On a Unified Form of Fractional Volterra-type Integro-Differential Equations and its applications

• Autorzy: Pandey Shared Chander , Chaudhary Kaushalya

Identyfikatory

DOI

10.7862/rf.2024.5

• Języki publikacji: EN

Abstrakty

EN

In the present article, an extension for the family of Volterra-type integro-differential equations, involving a generalization of Hilfer fractional derivative with the Lorenzo-Hartley’s G-function (LHGF) in the kernel, is proposed. A compact and computable solution of the considered family of integro-differential equations is established in terms of an infinite series of LHGF. Further, certain known and new special cases of the proposed family are also established. Furthermore, some examples of the integro-differential equation are also discussed. Moreover, from the application point of view, generalized fractional free-electron laser equations involving the Caputo and the Riemann-Liouville fractional derivatives are also determined. Finally, the graphical illustrations for the solutions of the studied generalized fractional free-electron laser equations are demonstrated.

Słowa kluczowe

PL

fractional-order integro-differential equation; Hilfer-Prabhakar fractional derivative; Lorenzo-Hartley’s G-function; równanie całkowo-różniczkowe rzędu ułamkowego; pochodna ułamkowa Hilfera-Prabhakara; funkcja G Lorenzo-Hartleya

• Wydawca: Oficyna Wydawnicza Politechniki Rzeszowskiej
• Czasopismo: Journal of Mathematics and Applications
• Rocznik: 2024
• Tom: Vol. 47
• Strony: 61--89
• Opis fizyczny: Bibliogr. 65 poz., rys.

Twórcy

autor

Pandey Shared Chander

• Faculty of Mathematics and Computing, Department of Mathematics and Statistics, Banasthali Vidyapith, Rajasthan, India

• https://orcid.org/000-0001-8329-3491

autor

Chaudhary Kaushalya

• Faculty of Mathematics and Computing, Department of Mathematics and Statistics, Banasthali Vidyapith, Rajasthan, India

