An Order Reduction Method for the Nonlinear Caputo-Hadamard Fractional Diffusion-Wave Model

doi:10.1007/s42967-023-00295-5

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (1): 392-408.doi: 10.1007/s42967-023-00295-5

An Order Reduction Method for the Nonlinear Caputo-Hadamard Fractional Diffusion-Wave Model

Jieying Zhang¹, Caixia Ou¹, Zhibo Wang¹, Seakweng Vong²

1 School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou 510006, Guangdong, China;
2 Department of Mathematics, University of Macau, Macao 999078, China

Received:2023-03-06 Revised:2023-06-02 Accepted:2023-06-29 Online:2025-04-21 Published:2025-04-21
Supported by:
This research is supported by the National Natural Science Foundation of China (No. 11701103), the Natural Science Foundation of Guangdong Province of China (Nos. 2022A1515012147, 2023A1515011504), and University of Macau (File Nos. MYRG2020-00035-FST, MYRG2018-00047-FST).

Abstract

Abstract: In this paper, the numerical solutions of the nonlinear Hadamard fractional diffusion-wave model with the initial singularity are investigated. Firstly, the model is transformed into coupled equations by virtue of a symmetric fractional-order reduction method. Then the L_log,2-1_σ formula on nonuniform grids is applied to approach to the time fractional derivative. In addition, the discrete fractional Grönwall inequality is used to analyze the optimal convergence of the constructed numerical scheme by the energy method. The accuracy of the theoretical analysis will be demonstrated by means of a numerical experiment at the end.

Key words: Caputo-Hadamard fractional differential equations (FDEs), Symmetric fractional-order reduction method, Nonuniform mesh, Stability and convergence

CLC Number:

Jieying Zhang, Caixia Ou, Zhibo Wang, Seakweng Vong. An Order Reduction Method for the Nonlinear Caputo-Hadamard Fractional Diffusion-Wave Model[J]. Communications on Applied Mathematics and Computation, 2025, 7(1): 392-408.

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References

1. Abbas, S., Benchohra, M., Hamidi, N., Henderson, J.: Caputo-Hadamard fractional differential equations in Banach spaces. Fract. Calc. Appl. Anal. 21(4), 1027–1045 (2018)
2. Cen, D., Wang, Z.: Time two-grid technique combined with temporal second order difference method for two-dimensional semilinear fractional sub-diffusion equations. Appl. Math. Lett. 129, 107919 (2022)
3. Cen, D., Wang, Z., Mo, Y.: Second order difference schemes for time-fractional KdV-Burgers equation with initial singularity. Appl. Math. Lett. 112, 106829 (2021)
4. Chen, H., Xu, D., Zhou, J.: A second-order accurate numerical method with graded meshes for an evolution equation with a weakly singular kernel. J. Comput. Appl. Math. 356, 152–163 (2019)
5. Denisov, S., Kantz, H.: Continuous-time random walk theory of super-slow diffusion. Europhys. Lett. 92(3), 30001 (2010)
6. Fan, E., Li, C., Li, Z.: Numerical approaches to Caputo-Hadamard fractional derivatives with applications to long-term integration of fractional differential systems. Commun. Nonlinear Sci. Numer. Simul. 106, 106096 (2022)
7. Gambo, Y., Jarad, F., Baleanu, D., Abdeljawad, T.: On Caputo modification of the Hadamard fractional derivatives. Adv. Difference Equ. 2014, 10 (2014)
8. Garra, R., Mainardi, F., Spada, G.: A generalization of the Lomnitz logarithmic creep law via Hadamard fractional calculus. Chaos Solitons Fractals 102, 333–338 (2017)
9. Gohar, M., Li, C., Yin, C.: On Caputo-Hadamard fractional differential equations. Int. J. Comput. Math. 97, 1459–1483 (2020)
10. Hadamard, J.: Essai sur létude des fonctions données par leur développement de Taylor. J. Math. Pures Appl. 8, 101–186 (1892)
11. Jarad, F., Abdeljawad, T., Baleanu, D.: Caputo-type modification of the Hadamard fractional derivatives. Adv. Difference Equ. 2012, 142 (2012)
12. Jin, B., Lazarov, R., Zhou, Z.: Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data. SIAM J. Sci. Comput. 38, 146–170 (2016)
13. Kilbas, A., Srivastava, H., Trujillo, J.: Theory and Applications of Fractional Differential Equations. Elsevier Science, Amsterdam (2006)
14. Li, C., Li, Z.: Stability and logarithmic decay of the solution to Hadamard-type fractional differential equation. J. Nonlinear Sci. 31, 31 (2021)
15. Li, C., Li, Z., Wang, Z.: Mathematical analysis and the local discontinuous Galerkin method for Caputo-Hadamard fractional partial differential equation. J. Sci. Comput. 85, 41 (2020)
16. Liao, H., Li, D., Zhang, J.: Sharp error estimate of the nonuniform L1 formula for linear reaction subdiffusion equations. SIAM J. Numer. Anal. 56, 1112–1133 (2018)
17. Liao, H., McLean, W., Zhang, J.: A discrete Grönwall inequality with applications to numerical schemes for subdiffusion problems. SIAM J. Numer. Anal. 57, 218–237 (2019)
18. Liao, H., Mclean, W., Zhang, J.: A second-order scheme with nonuniform time steps for a linear reaction-sudiffusion problem. Commun. Comput. Phys. 30, 567–601 (2021)
19. Liao, H., Sun, Z.: Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations. Numer. Methods Partial Differential Equations 26, 37–60 (2010)
20. Liao, H., Yan, Y., Zhang, J.: Unconditional convergence of a two-level linearized fast algorithm for semilinear subdiffusion equations. J. Sci. Comput. 80(1), 1–25 (2019)
21. Lyu, P., Vong, S.: A graded scheme with bounded grading for a time-fractional Boussinesq type equation. Appl. Math. Lett. 92, 35–40 (2019)
22. Lyu, P., Vong, S.: A symmetric fractional-order reduction method for direct nonuniform approximations of semilinear diffusion-wave equations. J. Sci. Comput. 93, 34 (2022)
23. Ma, L.: Comparison theorems for Caputo-Hadamard fractional differential equations. Fractals 27(03), 1950036 (2019)
24. Ren, J., Chen, H.: A numerical method for distributed order time fractional diffusion equation with weakly singular solutions. Appl. Math. Lett. 96, 159–165 (2019)
25. Sun, Z., Wu, X.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193–209 (2006)
26. Wang, Z., Cen, D., Mo, Y.: Sharp error estimate of a compact L1-ADI scheme for the two-dimensional time-fractional integro-differential equation with singular kernels. Appl. Numer. Math. 159, 190–203 (2021)
27. Wang, Z., Ou, C., Vong, S.: A second-order scheme with nonuniform time grids for Caputo-Hadamard fractional sub-diffusion equations. J. Comput. Appl. Math. 414, 114448 (2022)
28. Yukunthorn, W., Ahmad, B., Ntouyas, S., Tariboon, J.: On Caputo-Hadamard type fractional impulsive hybrid systems with nonlinear fractional integral conditions. Nonlinear Anal. Hybrid. Syst. 19, 77–92 (2016)
29. Zhang, X.: The general solution of differential equations with Caputo-Hadamard fractional derivatives and impulsive effect. Adv. Difference Equ. 2015, 215 (2015)
30. Zorich, V.: Mathematical Analysis I. Springer, Berlin (2004)

An Order Reduction Method for the Nonlinear Caputo-Hadamard Fractional Diffusion-Wave Model

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