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On Certain New Results of Fractional Calculus Involving Product of Generalized Special Functions

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Abstract

The key aim of the present work is to develop extended fractional calculus results associated with product of the generalized extended Mittag–Leffler function, S-function, general class of polynomials and \({\overline{\text{H}}}\)-function. Some special cases involving various simpler and useful special functions are given to show the importance and utilizations of our main findings. The outcomes are very general in characteristic and can be utilized to derive numerous interesting fractional integral operators involving simpler special functions and polynomials having uses in scientific and engineering problems.

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References

  1. Kumar, D., Singh, J., Tanwar, K., Baleanu, D.: A new fractional exothermic reactions model having constant heat source in porous media with power, exponential and Mittag-Leffler Laws. Int. J. Heat Mass Transf. 138, 1222–1227 (2019)

    Article  Google Scholar 

  2. Kumar, D., Singh, J., Baleanu, D.: On the analysis of vibration equation involving a fractional derivative with Mittag-Leffler law. Math. Methods Appl. Sci. (2019). https://doi.org/10.1002/mma.5903

    Article  MATH  Google Scholar 

  3. Bhatter, S., Mathur, A., Kumar, D., Singh, J.: A new analysis of fractional Drinfeld–Sokolov–Wilson model with exponential memory. Physica A. 537, 122578 (2020)

    Article  MathSciNet  Google Scholar 

  4. Singh, J., Kumar, D., Baleanu, D.: New aspects of fractional Biswas-Milovic model with Mittag-Leffler law. Math. Model. Natural Phenomena 14, 303 (2019)

    Article  MathSciNet  Google Scholar 

  5. Goswami, A., Singh, J., Kumar, D.: An efficient analytical approach for fractional equal width equations describing hydro-magnetic waves in cold plasma. Physica A 524, 563–575 (2019)

    Article  MathSciNet  Google Scholar 

  6. Love, E.R.: Some integral equations involving hypergeometric functions. Proc. Edin. Math. Soc. 15, 169–198 (1967)

    Article  MathSciNet  Google Scholar 

  7. McBride, A.C.: Fractional powers of a class of ordinary differential operators. Proc. London Math. Soc. (III) 45, 519–546 (1982)

    Article  MathSciNet  Google Scholar 

  8. Kalla, S.L.: Integral operators involving Fox’s H-function I. Acta Mexicana Cienc. Tecn. 3, 117–122 (1969)

    MathSciNet  MATH  Google Scholar 

  9. Kalla, S.L.: Integral operators involving Fox’s H-function II. Acta Mexicana Cienc. Tecn. 7, 72–79 (1969)

    MathSciNet  MATH  Google Scholar 

  10. Kalla, S.L., Saxena, R.K.: Integral operators involving hypergeometric functions. Math. Z. 108, 231–234 (1969)

    Article  MathSciNet  Google Scholar 

  11. Kalla, S.L., Saxena, R.K.: Integral operators involving hypergeometric functions II. Univ. Nac. Tucuman Rev. Ser A 24, 31–36 (1974)

    MathSciNet  MATH  Google Scholar 

  12. Saigo, M.: A remark on integral operators involving the Gauss hypergeometric functions. Math. Rep. Kyushu Univ. 11, 135–143 (1978)

    MathSciNet  MATH  Google Scholar 

  13. Saigo, M.: A certain boundary value problem for the Euler-Darboux equation I. Math. Japonica 24, 377–385 (1979)

    MathSciNet  MATH  Google Scholar 

  14. Saigo, M.: A certain boundary value problem for the Euler-Darboux equation II. Math. Japonica 25, 211–220 (1980)

    MathSciNet  MATH  Google Scholar 

  15. Saigo, M. and Maeda, N.: More generalization of fractional calculus. In Transform Methods and special functions, Verna, 1996 (Proc. 2nd Intern. Workshop Eds. P.Rusev, I. Dimovski, V. Kiryakova), IMI-BAS, Sofia, 386–400 (1998)

  16. Saxena, R.K., Saigo, M.: Generalized fractional calculus of the H-function associated with the Appell function. J. Frac. Calc. 19, 89–104 (2001)

    MathSciNet  MATH  Google Scholar 

  17. Bansal, M.K., Kumar, D., Jain, R.: Interrelationships Between Marichev Saigo Maeda fractional integral operators, the Laplace transform and the H(bar)-function. Int. J. Appl. Comput. Math. 5, 103 (2019)

    Article  Google Scholar 

  18. Bansal, M.K., Jain, R., Kumar, D.: A study of Marichev Saigo Maeda fractional integral operators associated with S-generalized Gauss hypergeometric function. Kyungpook Math. J. 59(3), 433–443 (2019)

    MathSciNet  MATH  Google Scholar 

  19. Choudhary, A., Kumar, D., Singh, J.: Some fractional calculus results pertaining to Mittag-Leffler type functions. J. Appl. Math. Statistics Informatics 13(1), 31–48 (2017)

    Article  MathSciNet  Google Scholar 

  20. Purohit, S.D., Suthar, D.L., Kalla, S.L.: Merichev-Saigo-Maeda fractional integration operators of the Bessel functions. Matematiche (Catania) 67, 21–32 (2012)

    MathSciNet  MATH  Google Scholar 

  21. Miller, K.S., Ross, B.: An introduction to the fractional Calculus and differential equations, a Wiley intersciences publilcation. Wiley, New York (1993)

    Google Scholar 

  22. Singh, H., Kumar, D., Baleanu, D.: Methods of mathematical modelling: fractional differential equations. CRC Press, Boca Ratan London New York (2019)

