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M-Lauricella hypergeometric functions: integral representations and solutions of fractional differential equations

Yıl 2023, Cilt: 72 Sayı: 2, 512 - 529, 23.06.2023

Enes Ata

https://doi.org/10.31801/cfsuasmas.1144644
Cited By: 3

Öz

In this paper, using the modified beta function involving the generalized M-series in its kernel, we described new extensions for the Lauricella hypergeometric functions $F_{A}^{(r)}$, $F_{B}^{(r)}$, $F_{C}^{(r)}$ and $F_{D}^{(r)}$. Furthermore, we obtained various integral representations for the newly defined extended Lauricella hypergeometric functions. Then, we obtained solution of fractional differential equations involving new extensions of Lauricella hypergeometric functions, as examples.

Anahtar Kelimeler

Fractional derivatives and integrals beta function confluent hypergeometric function Lauricelle functions fractional differential equations Laplace transform

Teşekkür

This work was partly presented in the 4th International Conference on Pure and Applied Mathematics (ICPAM-2022) which organized by Van Yüzüncü Yıl University on June 22-23, 2022 in Van-Turkey.

Kaynakça

  • Abubakar, U. M., A study of extended beta and associated functions connected to Fox-Wright function, J. Frac. Calc. Appl., 12(3)(13) (2021), 1-23.
  • Agarwal, R. P., Luo, M. J., Agarwal, P., On the extended Appell-Lauricella hypergeometric functions and their applications, Filomat, 31(12) (2017), 3693-3713. https://doi.org/10.2298/FIL1712693A
  • Agarwal, P., Agarwal, R. P., Ruzhansky, M., Special Functions and Analysis of Differential Equations, Chapman and Hall/CRC, New York, 2020.
  • Andrews, G. E., Askey, R., Roy, R., Special Functions, Cambridge Univ. Press, Cambridge, 1999. https://doi.org/10.1017/CBO9781107325937
  • 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. https://doi.org/10.17776/csj.1005486
  • 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. https://doi.org/10.2478/amns.2020.1.00014
  • Ata, E., Generalized beta function defined by Wright function, arXiv preprint https://arxiv.org/abs/1803.03121v3 [math.CA], (2021). https://doi.org/10.48550/arXiv.1803.03121
  • Ata, E., Modified special functions defined by generalized M-series and their properties, arXiv preprint https://arxiv.org/abs/2201.00867v1 [math.CA], (2022). https://doi.org/10.48550/arXiv.2201.00867
  • Chaudhry, M. A., Zubair, S. M., Generalized incomplete gamma functions with applications, J. Comput. Appl. Math., 55 (1994), 99-124. https://doi.org/10.1016/0377-0427(94)90187-2
  • Chaudhry, M. A., Qadir, A., Rafique, M., Zubair, S. M., Extension of Euler’s beta function, J. Comput. Appl. Math., 78 (1997), 19-32. https://doi.org/10.1016/S0377-0427(96)00102-1
  • Chaudhry, M. A., Qadir, A., Srivastava, H. M., Paris, R. B., Extended hypergeometric and confluent hypergeometric functions, Appl. Math. Comput., 159 (2004), 589-602. https://doi.org/10.1016/j.amc.2003.09.017
  • Choi, J., Rathie, A. K., Parmar, R. K., Extension of extended beta, hypergeometric and confluent hypergeometric functions, Honam Math. J., 36 (2014), 357-385. https://doi.org/10.5831/HMJ.2014.36.2.357
  • 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. https://doi.org/10.1186/s13662-018-1612-0
