| Citation: |
Jianyu Wang, Chunhua Fang, Guifeng Zhang, Zaiyun Zhang. MODIFIED COLLOCATION METHODS FOR SECOND KIND OF VOLTERRA INTEGRAL EQUATIONS WITH WEAKLY SINGULAR HIGHLY OSCILLATORY BESSEL KERNELS[J]. Journal of Applied Analysis & Computation, 2023, 13(6): 3231-3252. doi: 10.11948/20220559
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MODIFIED COLLOCATION METHODS FOR SECOND KIND OF VOLTERRA INTEGRAL EQUATIONS WITH WEAKLY SINGULAR HIGHLY OSCILLATORY BESSEL KERNELS
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Abstract
In this paper, we investigate the second kind of Volterra integral equations with weakly sinular highly oscillatory Bessel kernels by using two collocation methods: direct high-order interpolationorder (DO) and direct Hermite interpolation (DH). Based on hypergeometric and Gamma functions, we obtain a method for solving the modified moments $ \int_{0}^{1}x^{\alpha}(1-x)^{\beta}J_{v}(\omega x)dx $. Compared with the Filon-type $ (Q_{N}^{F}) $ method, piecewise constant collocation $ (Q_{N}^{L, 0}) $ method and linear collocation $ (Q_{N}^{L, 1}) $ method, we verified the efficiency of the method through error analysis and numerical examples.
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References
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Jianyu Wang, Chunhua Fang, Guifeng Zhang, Zaiyun Zhang. MODIFIED COLLOCATION METHODS FOR SECOND KIND OF VOLTERRA INTEGRAL EQUATIONS WITH WEAKLY SINGULAR HIGHLY OSCILLATORY BESSEL KERNELS[J]. Journal of Applied Analysis & Computation, 2023, 13(6): 3231-3252. doi: 10.11948/20220559
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Figure 1.
The error curves with
$ \omega $ $ x=0.2, 0.5 $ $ 0.8 $ -
Figure 2.
Comparison of the error at point
$ x=0.2 $ -
Figure 3.
The error curves with
$ \omega $ $ x=0.2, 0.5 $ $ 0.8 $ -
Figure 4.
Comparison of the error at point
$ x=0.2 $ -
Figure 5.
The error curves with
$ \omega $ $ x=0.2, 0.5 $ $ 0.8 $ -
Figure 6.
Comparison of the error at point
$ x=0.2 $ -
Figure 7.
The error curves with
$ \omega $ $ x=0.2, 0.5 $ $ 0.8 $ -
Figure 8.
Comparison of the error at point
$ x=0.2 $ -
Figure 9.
The error curves with
$ \omega $ $ s=0.2, 0.5 $ $ 0.8 $ -
Figure 10.
Comparison of the error at point
$ x=0.2 $ -
Figure 11.
The error curves with
$ \omega $ $ s=0.2, 0.5 $ $ 0.8 $ -
Figure 12.
Comparison of the error at point
$ x=0.2 $