Some Integral Transform Results for Hilfer–Prabhakar Fractional Derivative and Analysis of Free-Electron Laser Equation

An J, Hese EV, Baes M (2012) Phase-space consistency of stellar dynamical models determined by separable augmented densities. Mon Not R Astron Soc 422(1):652–664

Bazhlekova E, Dimovski I (2014) Exact solution of two-term time-fractional Thornley’s problem by operational method. Integral Transforms Spec Funct 25(1):61–74

Belgacem FBM, Silambarasan R (2012a) Advances in the natural transform. In: proceedings of the 9th international conference on mathematical problems in engineering, aerospace and sciences, Vol. 1493 of AIP Conference Proceedings, pp 106–110, Vienna, Austria

Belgacem FBM, Silambarasan R (2012b) Theory of the natural transform. Math Eng Sci Aerosp 3(1):99–124

Boyadjiev L, Dobner H-J (2001) Fractional free electron laser equations. Integral Transform Spec Funct 11(2):113–136

Chamati H, Tonchev NS (2006) Generalized Mittag-Leffler functions in the theory of finite-size scaling for systems with strong anisotropy and/or long-range interaction. J Phys A Math Gen 39(3):469

Chu Y-M, Inc M, Hashemi MS, Eshaghi S (2022) Analytical treatment of regularized Prabhakar fractional differential equations by invariant subspaces. Comput Appl Math 41:271. https://doi.org/10.1007/s40314-022-01977-1

D’Ovidio M, Polito F (2018) Fractional diffusion-telegraph equations and their associated stochastic solutions. Theory Probab Appl. https://doi.org/10.1137/S0040585X97T988812

Dattoli G, Giannessi L, Mezi L, Torre A, Caloi R (1991) FEL time-evolution operator. Nucl Instrum Methods Phys Res A 304(1–3):541–544

Dattoli G, Renieri A, Torre A (1993) Lectures on the Free Electron Laser theory and related topics. World Scientific Publ, Singapore

Dattoli G, Lorenzutta S, Maino G, Torre A (1996) Analytical treatment of the high-gain free electron laser equation. Radiat Phys Chem 48:29–40

Debnath L, Bhatta D (2007) Integral transforms and their applications. Chapman and Hall/CRC, Taylor and Francis Group, New York

Dubey VP, Singh J, Alshehri AM, Dubey S, Kumar D (2022a) Forecasting the behaviour of fractional order Bloch equations appearing in NMR flow via a hybrid computational technique. Chaos Solitons Fract 164:112691

Dubey VP, Kumar D, Alshehri HM, Singh J, Baleanu D (2022b) Generalized invexity and duality in multiobjective variational problems involving non-singular fractional derivative. Open Phys 20(1):939–962

Erdogan ME (2000) A note on an unsteady flow of a viscous fluid due to an oscillating plane wall. Inter J Non Linear Mech 35:1–6

Eshaghi S, Ansari A (2016) Lyapunov inequality for fractional differential equations with Prabhakar derivative. Math Inequal Appl 19(1):349–358

Garra R, Gorenflo R, Polito F, Tomovski Z (2014) Hilfer-Prabhakar derivatives and some applications. Appl Math Comput 242(1):576–589

Garrappa R, Mainardi F, Maione G (2016) Models of dielectric relaxation based on completely monotone functions. Fract Calc Appl Anal 19(5):1105–1160

Giusti A, Colombaro I (2018) Prabhakar-like fractional viscoelasticity. Commun Nonlinear Sci Numer Simul 56:138–143

Hashemi MS, Atangana A, Hajikhah S (2020) Solving fractional pantograph delay equations by an effective computational method. Math Comput Simul 177:295–305

Hashemi MS, Ashpazzadeh E, Moharrami M, Lakestani M (2021) Fractional order Alpert multiwavelets for discretizing delay fractional differential equation of pantograph type. Appl Numer Math 170:1–13

Hilfer R (2000) Fractional calculus and regular variation in thermodynamics. In: Hilfer R (ed) Applications of Fractional Calculus in Physics. World Scientific, Singapore, p 429

Hilfer R (2008) Threefold introduction to fractional derivatives. Anomalous transport foundations and applications. Germany Publishing, Weinheim

Khan ZH, Khan WA (2008) N-transform properties and applications. NUST J Eng Sci 1:127–133

Kilbas AA, Saigo M, Saxena RK (2004a) Generalized Mittag-Leffler function and generalized fractional calculus operators. Integr Transf Spec Funct 15(1):31–49

Kilbas AA, Saigo M, Saxena RK (2004b) Generalized Mittag-Leffler function and generalized fractional calculus operators. Integral Transform Spec Funct 15(1):31–49

Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of the fractional differential equations, North-Holland. Math Stud 204

Mainardi F (1997) Fractional calculus. Some basic problems in continuum and statistical mechanics. In: Carpinteri A, Mainardi F (eds) Fractals and fractional calculus in continuum mechanics. Springer Verlag, Wien, pp 291–348

Panchal SK, Khandagale AD, Dole PV (2016) Sumudu transform of Hilfer-Prabhakar fractional derivatives and applications. Proc Natl Conf Recent Trends Math 1:60–66

Pandey SC (2018) The Lorenzo–Hartley’s function for fractional calculus and its applications pertaining to fractional order modelling of anomalous relaxation in dielectrics. Comput Appl Math 37:2648–2666

Podlubny I (1999) Fractional differential equations. Academic Press, New York

Pogány TK, Tomovski Ž (2016) Probability distribution built by Prabhakar function. Related Turán and Laguerre inequalities. Integral Transforms Spec Funct 27(10):783–793

Polito F, Scalas E (2016) A generalization of the space-fractional Poisson process and its connection to some L´evy processes. Electron Commun Probab 21:1–14. https://doi.org/10.1214/16-ECP4383

Prabhakar TR (1971) A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math J 19:7–15

Saxena RK, Kalla SL (2003) On a fractional generalization of the free electron laser equation. Appl Math Comput 143:89–97

Silambarasan R, Belgacem FBM (2011) Application of the natural transform to Maxwell’s equations. In: proceedings of the progress in electromagnetics research symposium proceedings pp 899–902, Suzhou, China

Singh Y, Gill V, Kundu S, Kumar D (2019) On the Elzaki transform and its applications in fractional free electron laser equation. Acta Univ Sapientiae Math 11(2):419–429. https://doi.org/10.2478/ausm-2019-0030

Stanislavsky A, Weron K (2016) Atypical case of the dielectric relaxation responses and its fractional kinetic equation. Fract Calc Appl Anal 19(1):212–228

Tamarkin JD (1930) On integrable solutions of Abel’s integral equations. Ann Math 31:219–229

Tonchev NS (2007) Finite-size scaling in anisotropic systems. Phys Rev E 75:031110

Xia F-L, Jarad F, Hashemi MS, Riaz MB (2022) A reduction technique to solve the generalized nonlinear dispersive mK(m, n) equation with new local derivative. Results Phys 38:105512