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October  2021, 14(10): 3803-3819. doi: 10.3934/dcdss.2021019

Class of integrals and applications of fractional kinetic equation with the generalized multi-index Bessel function

1. 

Department of Mathematics, Wollo University, P.O. Box 1145, Dessie, Ethiopia

2. 

Department of HEAS (Mathematics), Rajasthan Technical University, Kota, 324010, India

3. 

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

* Corresponding author: Jagdev Singh

Received  October 2019 Revised  December 2019 Published  October 2021 Early access  March 2021

In this article, we have investigated certain definite integrals and various integral transforms of the generalized multi-index Bessel function, such as Euler transform, Laplace transform, Whittaker transform, K-transform and Fourier transforms. Also found the applications of the problem on fractional kinetic equation pertaining to the generalized multi-index Bessel function using the Sumudu transform technique. Mittage-Leffler function is used to express the results of the solutions of fractional kinetic equation as well as its special cases. The results obtained are significance in applied problems of science, engineering and technology.

Citation: Dayalal Suthar, Sunil Dutt Purohit, Haile Habenom, Jagdev Singh. Class of integrals and applications of fractional kinetic equation with the generalized multi-index Bessel function. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3803-3819. doi: 10.3934/dcdss.2021019
References:
[1]

P. AgarwalM. ChandD. BaleanuD. $\acute{O}$regan and S. Jain, On the solutions of certain fractional kinetic equations involving $k$-Mittag-Leffler function, Adv. Difference Equ., 249 (2018), 249-262.  doi: 10.1186/s13662-018-1694-8.

[2]

P. AgarwalS. K. NtouyasS. JainM. Chand and G. Singh, Fractional kinetic equations involving generalized $k$-Bessel function via Sumudu transform, Alexandria Engineering Journal, 57 (2018), 1937-1942.  doi: 10.1016/j.aej.2017.03.046.

[3]

N. Abeye and D. L. Suthar, The $\overline H$-function and Srivastava's polynomials involving the generalized Mellin-Barnes contour integrals, J. Fract. Calc. Appl., 10 (2019), 290-297. 

[4]

M. Chand, Z. Hammouch, J. K. Asamoah and D. Baleanu, Certain fractional integrals and solutions of fractional kinetic equations involving the product of S-function, Mathematical Methods in Engineering, Nonlinear Syst. Complex., Springer, Cham, 24 (2019), 213–244.

[5]

M. ChandJ. C. Prajapati and E. Bonyah, Fractional integrals and solution of fractional kinetic equations involving $k$-Mittag-Leffler function, Transactions of A. Razmadze Mathematical Institute, 171 (2017), 144-166.  doi: 10.1016/j.trmi.2017.03.003.

[6]

M. ChandJ. C. PrajapatiE. Bonyah and J. K. Bansal, Fractional calculus and applications of family of extended generalized Gauss hypergeometric functions, Discrete & Continuous Dynamical Systems - S, 13 (2020), 539-560.  doi: 10.3934/dcdss.2020030.

[7]

J. Choi and P. Agarwal, A note on fractional integral operator associated multi-index Mittag-Leffler functions, Filomat, 30 (2016), 1931-1939.  doi: 10.2298/FIL1607931C.

[8]

J. Choi and D. Kumar, Solutions of generalized fractional kinetic equations involving Aleph functions, Math. Commun., 20 (2015), 113-123. 

[9]

J. DaiyaJ. Ram and D. Kumar, The multivariable H-function and the general class of Srivastava polynomials involving the generalized Mellin-Barnes contour integrals, Filomat, 30 (2016), 1457-1464.  doi: 10.2298/FIL1606457D.

[10]

G. Dorrego and D. Kumar, A generalization of the kinetic equation using the Prabhakar-type operators, Honam Math. J., 39 (2017), 401-416. 

[11]

A. Erd$\acute{e}$lyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York, 1953.

[12]

A. Erd$\acute{e}$lyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York, 1954.

[13]

C. Fox, The asymptotic expansion of generalized hypergeometric functions, Proc. London Math. Soc., 27 (1928), 389-400.  doi: 10.1112/plms/s2-27.1.389.

