Exploration of Two Modified L1 Formulae to Approximate the Caputo Fractional Derivative

doi:10.1007/s42967-025-00510-5

Communications on Applied Mathematics and Computation ›› 2026, Vol. 8 ›› Issue (4): 1550-1586.doi: 10.1007/s42967-025-00510-5

Exploration of Two Modified L1 Formulae to Approximate the Caputo Fractional Derivative

Ya-ru Liu, Guang-hua Gao, Jun-zhen Lu

College of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, Jiangsu, China

收稿日期:2025-02-12 修回日期:2025-05-14 接受日期:2025-06-02 出版日期:2026-08-20 发布日期:2026-09-17
通讯作者: Guang-hua Gao, Email: gaogh@njupt.edu.cn E-mail:gaogh@njupt.edu.cn
基金资助:
The authors sincerely thank two anonymous reviewers for their constructive comments, which have helped us to polish and improve this work.

Exploration of Two Modified L1 Formulae to Approximate the Caputo Fractional Derivative

Ya-ru Liu, Guang-hua Gao, Jun-zhen Lu

College of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, Jiangsu, China

Received:2025-02-12 Revised:2025-05-14 Accepted:2025-06-02 Online:2026-08-20 Published:2026-09-17
Supported by:
The authors sincerely thank two anonymous reviewers for their constructive comments, which have helped us to polish and improve this work.

摘要/Abstract

摘要： In this work, we revisit two modified L1 interpolation approximations to the values of the Caputo fractional derivative of order α ∈ (0, 1) at the midpoints of the mesh. These approximations are distinct from the classical L1 formula which typically yields interpolation approximations at the mesh points themselves. Although these two formulae on uniform meshes have been proposed in previous literature (Dong et al. in Commun Appl Math Comput 5(4): 1446–1468, 2023; Osman in Numerical solution methods for fractional partial differential equations. PhD thesis, University of Southern Queensland, 2017), given their practical applications in solving fractional differential systems with possible weak singularities and the inherent simplicity of the L1-type formulae compared to the existing higher-order numerical formulae such as the L2-1_σ formula and L2-type formulae, it is more essential to consider the corresponding results on non-uniform meshes. To this end, two modified L1 interpolation formulae on non-uniform meshes, subsequently referred to as Formula I and Formula II, respectively, are explored in the current work. On the basis of deriving the formulae, the properties of coefficients in the formulae and the truncation errors under the graded mesh are analyzed in detail. It is worth noting that Formula I is reduced to the classical central difference quotient formula as α→1^-, which is superior to Formula II and the classical L1 formula. Furthermore, we present several numerical examples to demonstrate the efficacy of these two formulae, as well as the effectiveness of their applications in solving the fractional sub-diffusion equation. The unique solvability and stability of the resultant schemes with respect to the initial values are briefly mentioned. As far as we know, there has been hardly any previous systematic investigation for this type of formulae on non-uniform meshes.

关键词: Caputo fractional derivative, Modified L1 formula, Nonuniform mesh, Truncation error, Sub-diffusion

Abstract: In this work, we revisit two modified L1 interpolation approximations to the values of the Caputo fractional derivative of order α ∈ (0, 1) at the midpoints of the mesh. These approximations are distinct from the classical L1 formula which typically yields interpolation approximations at the mesh points themselves. Although these two formulae on uniform meshes have been proposed in previous literature (Dong et al. in Commun Appl Math Comput 5(4): 1446–1468, 2023; Osman in Numerical solution methods for fractional partial differential equations. PhD thesis, University of Southern Queensland, 2017), given their practical applications in solving fractional differential systems with possible weak singularities and the inherent simplicity of the L1-type formulae compared to the existing higher-order numerical formulae such as the L2-1_σ formula and L2-type formulae, it is more essential to consider the corresponding results on non-uniform meshes. To this end, two modified L1 interpolation formulae on non-uniform meshes, subsequently referred to as Formula I and Formula II, respectively, are explored in the current work. On the basis of deriving the formulae, the properties of coefficients in the formulae and the truncation errors under the graded mesh are analyzed in detail. It is worth noting that Formula I is reduced to the classical central difference quotient formula as α→1^-, which is superior to Formula II and the classical L1 formula. Furthermore, we present several numerical examples to demonstrate the efficacy of these two formulae, as well as the effectiveness of their applications in solving the fractional sub-diffusion equation. The unique solvability and stability of the resultant schemes with respect to the initial values are briefly mentioned. As far as we know, there has been hardly any previous systematic investigation for this type of formulae on non-uniform meshes.

Key words: Caputo fractional derivative, Modified L1 formula, Nonuniform mesh, Truncation error, Sub-diffusion

Ya-ru Liu, Guang-hua Gao, Jun-zhen Lu. Exploration of Two Modified L1 Formulae to Approximate the Caputo Fractional Derivative[J]. Communications on Applied Mathematics and Computation, 2026, 8(4): 1550-1586.