    Book  Google Scholar 

  23. Debnath, L., Bhatta, D.: Integral transforms and their applications. Chapman and Hall/CRC Press, Boca Raton FL (2006)

    MATH  Google Scholar 

  24. Marichev, O.I.: Volterra equation of Mellin convolution type with a Horn function in the kernel (In Russian). Izv. ANBSSR Ser. Fiz.-Mat Nauk 1, 128–129 (1974)

    Google Scholar 

  25. Samko, S., Kilbas, A., and Marichev, O.: Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Sci. Publ., New York (1993)

  26. Veeresha, P., Baskonus, H.M., Prakasha, D.G., Gao, W., Yel, G.: Regarding new numerical solution of fractional Schistosomiasis disease arising in biological phenomena, Chaos. Solitons & Fractals 133, 109661 (2020)

    Article  MathSciNet  Google Scholar 

  27. Ravichandran, C., Jothimani, K., Baskonus, H.M., Valliammal, N.: New results on nondensely characterized integrodifferential equations with fractional order. Eur. Phys. J. Plus 133, 109 (2018)

    Article  Google Scholar 

  28. Subashini, R., Ravichandran, C., Jothimani, K., Baskonus, H. M.: Existence results of Hilfer integro-differential equations with fractional order, Discrete Contin. Dyn. Syst. - S, 13(3), 911-923 (2020)

  29. Shah, A., Khan, R.A., Khan, A., Khan, H., Gómez-Aguilar, J.F.: Investigation of a system of nonlinear fractional order hybrid differential equations under usual boundary conditions for existence of solution. Math. Methods Appl. Sci. 44(2), 1628–1638 (2021)

    Article  MathSciNet  Google Scholar 

  30. Khan, H., Gómez-Aguilar, J.F., Abdeljawad, T., Khan, A.: Existence results and stability criteria for ABC-fuzzy-Volterra integro-differential equation. Fractals 28(8), 2040048 (2020)

    Article  Google Scholar 

  31. Abdeljawad, T., Atangana, A., Gómez-Aguilar, J.F., Jarad, F.: On a more general fractional integration by parts formulae and applications. Physica A 536, 122494 (2019)

    Article  MathSciNet  Google Scholar 

  32. Khan, A., Khan, H., Gómez-Aguilar, J.F., Abdeljawad, T.: Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel. Chaos Solitons Fractals 127, 422–427 (2019)

    Article  MathSciNet  Google Scholar 

  33. Srivastava, H.M., Karlson, P.W.: Multiple Gaussian hypergeometric series. Ellis Horwood Limited, New York (1985)

    Google Scholar 

  34. Inayat-Hussain, A.A.: New properties of hypergeometric series derivable from Feynman integrals: I transformation and reduction formulae. J. Phys. A. Math. Gen. 20, 4109–4117 (1987)

    Article  MathSciNet  Google Scholar 

  35. Inayat-Hussain, A.A.: New properties of hypergeometric series derivable from Feynman integrals: II a generalization of H-function. J. Phys. A. Math. Gen. 20, 4119–4128 (1987)

    Article  MathSciNet  Google Scholar 

  36. Buschman, R.G., Srivastava, H.M.: The H-function associated with a certain class of Feynman integrals. J. Phys. A Math. Gen. 23, 4707–4710 (1990)

    Article  Google Scholar 

  37. Saxena, R.K., Daiya, J.: Integral transforms of the S-functions. Matematiche (Catania) 70(2), 147–159 (2015)

    MathSciNet  MATH  Google Scholar 

  38. Diaz, R., and Pariguan, E.: On hypergeometric functions and Pochhammer k-symbol, Divulgaciones Matem´aticas, 15(2), 179–192 (2007)

  39. Bansal, M.K., Jolly, N., Jain, R., Kumar, D.: An integral operator involving generalized Mittag-Leffler function and associated fractional calculus results. J. Anal. 27(3), 727–740 (2019)

    Article  MathSciNet  Google Scholar 

  40. Özergin, E., Özarslan, M.A., Altin, A.: Extension of gamma, beta and hypergeometric functions. J. Comput. Appl. Math. 235, 4601–4610 (2011)

    Article  MathSciNet  Google Scholar 

  41. Srivastava, H.M.: A contour integral involving Fox’s H-function. Indian J. Math. 14, 1–6 (1972)

    MathSciNet  MATH  Google Scholar 

  42. Sharma, M., Jain, R.: A note on a generalized M-Series as a special function of fractional calculus. J. Fract. Calc. Appl. Anal. 12(4), 449–452 (2009)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Malaviya National Institute of Technology, Jaipur, Rajasthan, 302017, India

    Sanjay Bhatter & Amit Mathur

  2. Department of Mathematics, University of Rajasthan, Jaipur, Rajasthan, 302004, India

    Devendra Kumar

  3. Department of Mathematics, JECRC University, Jaipur, Rajasthan, 303905, India

    Jagdev Singh

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  1. Sanjay Bhatter
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  2. Amit Mathur
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  3. Devendra Kumar
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  4. Jagdev Singh
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All the authors have worked equally in this manuscript. All the four authors have read and approved the final manuscript.

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Correspondence to Devendra Kumar.

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Bhatter, S., Mathur, A., Kumar, D. et al. On Certain New Results of Fractional Calculus Involving Product of Generalized Special Functions. Int. J. Appl. Comput. Math 8, 135 (2022). https://doi.org/10.1007/s40819-022-01253-0

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