Finite difference scheme on non-uniform meshes for third-kind Volterra integral equations with nonsmooth solutions and its error analysis

Jiaxu Liang; Ruiqi Yu; Yu Li; Yan Fan; Jiaxu Liang; Ruiqi Yu; Yu Li; Yan Fan

doi:10.3934/era.2026177

Electronic Research Archive

2026, Volume 34, Issue 6: 3945-3967. doi: 10.3934/era.2026177

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Research article

Finite difference scheme on non-uniform meshes for third-kind Volterra integral equations with nonsmooth solutions and its error analysis

1.
Department of Mathematics, Northeast Forestry University, Harbin 150040, China
2.
Xingzhi College, Zhejiang Normal University, Jinhua 321004, China

Received: 04 April 2026 Revised: 26 April 2026 Accepted: 30 April 2026 Published: 13 May 2026

This paper presents a high-order numerical method for nonlinear Volterra integral equations of the third kind (VIE3) with weakly singular kernels. To overcome the accuracy loss caused by the unbounded derivatives of the solution near the origin, we propose a fractional Adams-Simpson-type method on a graded mesh. The stability and convergence of the proposed scheme, along with detailed error estimates, are rigorously established. It is shown that the method achieves an optimal convergence order of $4 - \alpha - \lambda \beta$, provided the mesh grading exponent $\lambda$ is chosen appropriately. Numerical experiments are presented to validate the theoretical convergence results and to demonstrate the effectiveness of the method in resolving initial singularities.
- Volterra integral equation of the third kind,
- weakly singular,
- Adams-Simpson-type scheme,
- non-uniform meshes
Citation: Jiaxu Liang, Ruiqi Yu, Yu Li, Yan Fan. Finite difference scheme on non-uniform meshes for third-kind Volterra integral equations with nonsmooth solutions and its error analysis[J]. Electronic Research Archive, 2026, 34(6): 3945-3967. doi: 10.3934/era.2026177

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Abstract

This paper presents a high-order numerical method for nonlinear Volterra integral equations of the third kind (VIE3) with weakly singular kernels. To overcome the accuracy loss caused by the unbounded derivatives of the solution near the origin, we propose a fractional Adams-Simpson-type method on a graded mesh. The stability and convergence of the proposed scheme, along with detailed error estimates, are rigorously established. It is shown that the method achieves an optimal convergence order of $4 - \alpha - \lambda \beta$, provided the mesh grading exponent $\lambda$ is chosen appropriately. Numerical experiments are presented to validate the theoretical convergence results and to demonstrate the effectiveness of the method in resolving initial singularities.

References

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