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New Fractional Operators Including Wright Function in Their Kernels

Year 2023, Volume: 15 Issue: 1, 79 - 88, 30.06.2023
https://doi.org/10.47000/tjmcs.999775

Abstract

In this paper, we defined new two-fractional derivative operators with a Wright function in their kernels. We also gave their Laplace and inverse Laplace transforms. Then, we presented some connections between the new fractional operators. Furthermore, as examples, we obtained solutions of differential equations involving new fractional operators. Finally, we examined the relations of the new fractional operators with the fractional operators, which can be found in the literature.

References

  • Abubakar, U.M., A comparative analysis of modified extended fractional derivative and integral operators via modified extended beta function with applications to generating functions, C¸ ankaya Univ. J. Sci. Eng., 19(1)(2022), 40–50.
  • Abubakar, U.M., Tahir, H.M., Abdulmumini, I.S., Extended gamma, beta and hypergeometric functions: properties and applications, J. Kerala Stat. Assoc., 32(2021), 18–39.
  • Agarwal, P., Jain, S., Mansour, T., Further extended Caputo fractional derivative operator and its applications, Russian J. Math. Phys., 24(4)(2017), 415–425.
  • Agarwal, P., Choi, J., Paris, R.B., Extended Riemann-Liouville fractional derivative operator and its applications, J. Nonlinear Sci. Appl. (JNSA), 8(5)(2015), 451–466.
  • Andrews, G.E., Askey, R., Roy, R., Special Functions, Cambridge University Press, Cambridge, 1999.
  • Ata, E., Generalized beta function defined by Wright function, arXiv:1803.03121v3 [math.CA], (2021).
  • Ata, E., Kıymaz, İ.O., A study on certain properties of generalized special functions defined by Fox-Wright function, Appl. Math. Nonlinear Sci., 5(1)(2020), 147–162.
  • Ata, E., Modified special functions defined by generalized M-series and their properties, arXiv:2201.00867v1 [math.CA], (2022).
  • Ata, E., Kıymaz, İ.O., Generalized gamma, beta and hypergeometric functions defined by Wright function and applications to fractional differential equations, Cumhuriyet Sci. J., 43(4)(2022), 684-695.
  • Atangana, A., Baleanu, D., New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Thermal Sci., 20(2)(2016), 763–769.
  • Baleanu, D., Agarwal, R.P., Parmar, R.K., Alqurashi, M., Salahshour, S., Extension of the fractional derivative operator of the Riemann-Liouville, J. Nonlinear Sci. Appl., 10(2017), 2914–2924.
  • Caputo, M., Fabrizio, M., A new definition of fractional derivative without singular kernel, Progr. Frac. Differ. Appl., 1(2)(2015), 73–85.
  • Chaudhry, M.A., Zubair, S.M., Generalized incomplete gamma functions with applications, J. Comput. Appl. Math., 55(1994), 99–124.
  • Çetinkaya, A., Kıymaz, İ.O., Agarwal, P., Agarwal, R., A comparative study on generating function relations for generalized hypergeometric functions via generalized fractional operators, Adv. Differ. Equ., 2018(1)(2018), 1–11.
  • Debnath, L., Bhatta, D., Integral Transforms and Their Applications, Third Edition, CRC Press, Boca Raton, London, New York, 2015.
  • Gomez-Aguilar, J.F., Atangana, A., New insight in fractional differentiation: Power, exponential decay and Mittag-Leffler laws and applications, EPJ Plus, 132(13)(2017), 1–21.
  • İlhan, E., Genelleştirilmiş Özel Fonksiyonlar Yardımıyla Tanımlanan Kesirli Operatörler ve Uygulamaları, Kırşehir Ahi Evran Üniversitesi, Fen Bilimleri Enstitüsü, 2020.
  • İlhan, E., Kıymaz, İ.O., A generalization of truncated M-fractional derivative and applications to fractional differential equations, Appl. Math. Nonlinear Sci., 5(1)(2020), 171–188.
  • Kıymaz, İ.O., Çetinkaya, A., Agarwal, P., An extension of Caputo fractional derivative operator and its applications, J. Nonlinear Sci. Appl., 9(2016), 3611–3621.
  • Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., Theory and Applications of Fractional Differential, North-Holland Mathematics Studies 204, 2006.
  • Losada, J., Nieto, J.J., Properties of a new fractional derivative without singular kernel, Progr. Frac. Differ. Appl., 1(2)(2015), 87–92.
  • Özarslan, M.A., Özergin, E., Some generating relations for extended hypergeometric functions via generalized fractional derivative operator, Math. Comput. Model., 52(9-10)(2010), 1825–1833.
  • Özergin, E., Özarslan M.A., Altın, A., Extension of gamma, beta and hypergeometric functions, J. Comput. Appl. Math., 235(2011), 4601–4610.
  • Parmar, R.K., Some generating relations for generalized extended hypergeometric functions involving generalized fractional derivative operator, Concr. Appl. Math., 12(2014), 217–228.
  • Parmar, R.K., A new generalization of gamma, beta, hypergeometric and confluent hypergeometric functions, Le Matematiche, 68(2013), 33–52.
  • Podlubny, I., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications, Academic Press, 1999.
  • Samko, S.G., Kilbas, A.A., Marichev, O.I., Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, 1993.
  • Şahin, R., Yağcı, O., Yağbasan, M.B., Kıymaz, İ.O., Çetinkaya, A., Further generalizations of gamma, beta and related functions, J. Ineq. Spec. Func., 9(4)(2018), 1–7.
  • Şahin, R., Yağcı, O., Fractional calculus of the extended hypergeometric function, Appl. Math. Nonlinear Sci., 5(1)(2020), 369-384.
  • Yang, X.J., Srivastava, H.M., Macchado, A.T., A new fractional derivative without singular kernel, Thermal Sci., 20(2)(2016), 753–756.
Year 2023, Volume: 15 Issue: 1, 79 - 88, 30.06.2023
https://doi.org/10.47000/tjmcs.999775