• https://orcid.org/009-0004-3652-2000

Bibliografia

• [1] M. Abramowitz, I.A. Stegun, D. Miller, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series No. 55 (1965).
• [2] B.N. Al-Saqabi, V.K. Tuan, Solution of a fractional differential equation, Integral Transform. Spec. Funct. 4 (1996) 321–326.
• [3] A.H. Al-Shammery, S.L. Kalla, H.G. Khajah, A fractional generalization of the free electron laser equation, Fract. Calc. Appl. Anal. 2 (4) (1999) 501–508.
• [4] A.H. Al-Shammery, S.L. Kalla, H.G. Khajah, On a generalized fractional integro-differential equation of Volterra-type, Integral Transform. Spec. Funct. 9 (2) (2000) 81–90.
• [5] T.M. Apostol, Mathematical Analysis, Addison-Wesley Publ. Co., Reading Mass and London, 1960.
• [6] D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific Publishing, Singapore, 2017.
• [7] D. Baleanu, J.A.T. Machado, A.C.J. Luo, Fractional Dynamics and Control, Springer, New York, 2012.
• [8] M. Berry, Why are special functions special?, Phys. Today 54 (4) (2001) 11–12.
• [9] L. Boyadjiev, H.J. Dobner, S.L. Kalla, A fractional integro-differential equation of Volterra type, Math. Comput. Modelling 28 (10) (1998) 103–113.
• [10] L. Boyadjiev, S.L. Kalla, H.G. Khajah, Analytical and numerical treatment of a fractional integro-differential equation of Volterra-type, Math. Comput. Modelling 25 (12) (1998) 1–9.
• [11] V.B.L. Chaurasia, S.C. Pandey, On a new computable solution of generalized fractional kinetic equations involving the generalized function for the fractional calculus and related functions, Astrophys. Space Sci. 317 (2008) 213–219.
• [12] V.B.L. Chaurasia, S.C. Pandey, Computable extensions of generalized fractional kinetic equations in astrophysics, Res. Astron. Astrophys. 10 (1) (2010) 22–32.
• [13] G. Dattoli, A. Renieri, A. Torre, Lectures on the Free Electron Laser Theory and Related Topics, World Scientific, Singapore, 1993.
• [14] G. Dattoli, S. Lorenzutta, G. Maino, A. Torre, Analytical treatment of the highgain free electron laser equation, Radiat. Phys. Chem. 48 (1) (1996) 29–40.
• [15] K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer Science & Business Media, 2010.
• [16] M.A.F. Dos Santos, Fractional Prabhakar derivative in diffusion equation with non-static stochastic resetting, Phys. 1 (2019) 40–58.
• [17] E.F. Doungmo Goufo, A biomathematical view on the fractional dynamics of cellulose degradation, Fract. Calc. Appl. Anal. 18 (3) (2015) 554–564.
• [18] M. D’Ovidio, F. Polito, Fractional diffusion-telegraph equations and their associated stochastic solutions, arXiv:1307.1696v3 [math.PR], 22 pages (2013).
• [19] R. Garra, R. Gorenflo, F. Polito, Ž. Tomovski, Hilfer-Prabhakar derivatives and some applications, Appl. Math. Comput. 242 (2014) 576–589.
• [20] A. Giusti, I. Colombaro, Prabhakar-like fractional viscoelasticity, Commun. Nonlinear Sci. Numer. Simul. 56 (2018) 138–143.
• [21] A. Giusti, General fractional calculus and Prabhakar’s theory, Commun. Nonlinear Sci. Numer. Simul. 83 (2020) 105114.
• [22] R. Gorenflo, A.A. Kilbas, F. Mainardi, S.V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer-Verlag Berlin, Heidelberg, 2014.
• [23] H.J. Haubold, A.M. Mathai, The fractional kinetic equation and thermonuclear functions, Astrophys. Space Sci. 273 (2000) 53–63.
• [24] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Company, Singapore, 2000.
• [25] R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials, J. Chem. Phys. 284 (2002) 399–408.
• [26] R. Hilfer, Threefold introduction to fractional derivatives, Anomalous Transport: Foundations and Applications, Wiley-VCH Verlag, Weinheim (2008) 17–73.
• [27] R. Hilfer, Y. Luchko, Ž. Tomovski, Operational method for solution of the fractional differential equations with the generalized Riemann-Liouville fractional derivatives, Fract. Calc. Appl. Anal. 12 (2009) 299–318.
• [28] A.A. Kilbas, M. Saigo, R.K. Saxena, Solution of Volterra integro-differential equations with generalized Mittag-Leffler functions in the kernels, J. Integral Equations Appl. 14 (2002) 377–396.
• [29] A.A. Kilbas, M. Saigo, R.K. Saxena, Generalized Mittag-Leffler function and generalized fractional calculus operators, Integral Transform. Spec. Funct. 15 (2004) 31–49.
• [30] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies Vol. 204, Elsevier, Amsterdam, 2006.
• [31] V. Kiryakova, Generalized Fractional Calculus and Applications, CRC Press, 1993.
• [32] V. Kiryakova, The special functions of fractional calculus as generalized fractional calculus operators of some basic functions, Comput. Math. Appl. 59 (2010) 1128–1141.
• [33] J. Liouville, Mémoire sur quelques questions de géométrie et de mécanique et sur un nouveau genre de calcul pour résoudre ces questions, J. École Polytech. 13 (21) (1832) 1–69.
