A CRANK-NICOLSON <i>L</i>1/PI DIFFERENCE SCHEME FOR NONLINEAR TIME FRACTIONAL INTEGRO-DIFFERENTIAL EQUATION ON GRADED MESHES

Emadidin Gahalla Mohmed Elmahdi; Jianfei Huang

doi:10.11948/20250119

2026 Volume 16 Issue 1

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Emadidin Gahalla Mohmed Elmahdi, Jianfei Huang. A CRANK-NICOLSON L1/PI DIFFERENCE SCHEME FOR NONLINEAR TIME FRACTIONAL INTEGRO-DIFFERENTIAL EQUATION ON GRADED MESHES[J]. Journal of Applied Analysis & Computation, 2026, 16(1): 347-361. doi: 10.11948/20250119

Citation:

Emadidin Gahalla Mohmed Elmahdi, Jianfei Huang. A CRANK-NICOLSON L1/PI DIFFERENCE SCHEME FOR NONLINEAR TIME FRACTIONAL INTEGRO-DIFFERENTIAL EQUATION ON GRADED MESHES[J]. Journal of Applied Analysis & Computation, 2026, 16(1): 347-361. doi: 10.11948/20250119

A CRANK-NICOLSON L1/PI DIFFERENCE SCHEME FOR NONLINEAR TIME FRACTIONAL INTEGRO-DIFFERENTIAL EQUATION ON GRADED MESHES

Emadidin Gahalla Mohmed Elmahdi^1,2,,
Jianfei Huang^1, ,

1.
College of Mathematical Sciences, Yangzhou University, 225002 Yangzhou, China
2.
Faculty of Education, University of Khartoum, P. O. Box 321 Khartoum, Sudan

Author Bio: Email: emadgahalla11@gmail.com(E. G. M. Elmahdi)
Corresponding author: Email: jfhuang@lsec.cc.ac.cn(J. Huang)
Fund Project: The authors were supported by Natural Science Foundation of Jiangsu Province of China (Grant No. BK20201427) and by National Natural Science Foundation of China (Grant No. 11701502)

Abstract

Abstract

In this paper, a Crank-Nicolson $ L1 $/trapezoidal Product integration (PI) difference scheme is constructed to numerically solve a Volterra-type nonlinear integro-differential equation. Assuming the exact solution exhibits a weak singularity at $ t=0 $, the convergence order of the fully discrete scheme is $ \mathcal{O}\left(N^{-\min\{r\sigma, 2-\alpha, 2\}}+M^{-2}\right) $, and the stability is analyzed using an improved Gr$ \ddot{\text{o}} $nwall inequality in terms of the $ L^2 $-norm. Finally, the theoretical results are verified by numerical experiments.
- Volterra-type integro-differential equations /
- nonuniform meshes /
- finite difference scheme /
- stability and convergence
MSC: 65M06, 65M12

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References

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