THIRD-ORDER ACCURATE IMPLICIT-EXPLICIT TIME-STEPPING SCHEME FOR A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION WITH FRACTIONAL ORDER $\alpha\in(1, 2)$

Lijuan Nong; Chen Wang; Meng Wang; An Chen

doi:10.11948/20250365

2027 Volume 17 Issue 1

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Lijuan Nong, Chen Wang, Meng Wang, An Chen. THIRD-ORDER ACCURATE IMPLICIT-EXPLICIT TIME-STEPPING SCHEME FOR A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION WITH FRACTIONAL ORDER $\alpha\in(1, 2)$[J]. Journal of Applied Analysis & Computation, 2027, 17(1): 163-183. doi: 10.11948/20250365

Citation:

Lijuan Nong, Chen Wang, Meng Wang, An Chen. THIRD-ORDER ACCURATE IMPLICIT-EXPLICIT TIME-STEPPING SCHEME FOR A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION WITH FRACTIONAL ORDER $\alpha\in(1, 2)$[J]. Journal of Applied Analysis & Computation, 2027, 17(1): 163-183. doi: 10.11948/20250365

THIRD-ORDER ACCURATE IMPLICIT-EXPLICIT TIME-STEPPING SCHEME FOR A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION WITH FRACTIONAL ORDER $\alpha\in(1, 2)$

1.
School of Mathematics and Statistic, Guangxi Normal University, No. 1 Yanzhong Road, 541006 Guilin, China
2.
The Center for Applied Mathematics of Guangxi (GXNU), No. 1 Yanzhong Road, 541006 Guilin, China
3.
Guangxi Colleges and Universities Key Laboratory of Mathematical Model and Application (GXNU), No. 1 Yanzhong Road, 541006 Guilin, China
4.
School of Mathematics and Statistic, Guilin University of Technology, No. 12 Jiangan Road, 541004 Guilin, China

Author Bio: Email: ljnong2013@163.com(L. Nong); Email: 3549597670@qq.com(C. Wang); Email: 1900531748@qq.com(M. Wang)
Corresponding author: Email: chena@glut.edu.cn(A. Chen)

Abstract

Abstract

This study focuses on designing a high-order accurate method to solve the nonlinear fractional differential equation with the Caputo derivative of fractional order $\alpha\in(1, 2)$. First, we design two third-order accurate numerical approximations using the graded meshes for the fractional integral, and then employ them to establish the implicit-explicit time-stepping scheme for the equivalent Volterra integral form of the original equation. The error estimates of the two third-order numerical approximations, as well as the stability and the error estimates of the proposed time-stepping scheme, are presented rigorously. This method is further applied to solve a class of multi-term differential equations as well as a nonlinear time-fractional diffusion–wave equation. Numerical examples are extensively provided to demonstrate the effectiveness of the proposed schemes.
- Third-order accurate /
- nonlinear fractional differential equation /
- implicit-explicit time-stepping scheme /
- stability /
- error estimate
MSC: 45D05, 65L20, 65R20

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References

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