[14]

H. Habenom, D. L. Suthar and M. Gebeyehu, Application of Laplace transform on fractional kinetic equation pertaining to the generalized Galué type Struve function, Adv. Math. Phys., (2019), 5074039, 8 pp. doi: 10.1155/2019/5074039.

[15]

H. J. Haubold and A. M. Mathai, The fractional kinetic equation and thermonuclear functions, Astrophys. Space Sci., 273 (2000), 53-63. 

[16]

R. JanM. A. KhanP. Kumam and P. Thounthong, Modeling the transmission of dengue infection through fractional derivatives, Chaos Solitons Fractals, 127 (2019), 189-216.  doi: 10.1016/j.chaos.2019.07.002.

[17]

V. Kourganoff, Introduction to the Physics of Stellar Interiors, D. Reidel Publishing Company Dordrecht, Holland, 1973. doi: 10.1007/978-94-010-2539-3.

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M. A. Khan, A. Khan and A. Elsonbaty, Modeling and simulation results of a fractional dengue model, Eur. Phys. J. Plus, 134 (2019), 329. doi: 10.1140/epjp/i2019-12765-0.

[19]

M. A. KhanZ. Hammouch and D. Baleanu, Modeling the dynamics of hepatitis E via the Caputo-Fabrizio derivative, Math. Model. Nat. Phenom., 14 (2019), 311-330.  doi: 10.1051/mmnp/2018074.

[20]

D. KumarJ. Choi and H. M. Srivastava, Solution of a general family of fractional kinetic equations associated with the generalized Mittag-Leffler function, Nonlinear Funct. Anal. Appl., 23 (2018), 455-471. 

[21]

D. Kumar, S. D. Purohit, A. Secer and A. Atangana, On generalized fractional kinetic equations involving generalized Bessel function of the first kind, Math. Probl. Eng., 2015 (2015), 289387, 7 pp. doi: 10.1155/2015/289387.

[22]

D. KumarJ. SinghK. Tanwar and D. Baleanu, A new fractional exothermic reactions model having constant heat source in porous media with power, exponential and Mittag-Leffler Laws, International Journal of Heat and Mass Transfer, 138 (2019), 1222-1227.  doi: 10.1016/j.ijheatmasstransfer.2019.04.094.

[23]

D. KumarJ. Singh and D. Baleanu, On the analysis of vibration equation involving a fractional derivative with Mittag-Leffler law, Mathematical Methods in the Applied Sciences, 43 (2019), 443-457.  doi: 10.1002/mma.5903.

[24]

D. KumarJ. Singh and D. Baleanu, Analysis of regularized long-wave equation associated with a new fractional operator with Mittag-Leffler type kernel, Physica A., 492 (2018), 155-167.  doi: 10.1016/j.physa.2017.10.002.

[25]

D. Kumar, F. Tchier, J. Singh and D. Baleanu, An efficient computational technique for fractal vehicular traffic flow, Entropy, 20 (2018), 259. doi: 10.3390/e20040259.

[26]

D. KumarJ. Singh and D. Baleanu, A new fractional model for convective straight fins with temperature-dependent thermal conductivity, Thermal Science, 22 (2018), 2791-2802.  doi: 10.2298/TSCI170129096K.

[27]

Y. LuchkoH. Martinez and J. Trujillo, Fractional Fourier transform and some of its applications, Fractional Calculus & Applied Analysis, 11 (2008), 457-470. 

[28]

A. M. Mathai, R. K. Saxena and H. J. Haubold, The H-Function Theory and Application, Springer, New York, 2010. doi: 10.1007/978-1-4419-0916-9.

[29]

N. MenariaD. Baleanu and S. D. Purohit, Integral formulas involving product of general class of polynomials and generalized Bessel function, Sohag J. Math., 3 (2016), 77-81.  doi: 10.18576/sjm/030205.

[30]

N. Menaria, K. S. Nisar and S. D. Purohit, On a new class of integrals involving product of generalized Bessel function of first kind and general class of polynomials, Acta Univ. Apulensis Math. Inform., (2016), 97–105. doi: 10.17114/j.aua.

[31]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New-York, NY, USA, 1993.

[32]

V. Namias, The fractional order Fourier and its application to quantum mechanics, J. Inst. Math. Appl., 25 (1980), 241-265.  doi: 10.1093/imamat/25.3.241.

