基于再生核和有限差分法求解变系数时间分数阶对流扩散方程

基于再生核和有限差分法求解变系数时间分数阶对流扩散方程

Combining RKM with FDM for Time Fractional Convection-Diffusion Equations with Variable Coefficients

摘要/Abstract

图/表 10

参考文献 28

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数学物理学报 ›› 2025, Vol. 45 ›› Issue (1): 153-164.

吕学琴^1,²,何松岩³,王世宇^2,^4,^*()

¹天津中德应用技术大学基础课部天津 300350
²哈尔滨师范大学数学科学学院哈尔滨 150025
³东北师范大学数学与统计学院长春 130024
⁴北京市第一〇一中学昌平实验学校北京 102206

收稿日期:2024-03-12 修回日期:2024-08-02 出版日期:2025-02-26 发布日期:2025-01-08
通讯作者: * 王世宇, E-mail: wsyhhhh2023@163.com
基金资助:
天津市高等教育委员会科技发展基金(2019KJ142);天津中德应用技术大学教学质量与教学改革研究计划项目(A2301)

Xueqin Lv^1,²,Songyan He³,Shiyu Wang^2,^4,^*()

¹College of Basic Science, Tianjin Sino-German University of Applied Sciences, Tianjin 300350
²School of Mathematics and Sciences, Harbin Normal University, Harbin 150025
³School of Mathematics and Statistics, Northeast Normal University, Changchun 130024
⁴Beijing No.101 Middle School Changping Experimental School, Beijing 102206

Received:2024-03-12 Revised:2024-08-02 Online:2025-02-26 Published:2025-01-08
Supported by:
Science & Technology Development Fund of Tianjin Education Commission for Higher Education(2019KJ142);Teaching Quality and Teaching Reform Research Project of Tianjin Sino-German University of Applied Technology(A2301)

摘要：

针对变系数的时间分数阶对流-扩散方程, 首先, 使用有限差分法, 得到了该方程的半离散格式. 之后再利用再生核方法, 得到了方程的精确解 $u(x,t_{n})$ , 将精确解 $u(x,t_{n})$ 取 $m$ 项截断, 可得到近似解 $u_{m}(x,t_{n})$ . 通过证明, 得到该方法是稳定的. 最后, 通过三个数值例子, 并与其他文献中的方法在同等条件下进行了比较, 证明该算法有效.

关键词: Caputo 分数阶导数, 再生核方法, 变系数时间分数阶对流扩散方程, 有限差分方法

Abstract:

In this paper, we will study the time fractional convection-diffusion equation with variable coefficients. First, we use the finite difference method. The time variable is discretized, and the semi-discrete scheme of the equation is obtained. The exact solution $u(x,t_{n})$ of the equation is obtained by using the theory of reproducing kernel method. Then the exact solution $u(x,t_{n})$ is truncated by $m$ term to obtain the approximate solution $u_{m}(x,t_{n})$ . By proving, we know that the method is stable. Moreover, $u_{m}^{(i)}(x,t_{n})$ converge uniformly to $u^{(i)}(x,t_{n})$ $(i=0,1,2)$ . Finally, we give several numerical examples and compare them with the methods in other literatures, which show that our algorithm is effective.

Key words: Caputo fractional derivative, reproducing kernel method, variable coefficient time fractional convection-diffusion equation, finite difference method

中图分类号:

吕学琴, 何松岩, 王世宇. 基于再生核和有限差分法求解变系数时间分数阶对流扩散方程[J]. 数学物理学报, 2025, 45(1): 153-164.

Xueqin Lv, Songyan He, Shiyu Wang. Combining RKM with FDM for Time Fractional Convection-Diffusion Equations with Variable Coefficients[J]. Acta mathematica scientia,Series A, 2025, 45(1): 153-164.

表 1

表 2

图1

不同 $n$ 下的绝对误差"

图1

表 3

不同 $\alpha$ 和 $m$ 下的最大绝对误差和收敛阶"

表 3

表 4

不同 $n$ 下的绝对误差"

表 4

图2

不同 $\alpha$ 下的绝对误差"

图2

表5

表 6

图3

精确解 $u(x,t)$ "

图3

图4

不同 $n$ 时的近似解 $u^{n+1}_{50}$ "