Abstract

References

  • Abubakar, U.M., A comparative analysis of modified extended fractional derivative and integral operators via modified extended beta function with applications to generating functions, C¸ ankaya Univ. J. Sci. Eng., 19(1)(2022), 40–50.
  • Abubakar, U.M., Tahir, H.M., Abdulmumini, I.S., Extended gamma, beta and hypergeometric functions: properties and applications, J. Kerala Stat. Assoc., 32(2021), 18–39.
  • Agarwal, P., Jain, S., Mansour, T., Further extended Caputo fractional derivative operator and its applications, Russian J. Math. Phys., 24(4)(2017), 415–425.
  • Agarwal, P., Choi, J., Paris, R.B., Extended Riemann-Liouville fractional derivative operator and its applications, J. Nonlinear Sci. Appl. (JNSA), 8(5)(2015), 451–466.
  • Andrews, G.E., Askey, R., Roy, R., Special Functions, Cambridge University Press, Cambridge, 1999.
  • Ata, E., Generalized beta function defined by Wright function, arXiv:1803.03121v3 [math.CA], (2021).
  • Ata, E., Kıymaz, İ.O., A study on certain properties of generalized special functions defined by Fox-Wright function, Appl. Math. Nonlinear Sci., 5(1)(2020), 147–162.
  • Ata, E., Modified special functions defined by generalized M-series and their properties, arXiv:2201.00867v1 [math.CA], (2022).
  • Ata, E., Kıymaz, İ.O., Generalized gamma, beta and hypergeometric functions defined by Wright function and applications to fractional differential equations, Cumhuriyet Sci. J., 43(4)(2022), 684-695.
  • Atangana, A., Baleanu, D., New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Thermal Sci., 20(2)(2016), 763–769.
  • Baleanu, D., Agarwal, R.P., Parmar, R.K., Alqurashi, M., Salahshour, S., Extension of the fractional derivative operator of the Riemann-Liouville, J. Nonlinear Sci. Appl., 10(2017), 2914–2924.
  • Caputo, M., Fabrizio, M., A new definition of fractional derivative without singular kernel, Progr. Frac. Differ. Appl., 1(2)(2015), 73–85.
  • Chaudhry, M.A., Zubair, S.M., Generalized incomplete gamma functions with applications, J. Comput. Appl. Math., 55(1994), 99–124.
  • Çetinkaya, A., Kıymaz, İ.O., Agarwal, P., Agarwal, R., A comparative study on generating function relations for generalized hypergeometric functions via generalized fractional operators, Adv. Differ. Equ., 2018(1)(2018), 1–11.
  • Debnath, L., Bhatta, D., Integral Transforms and Their Applications, Third Edition, CRC Press, Boca Raton, London, New York, 2015.
  • Gomez-Aguilar, J.F., Atangana, A., New insight in fractional differentiation: Power, exponential decay and Mittag-Leffler laws and applications, EPJ Plus, 132(13)(2017), 1–21.
  • İlhan, E., Genelleştirilmiş Özel Fonksiyonlar Yardımıyla Tanımlanan Kesirli Operatörler ve Uygulamaları, Kırşehir Ahi Evran Üniversitesi, Fen Bilimleri Enstitüsü, 2020.
  • İlhan, E., Kıymaz, İ.O., A generalization of truncated M-fractional derivative and applications to fractional differential equations, Appl. Math. Nonlinear Sci., 5(1)(2020), 171–188.
  • Kıymaz, İ.O., Çetinkaya, A., Agarwal, P., An extension of Caputo fractional derivative operator and its applications, J. Nonlinear Sci. Appl., 9(2016), 3611–3621.
  • Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., Theory and Applications of Fractional Differential, North-Holland Mathematics Studies 204, 2006.
  • Losada, J., Nieto, J.J., Properties of a new fractional derivative without singular kernel, Progr. Frac. Differ. Appl., 1(2)(2015), 87–92.
  • Özarslan, M.A., Özergin, E., Some generating relations for extended hypergeometric functions via generalized fractional derivative operator, Math. Comput. Model., 52(9-10)(2010), 1825–1833.
  • Özergin, E., Özarslan M.A., Altın, A., Extension of gamma, beta and hypergeometric functions, J. Comput. Appl. Math., 235(2011), 4601–4610.
  • Parmar, R.K., Some generating relations for generalized extended hypergeometric functions involving generalized fractional derivative operator, Concr. Appl. Math., 12(2014), 217–228.
  • Parmar, R.K., A new generalization of gamma, beta, hypergeometric and confluent hypergeometric functions, Le Matematiche, 68(2013), 33–52.
  • Podlubny, I., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of their Solution and some of their Applications, Academic Press, 1999.
  • Samko, S.G., Kilbas, A.A., Marichev, O.I., Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, 1993.
  • Şahin, R., Yağcı, O., Yağbasan, M.B., Kıymaz, İ.O., Çetinkaya, A., Further generalizations of gamma, beta and related functions, J. Ineq. Spec. Func., 9(4)(2018), 1–7.
  • Şahin, R., Yağcı, O., Fractional calculus of the extended hypergeometric function, Appl. Math. Nonlinear Sci., 5(1)(2020), 369-384.
  • Yang, X.J., Srivastava, H.M., Macchado, A.T., A new fractional derivative without singular kernel, Thermal Sci., 20(2)(2016), 753–756.
There are 30 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Enes Ata 0000-0001-6893-8693

