A higher-order uniform accuracy scheme for nonlinear $ \psi $-Volterra integral equations in two dimension with weakly singular kernel

Ziqiang Wang; Jiaojiao Ma; Junying Cao; Ziqiang Wang; Jiaojiao Ma; Junying Cao

doi:10.3934/math.2024697

AIMS Mathematics

2024, Volume 9, Issue 6: 14325-14357. doi: 10.3934/math.2024697

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Research article

A higher-order uniform accuracy scheme for nonlinear $ \psi $-Volterra integral equations in two dimension with weakly singular kernel

School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, 550025, China

Received: 01 March 2024 Revised: 10 April 2024 Accepted: 16 April 2024 Published: 19 April 2024
MSC : 65R20, 65D30

In this paper, we proposed a higher-order uniform accuracy scheme for nonlinear $ \psi $-Volterra integral equations in two dimension with weakly singular kernel by using the modified block-by-block method. First, we constructed a high order uniform accuracy scheme method in this paper by dividing the entire domain into some small sub-domains and approximating the integration function with biquadratic interpolation in each sub-domain. Second, we rigorously proved that the convergence order of the higher order uniform accuracy scheme was $ O(h_{s}^{3+\sigma_{1} }+h_{t}^{3+\sigma_{2} }) $ with $ 0 < \sigma_{1}, \sigma_{2} < 1 $ by using the discrete Gronwall inequality. Finally, two numerical examples were used to illustrate experimental results with different values of $ \psi $ to support the theoretical results.
- $ \psi $-Volterra integral equations,
- high-order uniform accuracy scheme,
- block-by-block method,
- convergence analysis
Citation: Ziqiang Wang, Jiaojiao Ma, Junying Cao. A higher-order uniform accuracy scheme for nonlinear $ \psi $-Volterra integral equations in two dimension with weakly singular kernel[J]. AIMS Mathematics, 2024, 9(6): 14325-14357. doi: 10.3934/math.2024697

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Abstract

In this paper, we proposed a higher-order uniform accuracy scheme for nonlinear $ \psi $-Volterra integral equations in two dimension with weakly singular kernel by using the modified block-by-block method. First, we constructed a high order uniform accuracy scheme method in this paper by dividing the entire domain into some small sub-domains and approximating the integration function with biquadratic interpolation in each sub-domain. Second, we rigorously proved that the convergence order of the higher order uniform accuracy scheme was $ O(h_{s}^{3+\sigma_{1} }+h_{t}^{3+\sigma_{2} }) $ with $ 0 < \sigma_{1}, \sigma_{2} < 1 $ by using the discrete Gronwall inequality. Finally, two numerical examples were used to illustrate experimental results with different values of $ \psi $ to support the theoretical results.

References

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