NUMERICAL SOLUTION OF FRACTIONAL SINGULAR PERTURBATION CAUCHY PROBLEM

Guocai Cai; Qingxia Liu

doi:10.11948/20250018

2025 Volume 15 Issue 6

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2025 > 15(6): 3207-3225

Next Article Previous Article

Guocai Cai, Qingxia Liu. NUMERICAL SOLUTION OF FRACTIONAL SINGULAR PERTURBATION CAUCHY PROBLEM[J]. Journal of Applied Analysis & Computation, 2025, 15(6): 3207-3225. doi: 10.11948/20250018

Citation:

Guocai Cai, Qingxia Liu. NUMERICAL SOLUTION OF FRACTIONAL SINGULAR PERTURBATION CAUCHY PROBLEM[J]. Journal of Applied Analysis & Computation, 2025, 15(6): 3207-3225. doi: 10.11948/20250018

NUMERICAL SOLUTION OF FRACTIONAL SINGULAR PERTURBATION CAUCHY PROBLEM

Guocai Cai^1,,
Qingxia Liu^1,2, ,

1.
School of Mathematical Sciences, Xiamen University, Xiamen, China
2.
Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing, Xiamen University, Xiamen, Fujian, China

Author Bio: Email: gotry@xmu.eud.cn(G. Cai)
Corresponding author: Email: liuqx@xmu.edu.cn(Q. Liu)
Fund Project: The authors were supported by the Natural Science Foundation of Fujian Province of China (Grant Nos. 2022J01035 and 2023J01026) and the National Natural Science Foundation of China (Grant Nos. 12371372, 12371081 and 12271450)

Abstract

Abstract

In this work, we consider a singularly perturbed Cauchy problem where the small parameter $ \varepsilon $ appears in the highest-order derivative term (i.e., the fractional derivative). First, we analyze some properties of the fractional singular perturbation problem. Then, a Shishkin mesh is introduced to address it, as traditional numerical methods for singularly perturbed problems may lead to numerical instability. Finally, we present numerical results demonstrating good stability and accuracy.
- Fractional singular perturbation /
- numerical solution /
- Shishkin meshes
MSC: 34A08, 34E15, 65Lxx

FullText(HTML)

References

[1]	E. Alvarez and C. Lizama, The super-diffusive singular perturbation problem, Mathematics, 2020, 8, 403, 1-14. DOI: 10.3390/math8030403. CrossRef Google Scholar
[2]	B. Asarel, M. Sah and R. K. Hona, Alternative methods of regular and singular perturbation problems, Applied Mathematics, 2024, 15(10), 687-708. doi: 10.4236/am.2024.1510040 CrossRef Google Scholar
[3]	X. Cai and F. W. Liu, Uniform convergence difference schemes for singularly perturbed mixed boundary problems, Journal of Computational and Applied Mathematics, 2004, 166, 31-54. doi: 10.1016/j.cam.2003.09.038 CrossRef Google Scholar
[4]	D. Chaikoskii and N. Mingkand, Internal layers for a singularly perturbed differential equation with Robin boundary value condition, Journal of East China Normal University (Natural Science), 2020, 2, 23-34. Google Scholar
[5]	V. Georgiev and M. Rastrelli, Sobolev spaces for singular perturbation of 2D Laplace operator, Nonlinear Anal. TMA, 2025, 251, 113710. doi: 10.1016/j.na.2024.113710 CrossRef Google Scholar
[6]	R. Gorenflo and F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, cited as arXiv: 0805.3823, and published on the page 223-276 of the book: A. Carpinteri and F. Mainardi (Editors): Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag, Wien and New York, 1997. DOI: 10.1007/978-3-7091-2664-6_5. Google Scholar
[7]	X. L. Han, L. F. Shi, Y. H. Xu and J. Q. Mo, Asymptotic solution for the fractional order singularly perturbed nonlinear differential equation with two parameters, Acta Mathematicae Applicatae Sinica, 2015, 38(4), 721-729. Google Scholar
[8]	K. Kumar, P. Chakravarthy and V. Aguiar, Numerical solution of time-fractional singularly perturbed convection–diffusion problems with a delay in time, Math Meth Appl Sci. Special Issue Paper, 2020, 1-18. Google Scholar
[9]	X. Y. Lin and F. Xie, Singular perturbation for a kind of nonlinear fractional differential equations, Journal of Donghua University, 2009, 35(2), 238-240. Google Scholar
[10]	Q. X. Liu, F. W. Liu, I. Turner and V. Anh, Numerical simulation for the 3D seepage flow with fractional derivatives in porous media, IMA Journal of Applied Mathematics, 2009, 74, 201-229. Google Scholar
[11]	Y. Luchko, Maximum principle for the generalized time-fractional diffusion equation, Journal of Mathematical Analysis and Applications, 2009, 351, 218-223. doi: 10.1016/j.jmaa.2008.10.018 CrossRef Google Scholar
[12]	M. Mariappan, An efficient computational algorithm for a class of nonlinear singular perturbation problems of convection diffusion type with Cauchy data, Journal of Nonlinear Science, 2025, 35(1), 1-19. doi: 10.1007/s00332-024-10095-7 CrossRef Google Scholar
[13]	J. J. H. Miller, E. O'Riordan and G. I. Shishkin, Fitted Numerical Methods for Singular perturbation Problems: Error Estimates in the Maximum Norm for Linear Problems in One and Two Dimensions, Singapore: World Scientific, 1996. Google Scholar
[14]	I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, California, 1999. Google Scholar
[15]	M. Rezapour, The Caputo-Fabrizio fractional derivative applied to a singular perturbation problem, International Journal of Mathematical Modelling and Numerical Optimisation, 2020, 9(3), 241-253. Google Scholar
[16]	Z. Sabir and M. R. Ali, Analysis of perturbation factors and fractional order derivatives for the novel singular model using the fractional Meyer wavelet neural networks, Chaos, Solitons and Fractals: X, 2023, 11, 100100. Google Scholar
[17]	D. Y. Shi and Y. M. Wu, Uniformly superconvergent analysis of an efficient two-grid method for nonlinear Bi-wave singular perturbation problem, Applied Mathematics and Computation, 2020, 367, 124772. Google Scholar
[18]	L. F. Shi and J. Q. Mo, Asymptotic solution for a class of semilinear singularly perturbed fractional differential equation, Chinese Physics B, 2010, 19(5), 050203. Google Scholar
[19]	G. I. Shishkin, Discrete Approximation of Singularly Perturbed Elliptic and Parabolic Equation (Russian), Second Doctorial Thesis, Keldysh Institure, Moscow, 1990. Google Scholar
[20]	E. R. El-Zahar, G. F. Al-Boqami and H. S. Al-Juaydi, Piecewise approximate analytical solutions of high-order reaction-diffusion singular perturbation problems with boundary and interior layers, AIMS Mathematics, 2024, 9(6), 2024756. Google Scholar
[21]	Y. N. Zhang, Z. Z. Sun and H. L. Liao, Finite difference methods for the time fractional diffusion equation on non-uniform meshes, Journal of Computational Physics, 2014, 265, 195-210. Google Scholar
[22]	H. B. Zhu, Singular perturbation boundary value problem for a fractional differential equation of order $ 2\alpha$, Journal of Anhui University of Technology (Natural Science), 2018, 35(1), 85-89. $ 2\alpha$" target="_blank">Google Scholar