  • Debnath, L., Bhatta, D., Integral Transforms and Their Applications, CRC Press, 2014. https://doi.org/10.1201/b17670
  • Goswami, A., Jain, S., Agarwal, P., Aracı, S., A note on the new extended beta and Gauss hypergeometric functions, Appl. Math. Infor. Sci., 12 (2018), 139-144. https://doi.org/10.18576/amis/120113
  • Hasanov, A., Srivastava, H. M., Some decomposition formulas associated with the Lauricella function $F_{A}^{(r)}$ and other multiple hypergeometric functions, Appl. Math. Lett., 19(2) (2006),113-121. https://doi.org/10.1016/j.aml.2005.03.009
  • Jain, S., Agarwal, P., Kıymaz, ˙I. O., Fractional integral and beta transform formulas for the extended Appell-Lauricella hypergeometric functions, Tamap J. Math, Stat., 2018 (2018), 1-5.
  • Kilbas, A. A., Srivastava, H. M., Trujillo, J. J., Theory and Applications of Fractional Differential, North-Holland Mathematics Studies 204, 2006.
  • Kulip, M. A. H., Mohsen, F. F., Barahmah, S. S., Further extended gamma and beta functions in term of generalized Wright function, Elec. J. Uni. of Aden for Basic and Appl. Sci., 1(2) (2020), 78-83.
  • Lee, D. M., Rathie, A. K., Parmar, R. K., Kim, Y. S., Generalization of extended beta function, hypergeometric and confluent hypergeometric functions, Honam Math. J., 33 (2011), 187-206. https://doi.org/10.5831/HMJ.2011.33.2.187
  • Mubeen, S., Rahman, G., Nisar, K. S., Choi, J., An extended beta function and its properties, J. Math. Sci., 102 (2017), 1545-1557. https://doi.org/10.17654/MS102071545
  • Özergin, E., Özarslan, M. A., Altın, A., Extension of gamma, beta and hypergeometric functions, J. Comput. Appl. Math., 235 (2011), 4601-4610. https://doi.org/10.1016/j.cam.2010.04.019
  • Padmanabham, P. A., Srivastava, H. M., Summation formulas associated with the Lauricella function $F_{A}^{(r)}$, Appl. Math. Lett., 13(1) (2000), 65-70. https://doi.org/10.1016/S0893-9659(99)00146-9
  • Parmar, R. K., A new generalization of gamma, beta, hypergeometric and confluent hypergeometric functions, Le Matematiche, 68 (2013), 33-52. https://doi.org/10.4418/2013.68.2.3
  • Podlubny, I., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier, 1998.
  • Rahman, G., Mubeen, S., Nisar, K. S., A new generalization of extended beta and hypergeometric functions, J. Frac. Calc. Appl., 11(2) (2020), 32-44.
  • Shadab, M., Saime, J., Choi, J., An extended beta function and its applications, J. Math. Sci., 103 (2018), 235-251. https://doi.org/10.17654/MS103010235
  • Sharma, M., Jain, R., A note on a generalized M-series as a special function of fractional calculus, Frac. Calc. Appl. Anal., 12(4) (2009), 449-452.
  • Srivastava, H. M., Agarwal, P., Jain, S., Generating functions for the generalized Gauss hypergeometric functions, Appl. Math. Comput., 247 (2014), 348-352. https://doi.org/10.1016/j.amc.2014.08.105
  • Srivastava, H. M., Karlsson, P. W., Multiple Gaussian Hypergeometric Series, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester, Halsted Press [John Wiley and Sons, Inc.], New York, 1985.
  • Srivastava, H. M., Manocha, H. L., A Treatise On Generating Functions, Halsted Press Wiley, New York, 1984.
  • Şahin R., An extension of some Lauricella hypergeometric functions, AIP Conf. Proc., 1558(1) (2013), 1140-1143. https://doi.org/10.1063/1.4825709
  • 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.