[33]

K. S. Nisar, S. D. Purohit, D. L. Suthar and J. Singh, Fractional Order Integration and Certain Integrals of Generalized Multi-index Bessel Function, Proceedings in Mathematics & Statistics, Springer, Singapore, 272 (2019). doi: 10.1007/978-981-13-9608-3_10.

[34]

M. I. QureshiK. A. Quraishi and R. Pal, Some definite integrals of Gradshteyn-Ryzhil and other integrals, Glo. J. Sci. Fron. Res., 24 (4) (2011), 75-80. 

[35]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, 1993.

[36]

R. K. Saxena and K. Nishimoto, N-fractional calculus of generalized Mittag-Leffler functions, J. Fract. Calc., 37 (2010), 43-52. 

[37]

S. A. A. Shah, A. A. Syed M. A. Khan, M. Farooq, S. Ullah and E. O. Alzahrani, A fractional order model for Hepatitis B virus with treatment via Atangana-Baleanu derivative, Physica A: Statistical Mechanics and its Applications, 530 (2020), 122636, 17 pp. doi: 10.1016/j.physa.2019.122636.

[38]

J. SinghA. KilicmanD. KumarR. Swroop and F. M. Ali, Numerical study for fractional model of nonlinear predator-prey biological population dynamical system, Thermal Science, 23 (2019), 2017-2025.  doi: 10.2298/TSCI190725366S.

[39]

J. SinghD. Kumar and D. Baleanu, New aspects of fractional Biswas-Milovic model with Mittag-Leffler law, Mathematical Modelling of Natural Phenomena, 14 (2019), 303-326.  doi: 10.1051/mmnp/2018068.

[40] L. J. Slater, Generalized Hypergeometric functions, Cambridge University Press, 1966. 
[41]

H. M. Srivastava and J. Choi, Zeta and $q$-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012. doi: 10.1016/B978-0-12-385218-2.00001-3.

[42]

D. L. Suthar and P. Agarwal, Generalized Mittag-Leffler function and the multivariable H-function involving the generalized Mellin-Barnes contour integrals, Commun. Numer. Anal., 2017 (2017), 25-33. 

[43]

D. L. Suthar and H. Amsalu, Certain integrals associated with the generalized Bessel-Maitland function, Appl. Appl. Math., 12 (2017), 1002-1016. 

[44]

D. L. Suthar, H. Amsalu and K. Godifey, Certain integrals involving multivariate Mittag-Leffler function, J. Inequal. Appl., (2019), 208–224. doi: 10.1186/s13660-019-2162-z.

[45]

D. L. Suthar and M. Ayene, Generalized fractional integral formulas for the $k$-Bessel function, J. Math., (2018), 5198621, 8 pp. doi: 10.1155/2018/5198621.

[46]

D. L. Suthar, D. Kumar and H. Habenom, Solutions of fractional kinetic equation associated with the generalized multi-index Bessel function via Laplace transform, Differ. Equ. Dyn. Syst., (2019). doi: 10.1007/s12591-019-00504-9.

[47]

D. L. SutharS. D. Purohit and R. K. Parmar, Generalized fractional calculus of the multi-index Bessel function, Math. Nat. Sci., 1 (2017), 26-32. 

[48]

D. L. Suthar and T. Tsagye, Riemann-Liouville fractional integrals and differential formula involving Multi-index Bessel-function, Math. Sci. Lett., 6 (2017), 233-237. 

[49]

S. UllahM. A. Khan and M. Farooq, A fractional model for the dynamics of TB virus, Chaos Solitons Fractals, 116 (2018), 63-71.  doi: 10.1016/j.chaos.2018.09.001.

[50]

W. WangM. A. KhanFa tmawatiP. Kumam and P. Thounthong, A comparison study of bank data in fractional calculus, Chaos Solitons Fractals, 126 (2019), 369-384.  doi: 10.1016/j.chaos.2019.07.025.

[51]

G. K. Watugala, Sumudu transform: A new integral transform to solve differential equations and control engineering problems, Int. J. Math. Educ. Sci. Technol., 24 (1993), 35-43.  doi: 10.1080/0020739930240105.