图4

[1]	Gupta S, Thakur B. Extended Legendre wavelet method for solving fractional order time hyperbolic partial differential equation. Int J Appl Comput Math, 2023, 9(3): 41-64
[2]	Osman M, Xia Y, Marwan M, Omer O A. Novel approaches for solving fuzzy fractional partial differential equations. Fractal Fract 2022, 6(11): Article 656
[3]	An X Y, Liu F W, Zheng M L, et al. A space-time spectral method for time-fractional Black-Scholes equation. Appl Numer Math, 2021, 165: 152-166
[4]	Prakash A, Veeresha P, Prakasha D G, Goyal M. A homotopy technique for a fractional order multi-dimensional telegraph equation via the Laplace transform. Eur Phys J Plus, 2019, 134: 1-18
[5]	Lin Y, Xu C. Finite difference/spectral approximations for the time-fractional diffusion equation. J Comput Phys, 2007, 225(2): 1533-1552
[6]	Chen Y, Wu Y, Cui Y, et al. Wavelet method for a class of fractional convection-diffusion equation with variable coefficients. J Comput Sci, 2010, 1(3): 146-149
[7]	Saadatmandi A, Dehghan M, Azizi M R. The Sinc-Legendre collocation method for a class of fractional convection-diffusion equations with variable coefficients. Commun Nonlinear Sci Numer Simul, 2012, 17(11): 4125-4136
[8]	Yaseen M, Abbas M, Nazir T, Baleanu D. A finite difference scheme based on cubic trigonometric B-splines for a time fractional diffusion-wave equation. Adv Difference Equ, 2017, 2017: 1-18
[9]	Ren L, Wang Y M. A fourth-order extrapolated compact difference method for time-fractional convection-reaction-diffusion equations with spatially variable coefficients. Appl Math Comput, 2017, 312: 1-22
[10]	Saw V, Kumar S. Collocation method for time fractional diffusion equation based on the Chebyshev polynomials of second kind. Int J Appl and Comput Math, 2020, 6(4): 117-129
[11]	Saw V, Kumar S. The Chebyshev collocation method for a class of time fractional convection-diffusion equation with variable coefficients. Math Methods Appl Sci, 2021, 44(8): 6666-6678
[12]	Uddin M, Haq S. RBFs approximation method for time fractional partial differential equations. Commun Nonlinear Sci Numer Simul, 2011, 16(11): 4208-4214
[13]	Zhang J, Zhang X, Yang B. An approximation scheme for the time fractional convection-diffusion equation. Appl Math Comput, 2018, 335: 305-312
[14]	Ravi Kanth A S V, Garg N. Numerical simulation of time fractional advection-diffusion-reaction equation using exponential B-Splines// Kumar A, Srivastava S, Singh S. Renewable Energy Towards Smart Grid: Select Proceedings of SGESC 2021. Singapore: Springer, 2022: 133-143
[15]	Li J C, Qin X Q. Finite point method for the time fractional convection-diffusion equation Xiong N, Xiao Z, Tong Z, et al. Advances in Computational Science and Computing. Cham: Springer, 2019: 28-36
[16]	傅博, 王世宇, 高婷婷, 吕学琴. 带有 Caputo-Fabrizio 导数的分数阶微分方程的快速高阶算法的研究. 数学物理学报, 2023, 43A(3): 896-912
	Fu B, Wang S Y, Gao T T, Lv X Q. A fast high order method for fractional differential equations with the Caputo-Fabrizio derivative. Acta Math Sci, 2023, 43A(3): 896-912
[17]	Cui M, Deng Z H. Numerical Functional Method in Reproducing Kernel Space. Harbin: The Publication of Harbin Institute of Technology, 1988
[18]	Cui M, Yan Y. The representation of the solution of a kind of operator equation Au=f. Numer Math J Chin Univ, 1995, 1: 82-86
[19]	Li X Y, Li H X, Wu B Y. Piecewise reproducing kernel method for linear impulsive delay differential equations with piecewise constant arguments. Appl Math Comput, 2019, 349: 304-313
[20]	Mei L. A novel method for nonlinear impulsive differential equations in broken reproducing kernel space. Acta Math Sci, 2020, 40: 723-733
[21]	Lv X, Cui M. An efficient computational method for linear fifth-order two-point boundary value problems. J Comput Appl Math, 2010, 234(5): 1551-1558
[22]	Sahihi H, Allahviranloo T, Abbasbandy S. Solving system of second-order BVPs using a new algorithm based on reproducing kernel Hilbert space. Appl Numer Math, 2020, 151: 27-39
[23]	Niu J, Sun L, Xu M, Hou J. A reproducing kernel method for solving heat conduction equations with delay. Appl Math Lett, 2020, 100: 106036
[24]	Chen Z, Jiang W, Du H. A new reproducing kernel method for Duffing equations. Int J Comput Math, 2021, 98(11): 2341-2354
[25]	Mei L, Wu B, Lin Y. Shifted-Legendre orthonormal method for high-dimensional heat conduction equations. AIMS Math, 2022, 7(5): 9463-9478
[26]	Fardi M, Al-Omari S K Q, Araci S. A pseudo-spectral method based on reproducing kernel for solving the time-fractional diffusion-wave equation. Adv Contin Discrete Models, 2022, 2022(1): 54-67
[27]	Li Z, Chen Q, Wang Y, Li X. Solving two-sided fractional super-diffusive partial differential equations with variable coefficients in a class of new reproducing kernel spaces. Fractal Fract, 2022, 6(9): 492-502
[28]	Mercer J. Functions of positive and negative type and their connection with theory of integral equations. Proc R Soc Lond, 1909: 83(559): 69-70

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