Applications of Generalized Integral Transforms

Ніна Опанасівна Вірченко, Марія Олександрівна Четвертак

Abstract


Background. The article studies the generalized integral transforms, such as generalizedLaplace’ integral transform, generalized Stieltjes’ integral transformation.

Objective. Investigation some applications of the new generalized classical integral transforms for solving integral and differential equations, for calculation integrals which are absent in reference and scientific literature.

Methods. We apply the methods the theory of functional variable, the theory of mathematical physics, the theory of special function and the methods the theory applied analysis.

Results. Some new forms of generalized Laplace’ integral transform are given. With help of the r-generalized confluent hypergeometric function the generalized Stieltjes’ integral transform is introduced. The inverse theorem of the generalized Stieltjes’ integral transform is proved. New properties of the r-generalized confluent hypergeometric function are explored.

Conclusions. New properties of the r-generalized confluent hypergeometric function are explored. These functions are expressing in the form by the Fox–Wright functions. Some forms of generalized Laplace’ integral transform are given. With help of the r-generalized confluent hypergeometric function the generalized Stieltjes’ integral transform is introduced. Interesting examples of applications of new generalized integral transforms in the theory of differential and integral equations, for calculation of integrals, which are absent in mathematical literature are given.

Keywords


Generalized integral transforms; Laplace’ integral transforms; Stieltjes’ integral transforms

References


A.A. Kilbas and M. Saigo, H-transforms. Charman and Hall, 2004, 390 p.

Higher Transcendental Functions, H. Bateman and A. Erdelyi, eds., vol. 1.New York: Mc. Grow-Hill 1953, 402 p.

Tables of Integral Transforms. vol. 1.New York,Toronto,London: MCGRAW-Hill Book Company, 1954, 346 p.

R.P. Agarwal, “Some properties of generalized Hankel transform”, Bull. Calcutta Math. Soc., no.43, pp. 153–167, 1951.

M.A. Al-Bassam, “H-R transform equations of Laguerre type”, Bull. College of Sci., Irag. vol. 9, pp. 181–184, 1966.

N.O. Virchenko et al., “Some results on a generalized hypergeometric function”, Integral Transf. Spec. Func., vol. 12, no. 3, pp. 89–100, 2001.

N.O. Virchenko “Fractional order integral Laguerre’s transform”, in 9th Integr. Colleguium Diff. Equations,Plovdiv,Bulgaria, 1998, 197 p.

N.O. Virchenko et al., “On generalized convolution of sin-, cos- and K-L integral transforms”, Ukr.Math. J., vol. 64, no.1, pp. 81–91, 2012.

B. Gonzalez et al., “A distributional inversion formula for a generalization of the Stieltjes and Poisson transforms”, Integr. Transforms Spec. Func., vol. 20, no. 12, pp. 897–903, 2009.

A.M. Mathai et al., Special Functions for Applied.New York: Springer, 2008, 464 p.

N.O. Virchenko, The Generalized Integral Transforms.Kyiv,Ukraine: Zadruga, 2013, 398 p.

M. Garg et al., “On a generalized finite Hankel transform”, Appl. Math. Comput., vol. 190, pp. 705–711, 2007.


GOST Style Citations


  1. Kilbas A., Saigo M., H-transforms. – Charman and Hall, 2004. – 390 p.

  2. Higher Transcendental Functions. Vol. 1 / Eds. H. Bateman, A. Erdelyi. – New York: Mc. Grow-Hill 1953. – 402 p.

  3. Tables of Integral Transforms. Vol. 1. – New York;Toronto;London: MCGRAW-Hill Book Company, 1954. – 346 p.

  4. Agarwal R.P. Some properties of generalized Hankel transform // Bull. Calcutta Math. Soc. – 1951. – 43. – P. 153–167.

  5. Al-Bassam M.A. H-R transform equations of Laguerre type // Bull. College Sci., Irag. – 1966. – 9. – P. 181–184.

  6. Virchenko N.O., Kalla S.L., Al-Zamel A. Some results on a generalized hypergeometric function // Integral Transf. Spec. Func. – 2001. – 12, № 3. – P. 89–100.

  7. Virchenko N.O. Fractional order integral Laguerre’s transform // 9th Integr. Colleguium Diff. Equations. – Plovdiv,Bulgaria, 1998. – P. 197.

  8. Virchenko N.O., Thao N.X. On generalized convolution of sin-, cos- and K-L integral transforms // Ukr. Math. J. – 2012. – 64, № 1. – P. 81–91.

  9. Gonzalez B., Negrin E. A distributional inversion formula for a generalization of the Stieltjes and Poisson transforms // Integr. Transforms Spec. Func. – 2009. – 20, № 12. – P. 897–903.

  10. Mathai A.M., Haubold H.J. Special Functions for Applied. – New York: Springer, 2008. – 464 p.

  11. Virchenko N.O. The Generalized Integral Transforms. – Kyiv: Zadruga, 2013. – 398 p.

  12. Garg M., Rao A., Kalla S. On a generalized finite Hankel transform // Appl. Math. Comput. – 2007. – 190. – P. 705–711.




DOI: https://doi.org/10.20535/1810-0546.2015.4.50514

Refbacks

  • There are currently no refbacks.




Copyright (c) 2017 NTUU KPI