[52]

G. K. Watugala, The Sumudu transform for functions of two variables, Math. Eng. Ind., 8 (2002), 293-302. 

[53]

A. Wiman, Über den Fundamentalsatz in der Teorie der Funktionen $E_\alpha(x)$, Acta Math., 29 (1905), 191-201.  doi: 10.1007/BF02403202.

[54] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, Cambridge, 1962.  doi: 10.1017/CBO9780511608759.
[55]

E. M. Wright, The asymptotic expansion of the generalized hypergeometric functions, J. London Math. Soc., 10 (1935), 286-293. 

show all references

References:
[1]

P. AgarwalM. ChandD. BaleanuD. $\acute{O}$regan and S. Jain, On the solutions of certain fractional kinetic equations involving $k$-Mittag-Leffler function, Adv. Difference Equ., 249 (2018), 249-262.  doi: 10.1186/s13662-018-1694-8.

[2]

P. AgarwalS. K. NtouyasS. JainM. Chand and G. Singh, Fractional kinetic equations involving generalized $k$-Bessel function via Sumudu transform, Alexandria Engineering Journal, 57 (2018), 1937-1942.  doi: 10.1016/j.aej.2017.03.046.

[3]

N. Abeye and D. L. Suthar, The $\overline H$-function and Srivastava's polynomials involving the generalized Mellin-Barnes contour integrals, J. Fract. Calc. Appl., 10 (2019), 290-297. 

[4]

M. Chand, Z. Hammouch, J. K. Asamoah and D. Baleanu, Certain fractional integrals and solutions of fractional kinetic equations involving the product of S-function, Mathematical Methods in Engineering, Nonlinear Syst. Complex., Springer, Cham, 24 (2019), 213–244.

[5]

M. ChandJ. C. Prajapati and E. Bonyah, Fractional integrals and solution of fractional kinetic equations involving $k$-Mittag-Leffler function, Transactions of A. Razmadze Mathematical Institute, 171 (2017), 144-166.  doi: 10.1016/j.trmi.2017.03.003.

[6]

M. ChandJ. C. PrajapatiE. Bonyah and J. K. Bansal, Fractional calculus and applications of family of extended generalized Gauss hypergeometric functions, Discrete & Continuous Dynamical Systems - S, 13 (2020), 539-560.  doi: 10.3934/dcdss.2020030.

[7]

J. Choi and P. Agarwal, A note on fractional integral operator associated multi-index Mittag-Leffler functions, Filomat, 30 (2016), 1931-1939.  doi: 10.2298/FIL1607931C.

[8]

J. Choi and D. Kumar, Solutions of generalized fractional kinetic equations involving Aleph functions, Math. Commun., 20 (2015), 113-123. 

[9]

J. DaiyaJ. Ram and D. Kumar, The multivariable H-function and the general class of Srivastava polynomials involving the generalized Mellin-Barnes contour integrals, Filomat, 30 (2016), 1457-1464.  doi: 10.2298/FIL1606457D.

[10]

G. Dorrego and D. Kumar, A generalization of the kinetic equation using the Prabhakar-type operators, Honam Math. J., 39 (2017), 401-416. 

[11]

A. Erd$\acute{e}$lyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York, 1953.

[12]

A. Erd$\acute{e}$lyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. 2, McGraw-Hill, New York, 1954.

[13]

C. Fox, The asymptotic expansion of generalized hypergeometric functions, Proc. London Math. Soc., 27 (1928), 389-400.  doi: 10.1112/plms/s2-27.1.389.

[14]

H. Habenom, D. L. Suthar and M. Gebeyehu, Application of Laplace transform on fractional kinetic equation pertaining to the generalized Galué type Struve function, Adv. Math. Phys., (2019), 5074039, 8 pp. doi: 10.1155/2019/5074039.

[15]

H. J. Haubold and A. M. Mathai, The fractional kinetic equation and thermonuclear functions, Astrophys. Space Sci., 273 (2000), 53-63. 

[16]

R. JanM. A. KhanP. Kumam and P. Thounthong, Modeling the transmission of dengue infection through fractional derivatives, Chaos Solitons Fractals, 127 (2019), 189-216.  doi: 10.1016/j.chaos.2019.07.002.

