Generalized Laplace transform and tempered Ψ-Caputo fractional derivative
Abstract
In this paper, images of the tempered Ψ-Hilfer fractional integral and the tempered Ψ-Caputo fractional derivative under the generalized Laplace transform are derived. The results are applied to find a solution to an initial value problem for a nonhomogeneous linear fractional differential equation with the tempered Ψ-Caputo fractional derivative of an order α for n− 1 <α<n∈N. An illustrative example is given for 0 <α< 1 comparing solutions to the same initial value problem but with different tempering and Ψ.
Keyword : Laplace transform, fractional derivative, fractional differential equation, representation of solution
How to Cite
Medveď, M., & Pospíšil, M. (2023). Generalized Laplace transform and tempered Ψ-Caputo fractional derivative. Mathematical Modelling and Analysis, 28(1), 146–162. https://doi.org/10.3846/mma.2023.16370
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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R. Almeida, A.B. Malinowska and M.T. Monteiro. Fractional differential equations with a Caputo derivative with respect to a Kernel function and their applications. Math. Methods Appl. Sci., 41(1):336–352, 2018. https://doi.org/10.1002/mma.4617
J. Chen, B. Zhuang, Y.Q. Chen and B. Cui. Diffusion control for a tempered anomalous diffusion system using fractional-order PI controllers. ISA Trans., 82(November):94–106, 2018. https://doi.org/10.1016/j.isatra.2017.04.005
A. Erdélyi. Higher Transcendental Functions, vol. 3. McGraw-Hill Book Company, Inc., 1955.
J. Hale. Theory of Functional Differential Equations, volume 3 of Applied Mathematical Sciences. Springer-Verlag, 1977. https://doi.org/10.1007/978-1-4612-9892-2
F. Jarad and T. Abdeljawad. A modified Laplace transform for certain generalized fractional operators. Results Nonlinear Anal., 1(2):88–98, 2018.
F. Jarad and T. Abdeljawad. Generalized fractional derivatives and Laplace transform. Discret. Contin. Dyn. Syst.-Ser. S, 13(3):709–722, 2020. https://doi.org/10.3934/dcdss.2020039
A.A. Kilbas, H.M. Srivastava and J.J. Trujillo. Theory and Applications of Fractional Differential Equations, volume 204 of North-Holland Mathematics Studies. Elsevier Science B.V., 2006.
M. Medveď and E. Brestovanská. Differential equations with tempered ψ-Caputo fractional derivative. Math. Model. Anal., 26(4):631–650, 2021. https://doi.org/10.3846/mma.2021.13252
I. Podlubny. Fractional Differential Equations. Academic Press, 1999.
M. Pospíšil. Laplace transform, Gronwall inequality and delay differential equations for general conformable fractional derivative. Commun. Math. Anal., 22(1):14–33, 2019.
M. Pospíšil and F. Jaroš. On the representation of solutions of delayed differential equations via Laplace transform. Electron. J. Qual. Theory Differ. Equ., (117):1– 13, 2016. https://doi.org/10.14232/ejqtde.2016.1.117
F. Sabzikar, M.M. Meerschaert and J. Chen. Tempered fractional calculus. J. Comput. Phys., 293:14–28, 2015. https://doi.org/10.1016/j.jcp.2014.04.024
J.L. Schiff. Laplace Transform: Theory and Applications. Springer-Verlag, 1999. https://doi.org/10.1007/978-0-387-22757-3
K. Shah, M.A. Alqudah, F. Jarad and T. Abdeljawad. Semi-analytical study of Pine Wilt Disease model with convex rate under Caputo–Febrizio fractional order derivative. Chaos Solitons Fractals, 135(June):109754, 2020. https://doi.org/10.1016/j.chaos.2020.109754
E.T. Whittaker and G.N. Watson. A Course of Modern Analysis, 4th ed. Cambridge University Press, 1963.
X.J. Yang, F. Gao and Y. Ju. General Fractional Derivatives with Applications in Viscoelasticity. Academic Press, 2020. https://doi.org/10.1016/B978-0-12-817208-7.00011-X
X.J. Yang and J.A. Tenreiro Machado. A new fractional operator of variable order: Application in the description of anomalous diffusion. Physica A, 481(September):276–283, 2017. https://doi.org/10.1016/j.physa.2017.04.054