[1] Alikhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424-438 (2015)
[2] Dimitrov, Y., Georgiev, S., Todorov, V.: Approximation of Caputo fractional derivative and numerical solutions of fractional differential equations. Fractal Fract. 7(10), 750 (2023)
[3] Dong, Z., Fan, E., Shen, A., Su, Y.: Three kinds of discrete formulae for the Caputo fractional derivative. Commun. Appl. Math. Comput. 5(4), 1446-1468 (2023)
[4] Du, R.L., Shen, J.: Second-order difference scheme for the time fractional mixed diffusion-wave equation with initial weak regularity. Math. Methods Appl. Sci. 48(10), 10254-10270 (2023). https://doi.org/10.1002/mma.9612
[5] Gao, G.H., Sun, Z.Z., Zhang, H.W.: A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications. J. Comput. Phys. 259(2), 33-50 (2014)
[6] Gómez-Aguilar, J.F., López-López, M.G., Alvarado-Martínez, V.M., Reyes-Reyes, J., Adam-Medina, M.: Modeling diffusive transport with a fractional derivative without singular kernel. Phys. A Stat. Mech. Appl. 447, 467-481 (2016)
[7] Hu, Y., Li, C., Li, H.: The finite difference method for Caputo-type parabolic equation with fractional Laplacian: one-dimension case. Chaos Solitons Fractals 102, 319-326 (2017)
[8] Ji, B.Q., Zhu, X.H., Liao, H.L.: Energy stability of variable-step L1-type schemes for time-fractional Cahn-Hilliard model. Commun. Math. Sci. 21, 1767-1789 (2023)
[9] Li, C., Cai, M.: Theory and Numerical Approximations of Fractional Integrals and Derivatives. SIAM, Philadelphia (2019)
[10] Li, C., Zeng, F.: Numerical Methods for Fractional Calculus. CRC Press, New York (2015)
[11] Liao, H.L., Liu, N., Zhao, X.: Asymptotically compatible energy of variable-step fractional BDF2 scheme for the time-fractional Cahn-Hilliard model. IMA J. Numer. Anal. 45(3), 1425-1454 (2024). https://doi.org/10.1093/imanum/drae034
[12] Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225(2), 1533-1552 (2007)
[13] Magin, R.L., Ingo, C., Colon-Perez, L., Triplett, W., Mareci, T.H.: Characterization of anomalous diffusion in porous biological tissues using fractional order derivatives and entropy. Microporous Mesoporous Mater. 178, 39-43 (2013)
[14] Mayrya, R.K., Singh, V.K.: A high-order adaptive numerical algorithm for fractional diffusion wave equation on non-uniform meshes. Numer. Algorithms 92, 1905-1950 (2023)
[15] Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)
[16] Osman, S.A.: Numerical solution methods for fractional partial differential equations. PhD thesis, University of Southern Queensland (2017)
[17] Osman, S.A., Langlands, T.A.M.: An implicit Keller box numerical scheme for the solution of fractional subdiffusion equations. Appl. Math. Comput. 348, 609-626 (2019)
[18] Osman, S.A., Langlands, T.A.M.: Numerical investigation of two models of nonlinear fractional reaction subdiffusion equations. Fract. Calculus Appl. Anal. 25(6), 2166-2192 (2022)
[19] Owolabi, K.M.: Mathematical modelling and analysis of two-component system with Caputo fractional derivative order. Chaos Solitons Fractals 103, 544-554 (2017)
[20] Owolabi, K.M., Atangana, A.: Robustness of fractional difference schemes via the Caputo subdiffusion-reaction equations. Chaos Solitons Fractals 111, 119-127 (2018)
[21] Quan, C.Y., Wu, X.: {H}^1-norm stability and convergence of an L2-type method on nonuniform meshes for sub-diffusion equation. SIAM J. Numer. Anal. 61(5), 2106-2132 (2023)
[22] Shen, J., Stynes, M., Sun, Z.Z.: Two finite difference schemes for multi-dimensional fractional wave equations with weakly singular solutions. Comput. Methods Appl. Math. 21(4), 913-928 (2021)
[23] Stynes, M., O’Riordan, E., Gracia, J.L.: Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J. Numer. Anal. 55(2), 1057-1079 (2017)
[24] Sun, Z.Z., Gao, G.H.: Fractional Differential Equations-Finite Difference Methods. Science Press, De Gruyter, Beijing (2020)
[25] Sun, Z.Z., Wu, X.N.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193-209 (2006)
[26] Wu, L., Zhai, S.: A new high order ADI numerical difference formula for time-fractional convection-diffusion equation. Appl. Math. Comput. 387, 124564 (2020)
[27] Zeng, F., Li, C.: A new Crank-Nicolson finite element method for the time-fractional subdiffusion equation. Appl. Numer. Math. 121, 82-95 (2017)

Exploration of Two Modified L1 Formulae to Approximate the Caputo Fractional Derivative

Exploration of Two Modified L1 Formulae to Approximate the Caputo Fractional Derivative

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