[17]

V. Kourganoff, Introduction to the Physics of Stellar Interiors, D. Reidel Publishing Company Dordrecht, Holland, 1973. doi: 10.1007/978-94-010-2539-3.

[18]

M. A. Khan, A. Khan and A. Elsonbaty, Modeling and simulation results of a fractional dengue model, Eur. Phys. J. Plus, 134 (2019), 329. doi: 10.1140/epjp/i2019-12765-0.

[19]

M. A. KhanZ. Hammouch and D. Baleanu, Modeling the dynamics of hepatitis E via the Caputo-Fabrizio derivative, Math. Model. Nat. Phenom., 14 (2019), 311-330.  doi: 10.1051/mmnp/2018074.

[20]

D. KumarJ. Choi and H. M. Srivastava, Solution of a general family of fractional kinetic equations associated with the generalized Mittag-Leffler function, Nonlinear Funct. Anal. Appl., 23 (2018), 455-471. 

[21]

D. Kumar, S. D. Purohit, A. Secer and A. Atangana, On generalized fractional kinetic equations involving generalized Bessel function of the first kind, Math. Probl. Eng., 2015 (2015), 289387, 7 pp. doi: 10.1155/2015/289387.

[22]

D. KumarJ. SinghK. Tanwar and D. Baleanu, A new fractional exothermic reactions model having constant heat source in porous media with power, exponential and Mittag-Leffler Laws, International Journal of Heat and Mass Transfer, 138 (2019), 1222-1227.  doi: 10.1016/j.ijheatmasstransfer.2019.04.094.

[23]

D. KumarJ. Singh and D. Baleanu, On the analysis of vibration equation involving a fractional derivative with Mittag-Leffler law, Mathematical Methods in the Applied Sciences, 43 (2019), 443-457.  doi: 10.1002/mma.5903.

[24]

D. KumarJ. Singh and D. Baleanu, Analysis of regularized long-wave equation associated with a new fractional operator with Mittag-Leffler type kernel, Physica A., 492 (2018), 155-167.  doi: 10.1016/j.physa.2017.10.002.

[25]

D. Kumar, F. Tchier, J. Singh and D. Baleanu, An efficient computational technique for fractal vehicular traffic flow, Entropy, 20 (2018), 259. doi: 10.3390/e20040259.

[26]

D. KumarJ. Singh and D. Baleanu, A new fractional model for convective straight fins with temperature-dependent thermal conductivity, Thermal Science, 22 (2018), 2791-2802.  doi: 10.2298/TSCI170129096K.

[27]

Y. LuchkoH. Martinez and J. Trujillo, Fractional Fourier transform and some of its applications, Fractional Calculus & Applied Analysis, 11 (2008), 457-470. 

[28]

A. M. Mathai, R. K. Saxena and H. J. Haubold, The H-Function Theory and Application, Springer, New York, 2010. doi: 10.1007/978-1-4419-0916-9.

[29]

N. MenariaD. Baleanu and S. D. Purohit, Integral formulas involving product of general class of polynomials and generalized Bessel function, Sohag J. Math., 3 (2016), 77-81.  doi: 10.18576/sjm/030205.

[30]

N. Menaria, K. S. Nisar and S. D. Purohit, On a new class of integrals involving product of generalized Bessel function of first kind and general class of polynomials, Acta Univ. Apulensis Math. Inform., (2016), 97–105. doi: 10.17114/j.aua.

[31]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New-York, NY, USA, 1993.

[32]

V. Namias, The fractional order Fourier and its application to quantum mechanics, J. Inst. Math. Appl., 25 (1980), 241-265.  doi: 10.1093/imamat/25.3.241.

[33]

K. S. Nisar, S. D. Purohit, D. L. Suthar and J. Singh, Fractional Order Integration and Certain Integrals of Generalized Multi-index Bessel Function, Proceedings in Mathematics & Statistics, Springer, Singapore, 272 (2019). doi: 10.1007/978-981-13-9608-3_10.

[34]

M. I. QureshiK. A. Quraishi and R. Pal, Some definite integrals of Gradshteyn-Ryzhil and other integrals, Glo. J. Sci. Fron. Res., 24 (4) (2011), 75-80. 

