The present article mainly focuses on the fractional derivatives with an exponential kernel (“exponential fractional derivatives” for brevity). First, several extended integral transforms of the exponential fractional derivatives are proposed, including the Fourier transform and the Laplace transform. Then, the L2 discretisation for the exponential Caputo derivative with α ∈ (1, 2) is established. The estimation of the truncation error and the properties of the coefficients are discussed. In addition, a numerical example is given to verify the correctness of the derived L2 discrete formula.
Enyu Fan, Jingshu Wu, Shaoying Zeng
. On the Fractional Derivatives with an Exponential Kernel[J]. Communications on Applied Mathematics and Computation, 2023
, 5(4)
: 1655
-1673
.
DOI: 10.1007/s42967-022-00233-x
1. Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J.: Fractional Calculus: Models and Numerical Methods. World Scientific, Singapore (2012)
2. Fan, E.Y., Li, C.P., Li, Z.Q.: Numerical methods for the Caputo-type fractional derivative with an exponential kernel. J. Appl. Anal. Comput. (2022). https://doi.org/10.11948/20220177
3. Jarad, F., Abdeljawad, T.: Generalized fractional derivatives and Laplace transform. Discrete Contin. Dyn. Syst. Ser. S 13(3), 709-722 (2020)
4. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science, Amsterdam (2006)
5. Li, C.P., Li, Z.Q.: Stability and ψ-algebraic decay of the solution to ψ-fractional differential system. Int. J. Nonlinear Sci. Numer. Simul. (2021). https://doi.org/10.1515/ijnsns-2021-0189
6. Li, C.P., Li, Z.Q.: On blow-up for a time-space fractional partial differential equation with exponential kernel in temporal derivative. J. Math. Sci. (2022). https://doi.org/10.1007/s10958-022-05894-w
7. Li, C.P., Li, Z.Q.: The finite-time blow-up for semi-linear fractional diffusion equations with time ψ-Caputo derivative. J. Nonlinear Sci. 32(6), 82 (2022)
8. Li, C.P., Li, Z.Q., Yin, C.T.: Which kind of fractional partial differential equations has solution with exponential Asymptotics? In: Dzielinski, A., Sierociuk, D., Ostalczyk, P. (eds.) Proceedings of the International Conference on Fractional Differentiation and Its Applications (ICFDA’21). Lecture Notes in Network Systems, vol. 452, pp. 112-117. Springer, Cham (2022)
9. Li, C.P., Zeng, F.H.: Numerical Methods for Fractional Calculus. CRC Press, Boca Raton (2015)
10. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)
11. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
12. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, London (1993)