• [34] C.F. Lorenzo, T.T. Hartley, Generalized functions for the fractional calculus, NASA TP 209424 (1999).
• [35] C.F. Lorenzo, T.T. Hartley, The Fractional Trigonometry: With Application to Fractional Differential Equations and Science, John Wiley & Sons, Hoboken, New Jersey, 2017.
• [36] A. Mahmood, S. Parveen, A. Ara, N.A. Khan, Exact analytic solution for the unsteady flow of a non-Newtonian fluid between two cylinders with fractional derivative model, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 3309–3319.
• [37] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010.
• [38] A.M. Mathai, H.J. Haubold, Special Functions for Applied Scientists, Springer, New York, 2008.
• [39] A.M. Mathai, H.J. Haubold, Introduction to Fractional Calculus, Nova Science Publishers, 2017.
• [40] D. Matignon, Stability properties for generalized fractional differential systems, ESAIM Proc. Systèmes Différentiels Fractionnaires – Modèles, Methods et Applications 5 (1998).
• [41] C. Milici, G. Drăgănescu, J.A.T. Machado, Introduction to Fractional Differential Equations, Nonlinear Syst. Complexity Vol. 25, Springer, Switzerland, 2019.
• [42] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
• [43] R. Nawaz, A. Khattak, M. Akbar, S. Ahsan, Z. Shah, A. Khan, Solution of fractional-order integro-differential equations using optimal homotopy asymptotic method, J. Thermal Anal. Calorimetry (2020) 1–13.
• [44] K.B. Oldham, J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, San Diego, 1974.
• [45] S.C. Pandey, The Lorenzo-Hartley function for fractional calculus and its applications pertaining to fractional order modelling of anomalous relaxation in dielectrics, Comput. Appl. Math. 37 (3) (2018) 2648–2666.
• [46] S.C. Pandey, On some computable solutions of unified families of fractional differential equations, São Paulo J. Math. Sci. (2022) 1280–1308.
• [47] I. Podlubny, The Laplace transform method for linear differential equations of the fractional order, Inst. Exper. Phys. Slovak Acad. Sci. 2 (1994) 1–35.
• [48] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
• [49] F. Polito, Ž. Tomovski, Some properties of Prabhakar-type fractional calculus operators, Fract. Differ. Calculus 6 (1) (2016) 73–94.
• [50] T.R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J. 19 (1971) 7–15.
• [51] A.P. Prudnikov, I.A. Brychkov, O.I. Marichev, Integrals and Series: Special Functions Vol. 2, CRC Press, 1986.
• [52] I.M. Ryzhik, I.S. Gradshteĭn, Table of Integrals, Series, and Products, Academic Press, New York, 1965.
• [53] U.K. Saha, L.K. Arora, A.K. Arora, On the relationships of the R-function of Lorenzo and Hartley with other special functions of fractional calculus, Fract. Calc. Appl. Anal. 12 (4) (2009) 453–458.
• [54] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993.
• [55] R.K. Saxena, S.L. Kalla, On a fractional generalization of the free electron laser equation, Appl. Math. Comput. 143 (1) (2003) 89–97.
• [56] R. Saxena, S.L. Kalla, L. Galue, On a fractional generalization of the free electron laser equations associated with generalized Mittag-Leffler function and a confluent hypergeometric function of two variables, J. Inequalities Spec. Funct. 8 (2) (2017) 157–165.
• [57] R.K. Saxena, J. Ram, D. Kumar, Alternative derivation of generalized fractional kinetic equations, J. Fract. Calc. Appl. 4 (2) (2013) 322–334.
• [58] A. Shakeel, S. Ahmad, H. Khan, H. Vieru, Solutions with Wright functions for time fractional convection flow near a heated vertical plate, Adv. Differ. Equations 51 (2016) 1–11.
• [59] H.M. Srivastava, Remarks on some families of fractional-order differential equations, Integral Transform. Spec. Funct. 28 (7) (2017) 560–564.
• [60] H.M. Srivastava, R.K. Saxena, Some Volterra-type fractional integro-differential equations with a multivariable confluent hypergeometric function as their kernel, J. Integral Equations Appl. 17 (2005) 199–217.
• [61] H.M. Srivastava, Ž. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel, Appl. Math. Comput. 211 (2009) 198–210.
• [62] H. Sun, Y. Zhang, D. Baleanu, W. Chen, Y. Chen, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci. Numer. Simul. 64 (2018) 213–231.
• [63] Ž. Tomovski, R. Hilfer, H.M. Srivastava, Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions, Integral Transform. Spec. Funct. 21 (11) (2010) 797–814.
• [64] X.J. Yang, General Fractional Derivatives: Theory, Methods and Applications, CRC Press, 2019.
• [65] X.J. Yang, F. Gao, Y. Ju, General Fractional Derivatives with Applications in Viscoelasticity, Academic Press, 2020.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025).