. F. Gómez-Aguilar, A new derivative with normal distribution kernel: theory, methods and applications, Phys. A, 476 (2017), 1–14.
  • View Article
  • MathSciNet
  • Google Scholar


• [4] 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


• [5] G. Doetsch, L. Debnath, Introduction to the theory and application of the Laplace transformation, Translated from the second German edition by Walter Nader, Springer-Verlag, New York-Heidelberg (1974)
  • MathSciNet


• [6] C. Donolato, Analytical and numerical inversion of the Laplace-Carson transform by a differential method, Comput. Phys. Comm., 145 (2002), 298–309.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] 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


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


• [9] T. M. Elzaki, Application of new transform Elzaki transform to partial differential equations, Global J. Pure Appl. Math., 7 (2011), 65–70.
  • Google Scholar


• [10] F. Jarad, E. Uğurlu, T. Abdeljawad, D. Baleanu, On a new class of fractional operators, Adv. Difference Equ., 2017 (2017), 16 pages.
  • View Article
  • MathSciNet
  • Google Scholar


• [11] M.-J. Kang, J.-K. Jeon, H.-J. Han, S.-M. Lee, Analytic solution for American strangle options using Laplace-Carson transforms, Commun. Nonlinear Sci. Numer. Simul., 47 (2017), 292–307.
  • View Article
  • MathSciNet
  • Google Scholar


• [12] A. Kılıçman, H. Eltayeb, K. A. M. Atan , A note on the comparison between Laplace and Sumudu transforms, Bull. Iranian Math. Soc., 37 (2011), 131–141.
  • MathSciNet
  • Google Scholar
  • MATH


• [13] 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


• [14] D. Kumar, J. Singh, D. Baleanu , A hybrid computational approach for Klein-Gordon equations on Cantor sets, Nonlinear Dynam., 87 (2017), 511–517.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] M. Lévesque, M. D. Gilchrist, N. Bouleau, K. Derrien, D. Baptiste, Numerical inversion of the Laplace–Carson transform applied to homogenization of randomly reinforced linear viscoelastic media , Comput. Mech., 40 (2008), 771–789.
  • View Article
  • Google Scholar
  • MATH


• [16] X. Liang, F. Gao, Y.-N. Gao, X.-J. Yang, Applications of a novel integral transform to partial differential equations, J. Nonlinear Sci. Appl. , 10 (2017), 528–534.
  • View Article
  • MathSciNet
  • Google Scholar


• [17] X. Liang, G. N. Liu, S. J. Su, Applications of a novel integral transform to the convection-dispersion equations , Therm. Sci., 21 (2017), S233–S240.
  • Google Scholar


• [18] J. Singh, D. Kumar, M. A. Qurashi, D. Baleanu, A novel numerical approach for a nonlinear fractional dynamical model of interpersonal and romantic relationships, Entropy, 19 (2017), 17 pages.
  • View Article
  • Google Scholar


• [19] J. Singh, R. Swroop, D. Kumar, A computational approach for fractional convection-diffusion equation via integral transforms, Ain Shams Eng. J., (2016), in press
  • View Article
  • Google Scholar


• [20] H. M. Srivastava, M.-J. Luo, R. K. Raina, A new integral transform and its applications, Acta Math. Sci. Ser. B Engl. Ed., 35 (2015), 1386–1400.
  • View Article
  • MathSciNet
  • Google Scholar


• [21] S. Weerakoon, Application of Sumudu transform to partial differential equations, Internat. J. Math. Ed. Sci. Tech., 25 (1994), 277–283.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] X.-J. Yang , A new integral transform method for solving steady heat-transfer problem, Therm. Sci., 20 (2016), S639–S642.
  • Google Scholar


• [23] X.-J. Yang, A new integral transform with an application in heat-transfer problem , Therm. Sci., 20 (2016), S677–S681.
  • View Article
  • Google Scholar


• [24] X.-J. Yang, New integral transform method for solving steady heat-transfer problem , Therm. Sci., 20 (2016), S639–S642.
  • View Article
  • Google Scholar


• [25] X.-J. Yang, A new integral transform operator for solving the heat-diffusion problem, Appl. Math. Lett., 64 (2017), 193– 197.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [26] X.-J. Yang, D. Baleanu, Y. Khan, S. T. Mohyud-Din, Local fractional variational iteration method for diffusion and wave equations on Cantor sets, Romanian J. Phys., 59 (2014), 36–48.
  • MathSciNet
  • Google Scholar


• [27] X.-J. Yang, Y.-G. Yang, C. Cattani, M.-Z. Zhu , A new technique for solving the 1-D Burgers equation, Therm. Sci., 21 (2017), S129–S136.
  • Google Scholar