Some Relations for the Generalized G ̃n,P ̃n Integral Transforms and Riemann-Liouville, Weyl Integral Operators

Abstract

In this paper, Parseval-Goldstein type theorems involving the G ̃n-integral transform which is modified from G_2n-integral transform [7] and the -integral transform [8] are examined. Then, theorems in this paper are shown to yield a number of new identities involving several well-known integral transforms. Using these theorems and their corollaries, a number of interesting infinite integrals of elementary and special functions are presented. Generalizations of Riemann-Liouville and Weyl fractional integral operators are also defined. Some theorems relating generalized Laplace transform, generalized Widder Potential transform, generalized Hankel transform and generalized Bessel transform are obtained. Some illustrative examples are given as applications of these theorems and their results.

Keywords

• Laplace transform
• Widder Potential transform
• Hankel transform
• Riemann-Liouville fractional integral
• Weyl fractional integral

References

1. [1] Debnath, L., and Bhatta, D., “Integral Transforms and Their Applications”, 2nd Edition, Chapman Hall/CRC, Boca Raton, FL, (2007).
2. [2] Miller, K.S., and Ross, B., “An Introduction to Fractional Calculus and Fractional Differential Equations”, John Wiley-Interscience Publication, New York, Toronto Singapore, (1993).
3. [3] Podlubny, I., “Fractional Differential Equations: an Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications”, Mathematics in Science and Engineering, Academic Press, London, (1999).
4. [4] Dernek, A.N., and Aylıkcı F., "Some results on the 𝒫ν,2n,𝒦ν,n and Hν,n-integral transforms", Turkish Journal of Mathematics, 41(2): 337-349, (2017). DOI:10.3906/mat-1501-79
5. [5] Yürekli, O., and Sadek, I., "A Parseval-Goldstein type theorem on the Widder potential transform and its applications", International Journal of Mathematics and Mathematical Sciences, 14(3): 517-524, (1991). DOI:10.1155/S0161171291000704
6. [6] Yürekli, O., "Identities on fractional integrals and various integral transforms", Applied Mathematics and Computation, 187: 559-566, (2007). DOI: 10.1016/j.amc.2006.09.001
7. [7] Dernek, N., Aylıkcı, F., and Balaban, G., "New identities for the generalized Glasser transform, the generalized Laplace transform and the ℇ2𝑛,1-transform", IECMSA-IV., Book of Abstracts: 135-138, (2015).
8. [8] Dernek, N., Ölçücü E.Ö., and Aylıkcı, F., "New identities and Parseval type relations for the generalized integral transforms ℒ4n,𝒫4n,ℱs,2n and ℱc,2n", Applied Mathematics and Computation, 269: 536-547, (2015). DOI: 10.1016/j.amc.2015.07.095.

9. [9] Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi F.G., “Tables of Integral Transforms. Vol. II”, McGraw-Hill Book Company, Inc., New York-Toronto-London, based, in part, on notes left by Harry Bateman, (1954).
10. [10] Widder, D.V., “An Introduction to Transform Theory”, New York: Academic Press, bibliography: 243-246, (1971).
11. [11] Glasser, M.L., “Some Bessel function integrals”, Kyungpook Mathematical Journal, 13(2): 171-174, (1973).
12. [12] Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G., “Higher Transcendental Functions. Vol. I”, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981, based on notes left by Harry Bateman, With a preface by Mina Rees, With a foreword by E. C. Watson, Reprint of the (1953) original.
13. [13]Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G., “Higher Transcendental Functions. Vol. II”, Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981, based on notes left by Harry Bateman, Reprint of the (1953) original.
14. [14] Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G., “Tables of Integral Transforms. Vol. I”, McGraw-Hill Book Company, Inc., New York-Toronto-London, (1954), based, in part, on notes left by Harry Bateman.
15. [15]Jangid, K., Parmar, R.K., Agarwal, R., and Purohit, S.D., “Fractional calculus and integral transforms of the product of a general class of polynomial and incomplete Fox–Wright functions”, Advances in Difference Equations, 606: 1-17, (2020). DOI:10.1186/s13662-020-03067-0
16. [16]Agarwal, R., Yadav, M.P., Baleanu, D., and Purohit, S.D., “Existence and uniqueness of miscible flow equation through porous media with a non singular fractional derivative”, AIMS Mathematics, 5(2): 1062–1073, (2020). DOI: 10.3934/math.2020074
17. [17] Song, L., Xu, S., and Yang, Y., “Dynamical models of happiness with fractional order”, Communications in Nonlinear Science and Numerical Simulation, 15(3): 616-628, (2010).
18. [18] De, Gaetano, A., Sakulrang, S., Borri, A., Pitocco, D., Sungnul, S., and Moore, E.J., “Modeling continuous glucose monitoring with fractional differential equations subject to shocks”, Journal of Theoretical Biology, 526: 110776, (2021).
19. [19] Shen, W.Y., Chu,Y.M., ur, Rahman, M., Mahariq, I., and Zeb, A., “Mathematical analysis of HBV and HCV co-infection model under nonsingular fractional order derivative”, Results in Physics, 28: 104582, (2021).
20. [20] Jafari, H., Ganji, R.M., Nkomo, N.S., and Lv, Y.P.A., “Numerical study of fractional order population dynamics model”, Results in Physics, 27: 104456, (2021).