On the generalized fractional Laplace transform

Kumar, Virendra

doi:10.2478/mjpaa-2021-0030

Abstract

In the present paper a generalization of the Laplace transform is introduced and studied. Its inversion formula is also obtained. As an application, we obtain the generalized fractional Laplace transform of a general class of functions and a product of the Fox’s H- function and general class of functions. The results obtained are of general nature and capable of yielding a large number of known or new results as special cases. For illustration, some special cases involving important special functions are mentioned.

References

[1] B. R. Bhonsle, A relation between Laplace and Hankel transforms, Proc. Glasgow Math. Assoc., 5, No. 3 (1962), 114-115.
Search in Google Scholar Back to article
[2] B. R. Bhonsle, A relation between Laplace and Hankel transforms, Math. Japon., 10 (1965), 85-89.
Search in Google Scholar Back to article
[3] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol I, McGraw-Hill Book Company, New York, Toronto and London (1953).
Search in Google Scholar Back to article
[4] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol II, McGraw-Hill Book Company, New york, Toronto and London (1953).
Search in Google Scholar Back to article
[5] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol III, McGraw-Hill Book Company, New york, Toronto and London (1953).
Search in Google Scholar Back to article
[6] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Tables of Integral Transforms, Vol I, McGraw-Hill Book Company, New York (1954).
Search in Google Scholar Back to article
[7] K. C. Gupta and S. M. Agrawal, Unified theorems involving H-function transform and Meijer Bessel function transform, Proc. Indian Acad. Sci., 96, No. 2 (1987), 125-130.
Search in Google Scholar Back to article
[8] S. P. Goyal and S. K. Vasishta, Certain relations between generalized Kontorovitch-Lebdev transform and H-function transform, Ranchi Univ. Math. Jour., 6 (1975), 95-102.
Search in Google Scholar Back to article
[9] S. P. Goyal and R. M. Jain, Certain results for two-dimensional Laplace transform with applications, Proc. Nat. Acad. Sci. India, 59(A), No. (III) (1989), 407-414.
Search in Google Scholar Back to article
[10] S. P. Goyal and R. K. Laddha, On the generalized Riemann zeta functions and the generalized Lambert transform, Ganita Sandesh, 11, No. 2 (1997), 99-108.
Search in Google Scholar Back to article
[11] P. Humbert and R. P. Agarwal, Sur la fonction de Mittag-Leffler et quelques unes de ses generalizations, Bull. Sci. Math., 77, No. 2 (1953), 180-185.
Search in Google Scholar Back to article
[12] V. Kumar, A general class of functions and N- fractional calculus, J. Rajasthan Acad. Phys. Sci., 11, No. 3 (2012), 223-230.
Search in Google Scholar Back to article
[13] V. Kumar, N-fractional calculus of general class of functions and Fox’s H-function, Proc. Natl. Acad. Sci., sect. A, Phys. Sci. 83, No. 3 (2013), 271-277.
Search in Google Scholar Back to article
[14] V. Kumar, The Euler transform of V- function, Afr. Mat., 29, No. (1-2) (2018), 23-2710.1007/s13370-017-0522-8
Search in Google Scholar Back to article
[15] V. Kumar, Theorems connecting Mellin and Hankel transforms, J. Classical Anal., 15, No. 2 (2019), 123-130.
Search in Google Scholar Back to article
[16] V. Kumar, Theorems connecting Fourier sine transform and Hankel transform, SeMA Journal, 77, No. 2 (2020), 219-225.
Search in Google Scholar Back to article
[17] V. Kumar, Theorems connecting Stieltjes transform and Hankel transform, São Paulo J. Math. Sci., DOI: 10.1007/s40863-020-00182-4.10.1007/s40863-020-00182-4
Search in Google Scholar Back to article
[18] V. Kumar, On a General Theorem Connecting Laplace Transform and Generalized Weyl fractional Integral Operator Involving Foxs H-Function and a General Class of Functions, J. Frac. Calc. and Appl., 11, No. 2 (2020), 270-280.
Search in Google Scholar Back to article
[19] V. Kumar, On a Generalized Fractional Fourier Transform, Palestine J. Maths., 9, No. 2 (2020), 903-907.
Search in Google Scholar Back to article
[20] A. A. Kilbas and M. Saigo, H-Transforms: Theory and Application, Chapman & Hall/CRC Press, Boca Raton, London, New york (2004).
Search in Google Scholar Back to article
[21] G. M. Mittag-Leffler, Sur la nouvelle function Eα(x), C. R. Acad. Sci. Paris, 137 (1903), 554-558.
Search in Google Scholar Back to article
[22] A. M. Mathai, R. K. Saxena and H. J. Haubold, The H-Function: Theory and Applications. Springer, New york (2010).10.1007/978-1-4419-0916-9
Search in Google Scholar Back to article
[23] G. D, Medina, N. R. Ojeda, J. H. Pereira and L. G. Romero, Fractional Laplace transform and fractional calculus, Inter. Math. Forum, 12, No. 20 (2017), 991-1000.
Search in Google Scholar Back to article
[24] A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and Series, vol. 3, More Special Functions, Gordon and Breach Science Publishers, New york (1990).
Search in Google Scholar Back to article
[25] T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J., 19 (1971), 7-15.
Search in Google Scholar Back to article
[26] L. G. Romero, G. D. Medina, N. R. Ojeda and J. H. Pereira, A new alfa-integral Laplace transform, Asian J. Current Engg. Maths., 5 (2016), 59-62.
Search in Google Scholar Back to article
[27] H. M. Srivastava, A relation between Meijer and Hankel transforms, Math. Japon., 11 (1966), 11-13.
Search in Google Scholar Back to article
[28] H. M. Srivastava, On a relation between Laplace and Hankel transforms, Matematiche (Catania), 21 (1966), 199-202.
Search in Google Scholar Back to article
[29] H. M. Srivastava, Some remarks on a generalization of the Stieltjes transform, Publ. Math. Debrecen, 23 (1976), 119-122.
Search in Google Scholar Back to article
[30] H. M. Srivastava, K. C. Gupta and S. P. Goyal, The H-function of one and two Variables with Applications, South Asian Publishers, New Delhi (1982).
Search in Google Scholar Back to article
[31] H. M. Srivastava and V. K. Tuan, A new convolution theorem for the Stieltjes transform and its application to a class of singular integral equations, Arch. Math.(Basel), 64, No. 2 (1995), 144-149.
Search in Google Scholar Back to article
[32] H. M. Srivastava and O. Yürekli, A theorem on a Stieltjes-type integral transform and its applications to obtain infinite integrals of elementary and special functions, Complex Vari., Theory Appl., 28, No. 2 (1995), 159-168.
Search in Google Scholar Back to article
[33] A. Wiman, Über den Fundamentalsatz in der Teorie der Funktionen Eα(x), Acta. Math., 29 (1905), 191-201.10.1007/BF02403202
Search in Google Scholar Back to article
[34] E. M. Wright, The asymptotic expansion of the generalized Bessel function, Proc. London Math. Soc., 38, No. 2 (1935), 257–270.
Search in Google Scholar Back to article
[35] S. Yakubovich and M. Martins, On the iterated Stieltjes transform and its convolution with applications to singular integral equations, Trans. Spec. Func., 25, No. (2013), DOI: 10.1080/10652469.2013.868457.10.1080/10652469.2013.868457
Search in Google Scholar Back to article

On the generalized fractional Laplace transform

Abstract

Paradigm

My account