Yıl 2023, Cilt: 72 Sayı: 2, 512 - 529, 23.06.2023

Enes Ata

https://doi.org/10.31801/cfsuasmas.1144644
Cited By: 3

Öz

Kaynakça

  • Abubakar, U. M., A study of extended beta and associated functions connected to Fox-Wright function, J. Frac. Calc. Appl., 12(3)(13) (2021), 1-23.
  • Agarwal, R. P., Luo, M. J., Agarwal, P., On the extended Appell-Lauricella hypergeometric functions and their applications, Filomat, 31(12) (2017), 3693-3713. https://doi.org/10.2298/FIL1712693A
  • Agarwal, P., Agarwal, R. P., Ruzhansky, M., Special Functions and Analysis of Differential Equations, Chapman and Hall/CRC, New York, 2020.
  • Andrews, G. E., Askey, R., Roy, R., Special Functions, Cambridge Univ. Press, Cambridge, 1999. https://doi.org/10.1017/CBO9781107325937
  • 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. https://doi.org/10.17776/csj.1005486
  • 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. https://doi.org/10.2478/amns.2020.1.00014
  • Ata, E., Generalized beta function defined by Wright function, arXiv preprint https://arxiv.org/abs/1803.03121v3 [math.CA], (2021). https://doi.org/10.48550/arXiv.1803.03121
  • Ata, E., Modified special functions defined by generalized M-series and their properties, arXiv preprint https://arxiv.org/abs/2201.00867v1 [math.CA], (2022). https://doi.org/10.48550/arXiv.2201.00867
  • Chaudhry, M. A., Zubair, S. M., Generalized incomplete gamma functions with applications, J. Comput. Appl. Math., 55 (1994), 99-124. https://doi.org/10.1016/0377-0427(94)90187-2
  • Chaudhry, M. A., Qadir, A., Rafique, M., Zubair, S. M., Extension of Euler’s beta function, J. Comput. Appl. Math., 78 (1997), 19-32. https://doi.org/10.1016/S0377-0427(96)00102-1
  • Chaudhry, M. A., Qadir, A., Srivastava, H. M., Paris, R. B., Extended hypergeometric and confluent hypergeometric functions, Appl. Math. Comput., 159 (2004), 589-602. https://doi.org/10.1016/j.amc.2003.09.017
  • Choi, J., Rathie, A. K., Parmar, R. K., Extension of extended beta, hypergeometric and confluent hypergeometric functions, Honam Math. J., 36 (2014), 357-385. https://doi.org/10.5831/HMJ.2014.36.2.357
  • 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. https://doi.org/10.1186/s13662-018-1612-0
  • Debnath, L., Bhatta, D., Integral Transforms and Their Applications, CRC Press, 2014. https://doi.org/10.1201/b17670
  • Goswami, A., Jain, S., Agarwal, P., Aracı, S., A note on the new extended beta and Gauss hypergeometric functions, Appl. Math. Infor. Sci., 12 (2018), 139-144. https://doi.org/10.18576/amis/120113
  • Hasanov, A., Srivastava, H. M., Some decomposition formulas associated with the Lauricella function $F_{A}^{(r)}$ and other multiple hypergeometric functions, Appl. Math. Lett., 19(2) (2006),113-121. https://doi.org/10.1016/j.aml.2005.03.009
  • Jain, S., Agarwal, P., Kıymaz, ˙I. O., Fractional integral and beta transform formulas for the extended Appell-Lauricella hypergeometric functions, Tamap J. Math, Stat., 2018 (2018), 1-5.
  • Kilbas, A. A., Srivastava, H. M., Trujillo, J. J., Theory and Applications of Fractional Differential, North-Holland Mathematics Studies 204, 2006.
  • Kulip, M. A. H., Mohsen, F. F., Barahmah, S. S., Further extended gamma and beta functions in term of generalized Wright function, Elec. J. Uni. of Aden for Basic and Appl. Sci., 1(2) (2020), 78-83.
  • Lee, D. M., Rathie, A. K., Parmar, R. K., Kim, Y. S., Generalization of extended beta function, hypergeometric and confluent hypergeometric functions, Honam Math. J., 33 (2011), 187-206. https://doi.org/10.5831/HMJ.2011.33.2.187
  • Mubeen, S., Rahman, G., Nisar, K. S., Choi, J., An extended beta function and its properties, J. Math. Sci., 102 (2017), 1545-1557. https://doi.org/10.17654/MS102071545
  • Özergin, E., Özarslan, M. A., Altın, A., Extension of gamma, beta and hypergeometric functions, J. Comput. Appl. Math., 235 (2011), 4601-4610. https://doi.org/10.1016/j.cam.2010.04.019
  • Padmanabham, P. A., Srivastava, H. M., Summation formulas associated with the Lauricella function $F_{A}^{(r)}$, Appl. Math. Lett., 13(1) (2000), 65-70. https://doi.org/10.1016/S0893-9659(99)00146-9
  • Parmar, R. K., A new generalization of gamma, beta, hypergeometric and confluent hypergeometric functions, Le Matematiche, 68 (2013), 33-52. https://doi.org/10.4418/2013.68.2.3
  • Podlubny, I., Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier, 1998.
  • Rahman, G., Mubeen, S., Nisar, K. S., A new generalization of extended beta and hypergeometric functions, J. Frac. Calc. Appl., 11(2) (2020), 32-44.
  • Shadab, M., Saime, J., Choi, J., An extended beta function and its applications, J. Math. Sci., 103 (2018), 235-251. https://doi.org/10.17654/MS103010235
  • Sharma, M., Jain, R., A note on a generalized M-series as a special function of fractional calculus, Frac. Calc. Appl. Anal., 12(4) (2009), 449-452.
  • Srivastava, H. M., Agarwal, P., Jain, S., Generating functions for the generalized Gauss hypergeometric functions, Appl. Math. Comput., 247 (2014), 348-352. https://doi.org/10.1016/j.amc.2014.08.105
  • Srivastava, H. M., Karlsson, P. W., Multiple Gaussian Hypergeometric Series, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester, Halsted Press [John Wiley and Sons, Inc.], New York, 1985.
  • Srivastava, H. M., Manocha, H. L., A Treatise On Generating Functions, Halsted Press Wiley, New York, 1984.
  • Şahin R., An extension of some Lauricella hypergeometric functions, AIP Conf. Proc., 1558(1) (2013), 1140-1143. https://doi.org/10.1063/1.4825709
  • 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.
Toplam 33 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Adi Diferansiyel Denklemler, Fark Denklemleri ve Dinamik Sistemler, Uygulamalı Matematik
Bölüm Research Article
Yazarlar