[35]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, 1993.

[36]

R. K. Saxena and K. Nishimoto, N-fractional calculus of generalized Mittag-Leffler functions, J. Fract. Calc., 37 (2010), 43-52. 

[37]

S. A. A. Shah, A. A. Syed M. A. Khan, M. Farooq, S. Ullah and E. O. Alzahrani, A fractional order model for Hepatitis B virus with treatment via Atangana-Baleanu derivative, Physica A: Statistical Mechanics and its Applications, 530 (2020), 122636, 17 pp. doi: 10.1016/j.physa.2019.122636.

[38]

J. SinghA. KilicmanD. KumarR. Swroop and F. M. Ali, Numerical study for fractional model of nonlinear predator-prey biological population dynamical system, Thermal Science, 23 (2019), 2017-2025.  doi: 10.2298/TSCI190725366S.

[39]

J. SinghD. Kumar and D. Baleanu, New aspects of fractional Biswas-Milovic model with Mittag-Leffler law, Mathematical Modelling of Natural Phenomena, 14 (2019), 303-326.  doi: 10.1051/mmnp/2018068.

[40] L. J. Slater, Generalized Hypergeometric functions, Cambridge University Press, 1966. 
[41]

H. M. Srivastava and J. Choi, Zeta and $q$-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012. doi: 10.1016/B978-0-12-385218-2.00001-3.

[42]

D. L. Suthar and P. Agarwal, Generalized Mittag-Leffler function and the multivariable H-function involving the generalized Mellin-Barnes contour integrals, Commun. Numer. Anal., 2017 (2017), 25-33. 

[43]

D. L. Suthar and H. Amsalu, Certain integrals associated with the generalized Bessel-Maitland function, Appl. Appl. Math., 12 (2017), 1002-1016. 

[44]

D. L. Suthar, H. Amsalu and K. Godifey, Certain integrals involving multivariate Mittag-Leffler function, J. Inequal. Appl., (2019), 208–224. doi: 10.1186/s13660-019-2162-z.

[45]

D. L. Suthar and M. Ayene, Generalized fractional integral formulas for the $k$-Bessel function, J. Math., (2018), 5198621, 8 pp. doi: 10.1155/2018/5198621.

[46]

D. L. Suthar, D. Kumar and H. Habenom, Solutions of fractional kinetic equation associated with the generalized multi-index Bessel function via Laplace transform, Differ. Equ. Dyn. Syst., (2019). doi: 10.1007/s12591-019-00504-9.

[47]

D. L. SutharS. D. Purohit and R. K. Parmar, Generalized fractional calculus of the multi-index Bessel function, Math. Nat. Sci., 1 (2017), 26-32. 

[48]

D. L. Suthar and T. Tsagye, Riemann-Liouville fractional integrals and differential formula involving Multi-index Bessel-function, Math. Sci. Lett., 6 (2017), 233-237. 

[49]

S. UllahM. A. Khan and M. Farooq, A fractional model for the dynamics of TB virus, Chaos Solitons Fractals, 116 (2018), 63-71.  doi: 10.1016/j.chaos.2018.09.001.

[50]

W. WangM. A. KhanFa tmawatiP. Kumam and P. Thounthong, A comparison study of bank data in fractional calculus, Chaos Solitons Fractals, 126 (2019), 369-384.  doi: 10.1016/j.chaos.2019.07.025.

[51]

G. K. Watugala, Sumudu transform: A new integral transform to solve differential equations and control engineering problems, Int. J. Math. Educ. Sci. Technol., 24 (1993), 35-43.  doi: 10.1080/0020739930240105.

[52]

G. K. Watugala, The Sumudu transform for functions of two variables, Math. Eng. Ind., 8 (2002), 293-302. 

[53]

A. Wiman, Über den Fundamentalsatz in der Teorie der Funktionen $E_\alpha(x)$, Acta Math., 29 (1905), 191-201.  doi: 10.1007/BF02403202.

[54] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, Cambridge, 1962.  doi: 10.1017/CBO9780511608759.
[55]

E. M. Wright, The asymptotic expansion of the generalized hypergeometric functions, J. London Math. Soc., 10 (1935), 286-293. 

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