İ. Onur Kıymaz 0000-0003-2375-0202

Publication Date June 30, 2023
Published in Issue Year 2023 Volume: 15 Issue: 1

Cite

APA Ata, E., & Kıymaz, İ. O. (2023). New Fractional Operators Including Wright Function in Their Kernels. Turkish Journal of Mathematics and Computer Science, 15(1), 79-88. https://doi.org/10.47000/tjmcs.999775
AMA Ata E, Kıymaz İO. New Fractional Operators Including Wright Function in Their Kernels. TJMCS. June 2023;15(1):79-88. doi:10.47000/tjmcs.999775
Chicago Ata, Enes, and İ. Onur Kıymaz. “New Fractional Operators Including Wright Function in Their Kernels”. Turkish Journal of Mathematics and Computer Science 15, no. 1 (June 2023): 79-88. https://doi.org/10.47000/tjmcs.999775.
EndNote Ata E, Kıymaz İO (June 1, 2023) New Fractional Operators Including Wright Function in Their Kernels. Turkish Journal of Mathematics and Computer Science 15 1 79–88.
IEEE E. Ata and İ. O. Kıymaz, “New Fractional Operators Including Wright Function in Their Kernels”, TJMCS, vol. 15, no. 1, pp. 79–88, 2023, doi: 10.47000/tjmcs.999775.
ISNAD Ata, Enes - Kıymaz, İ. Onur. “New Fractional Operators Including Wright Function in Their Kernels”. Turkish Journal of Mathematics and Computer Science 15/1 (June 2023), 79-88. https://doi.org/10.47000/tjmcs.999775.
JAMA Ata E, Kıymaz İO. New Fractional Operators Including Wright Function in Their Kernels. TJMCS. 2023;15:79–88.
MLA Ata, Enes and İ. Onur Kıymaz. “New Fractional Operators Including Wright Function in Their Kernels”. Turkish Journal of Mathematics and Computer Science, vol. 15, no. 1, 2023, pp. 79-88, doi:10.47000/tjmcs.999775.
Vancouver Ata E, Kıymaz İO. New Fractional Operators Including Wright Function in Their Kernels. TJMCS. 2023;15(1):79-88.