Enes Ata 0000-0001-6893-8693

Yayımlanma Tarihi 23 Haziran 2023
Gönderilme Tarihi 17 Temmuz 2022
Kabul Tarihi 7 Kasım 2022
Yayımlandığı Sayı Yıl 2023 Cilt: 72 Sayı: 2

Kaynak Göster

APA Ata, E. (2023). M-Lauricella hypergeometric functions: integral representations and solutions of fractional differential equations. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 72(2), 512-529. https://doi.org/10.31801/cfsuasmas.1144644
AMA Ata E. M-Lauricella hypergeometric functions: integral representations and solutions of fractional differential equations. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. Haziran 2023;72(2):512-529. doi:10.31801/cfsuasmas.1144644
Chicago Ata, Enes. “M-Lauricella Hypergeometric Functions: Integral Representations and Solutions of Fractional Differential Equations”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72, sy. 2 (Haziran 2023): 512-29. https://doi.org/10.31801/cfsuasmas.1144644.
EndNote Ata E (01 Haziran 2023) M-Lauricella hypergeometric functions: integral representations and solutions of fractional differential equations. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72 2 512–529.
IEEE E. Ata, “M-Lauricella hypergeometric functions: integral representations and solutions of fractional differential equations”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., c. 72, sy. 2, ss. 512–529, 2023, doi: 10.31801/cfsuasmas.1144644.
ISNAD Ata, Enes. “M-Lauricella Hypergeometric Functions: Integral Representations and Solutions of Fractional Differential Equations”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 72/2 (Haziran 2023), 512-529. https://doi.org/10.31801/cfsuasmas.1144644.
JAMA Ata E. M-Lauricella hypergeometric functions: integral representations and solutions of fractional differential equations. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2023;72:512–529.
MLA Ata, Enes. “M-Lauricella Hypergeometric Functions: Integral Representations and Solutions of Fractional Differential Equations”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, c. 72, sy. 2, 2023, ss. 512-29, doi:10.31801/cfsuasmas.1144644.
Vancouver Ata E. M-Lauricella hypergeometric functions: integral representations and solutions of fractional differential equations. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2023;72(2):512-29.

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