HIGH-ORDER NUMERICAL SCHEME AND THEORETICAL ANALYSIS FOR NONLINEAR TWO-DIMENSIONAL FRACTIONAL VOLTERRA INTEGRAL EQUATIONS WITH INITIAL VALUE SINGULARITY

Ziqiang Wang; Xiao Wang; Junying Cao

doi:10.11948/20240345

2025 Volume 15 Issue 6

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Article navigation > Journal of Applied Analysis & Computation > 2025 > 15(6): 3369-3402

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Ziqiang Wang, Xiao Wang, Junying Cao. HIGH-ORDER NUMERICAL SCHEME AND THEORETICAL ANALYSIS FOR NONLINEAR TWO-DIMENSIONAL FRACTIONAL VOLTERRA INTEGRAL EQUATIONS WITH INITIAL VALUE SINGULARITY[J]. Journal of Applied Analysis & Computation, 2025, 15(6): 3369-3402. doi: 10.11948/20240345

Citation:

Ziqiang Wang, Xiao Wang, Junying Cao. HIGH-ORDER NUMERICAL SCHEME AND THEORETICAL ANALYSIS FOR NONLINEAR TWO-DIMENSIONAL FRACTIONAL VOLTERRA INTEGRAL EQUATIONS WITH INITIAL VALUE SINGULARITY[J]. Journal of Applied Analysis & Computation, 2025, 15(6): 3369-3402. doi: 10.11948/20240345

HIGH-ORDER NUMERICAL SCHEME AND THEORETICAL ANALYSIS FOR NONLINEAR TWO-DIMENSIONAL FRACTIONAL VOLTERRA INTEGRAL EQUATIONS WITH INITIAL VALUE SINGULARITY

1.
School of Data Science and Information Engineering, Guizhou Minzu University, 550025 Guiyang, China
2.
Qiannan Polytechnic for Nationalities, 558022 Duyun, China

Author Bio: Email: wangzq@lsec.cc.ac.cn(Z. Wang); Email: wangxiaomath@126.com (X. Wang)
Corresponding author: Email: caojunying@gzmu.edu.cn(J. Cao)
Fund Project: Z. Q. Wang was supported by National Natural Science Foundation of China (Grant Nos. 12461077, 11961009), Foundation of Guizhou Science and Technology Department (Grant No. QHKJC-ZK[2024]YB497), Department of Education of Guizhou Province (Grant No. QJJ2023012), High-Level Innovative Talent Project of Guizhou Province (Grant No. QKHPTRC-GCC2023027), and the Science and Technology Innovation Talent Team Project of Data Science and Computing Intelligence of Guizhou Province (Grant no. QKHRCCXTD2025038). J. Y. Cao was supported by National Natural Science Foundation of China (Grant No. 12361083), Science research fund support project of the Guizhou Minzu University (Grant No. GZMUZK[2023]CXTD05) and Young Teacher Foundation of Henan Province (No. 2021GGJS080)

Abstract

Abstract

Various numerical methods have been proposed for solving one-dimensional weakly singular Volterra integral equations (VIEs) with smooth solutions. The main purpose of this paper is to propose and analyze a numerical method for the solution of two-dimensional nonlinear weakly singular VIEs with non-smooth solutions by involving the transformation of variables and modified Block-by-Block method. We rigorously prove that the new scheme can achieve an order of $ O(\tau_s^{3+\alpha}+\lambda_t^{3+\beta}) $ for non-smooth solutions with step size $ \tau_s,\lambda_t $ for $ 0<\alpha,\beta<1 $. Some numerical examples are conducted to support the theoretical results and demonstrate the effectiveness of the proposed method.
- Nonlinear Volterra integral equations /
- non-smooth solution /
- Blockby-Block method /
- convergence analysis
MSC: 65R20, 65D30

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References

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