Article Contents

2017, Volume 7, Issue 1: 77-88. Doi: 10.3934/naco.2017004

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Effective approximation method for solving linear Fredholm-Volterra integral equations

1.
Faculty of Science and Technology, University Sains Islam Malaysia (USIM), Negeri Sembilan, Malaysia
2.
Institute for Mathematical Research (INSPEM), University Putra Malaysia (UPM), Selangor, Malaysia
3.
Department of Mathematics, Faculty of Science, University Putra Malaysia (UPM)
4.
Institute for Mathematical Research (INSPEM), UPM, Selangor, Malaysia

* Corresponding author: Z. K. Eshkuvatov, zainidin@usim.edu.my
* Corresponding author: Z. K. Eshkuvatov, zainidin@usim.edu.my

Received: March 2016

Revised: February 2017

Published: May 2017

Abstract / Introduction Full Text(HTML) Figure(0) / Table(9) Related Papers Cited by

Abstract

An efficient approximate method for solving Fredholm-Volterra integral equations of the third kind is presented. As a basis functions truncated Legendre series is used for unknown function and Gauss-Legendre quadrature formula with collocation method are applied to reduce problem into linear algebraic equations. The existence and uniqueness solution of the integral equation of the 3rd kind are shown as well as rate of convergence is obtained. Illustrative examples revels that the proposed method is very efficient and accurate. Finally, comparison results with the previous work are also given.

Keywords:

Mathematics Subject Classification: Primary: 45B05; Secondary: 45A05, 45L05.

Citation:

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Table 1. The error term $Qe_{n} \left(s\right)=\left|x\left(s\right)-Qx_{n} \left(s\right)\right|$ for Example 1

s	Qe_n(s)(11)
	n=5	n=11	n=20
1	3.000e-19	1.000e-19	4.000e-19
0.8	2.000e-19	1.000e-19	4.000e-19
0.6	1.000e-19	0.000e-0	3.000e-19
0.4	0.000e-0	0.000e-0	1.000e-19
0.2	5.000e-20	7.000e-20	7.000e-20
0.1	3.000e-20	1.000e-19	7.000e-20
0	6.000e-20	1.100e-19	6.000e-20

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Table 2. The error term $Qe_{n} \left(s\right)=\left|x\left(s\right)-Qx_{n} \left(s\right)\right|$ for Example 2

s	Qe_n(s)(11)
	n=5	n=11	n=20
1.0	3.571e-3	2.676e-3	3.999e-4
0.9	9.534e-3	1.518e-3	2.031e-4
0.7	1.445e-3	1.501e-3	1.800e-4
0.5	1.014e-2	8.541e-4	1.368e-4
0.3	3.406e-3	5.103e-4	5.103e-4
0.1	2.674e-2	3.278e-3	3.673e-4
0.0	4.155e-1	2.331e-1	1.244e-1

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Table 3. Error comparison between $ Qe_{n} \left(s\right)$ and $ Ce_{n} \left(s\right)$ for Example 2

s	n=2
	Qe_n(s)(11)	Ce_n(s)[6]
1.00	9.943e-2	9.500e-2
0.75	1.874e-3	6.380e-2
0.50	3.183e-2	3.690e-2
0.25	9.565e-2	8.200e-2
0.00	7.925e-1	6.927e-1

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Table 4. The comparison of error terms $Qe_{n} \left(s\right)$ and $Me_{n} \left(s\right)$ for Example 3

s	n=5		n=7		n=9
	Qe_n(s)(11)	Me_n(s)[13]	Qe_n(s)(11)	Me_n(s)[13]	Qe_n(s)(11)	Me_n(s)[13]
0.999	6.935e-1	1.829e-1	2.908e-1	1.109e-1	4.768e-2	1.562e-2
0.753	2.829e-1	1.026e0	6.962e-2	2.700e-1	1.581e-2	1.445e-2
0.352	4.374e-1	1.009e0	1.355e-1	1.502e-1	1.387e-2	3.929e-2
0.001	6.416e-1	3.025e-1	1.412e-1	1.401e-1	1.750e-2	1.152e-2
-0.001	6.411e-1	3.058e-1	1.412e-1	1.386e-1	1.750e-2	1.123e-2
-0.352	5.173e-1	4.409e-1	1.370e-1	4.024e-2	1.388e-2	2.596e-2
-0.753	4.027e-1	1.165e-1	7.146e-2	1.532e-1	1.580e-2	6.695e-2
-0.999	7.916e-1	2.605e-1	2.913e-1	1.226e-1	4.768e-2	2.196e-2

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Table 5. The error comparisons between $Qe_{n} (s)$ and $ Me_{n} (s)$ for lager ''n''

s	Me_n(s)[13]			Qe_n(s)(11)
	n=13	n=15	n=19	n=13	n=15	n=19	n=20
0.999	6.092e-3	1.863e-2	9.825e0	3.002e-4	1.406e-5	1.385e-8	6.319e-10
0.753	6.688e-3	4.460e-2	1.083e0	9.126e-5	2.509e-7	8.004e-10	6.528e-11
0.352	6.233e-4	1.067e-2	3.697e-1	4.099e-5	3.483e-6	1.565e-9	5.362e-11
0.001	3.081e-3	4.409e-2	1.212e0	8.363e-5	3.644e-6	3.330e-9	4.196e-15
-0.001	3.111e-3	4.361e-2	1.208e0	8.363e-5	3.644e-6	3.330e-9	4.212e-15
-0.352	3.111e-3	2.288e-2	5.682e-1	4.099e-5	3.483e-6	1.565e-9	5.362e-11
-0.753	9.683e-3	5.025e-2	9.905e-1	9.126e-5	2.509e-7	8.004e-10	6.528e-11
-0.999	2.291e-3	1.083e-1	7.677e0	3.002e-4	1.406e-5	1.385e-8	6.319e-10

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Table 6. Comparison of error terms $Qe_{n} \left(s\right)$ and $Mue_{n} \left(s\right)$ for Example 4

s	n=5		n=10
	Qe_n(s)(11)	Mue_n(s)[18]	Qe_n(s)(11)	Mue_n(s)[18]
0.0	6.117e-5	0	5.662e-11	0
0.4	9.766e-6	4.593e-6	1.379e-11	2.213e-12
0.8	4.252e-6	8.389e-6	1.037e-11	5.683e-12
1.2	3.841e-6	1.378e-5	1.059e-11	1.009e-11
1.6	7.448e-6	2.153e-5	1.470e-11	1.612e-11
2.0	3.880e-5	3.179e-5	6.304e-11	2.376e-11

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Table 7. Comparison of error terms $Qe_{n} \left(s\right)$ and $Mue_{n} \left(s\right)$ for Example 4

s	n=9
	Qe_n(s)(11)	Mue_n(s)[18]
0.0	9.194e-10	0
0.2222	2.775e-10	5.694e-11
0.4444	2.343e-10	9.869e-11
0.6667	1.951e-10	1.478e-10
0.8889	7.917e-10	2.037e-10
1.1111	7.663e-10	2.698e-10
1.3333	1.775e-10	3.493e-10
1.5556	2.001e-10	4.458e-10
1.7778	2.224e-10	5.663e-10
2.0000	6.907e-10	7.011e-10

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Table 8. Comparison of error terms $Qe_{n} \left(s\right)$ and $Mue_{n} \left(s\right)$ for Example 5

s	n=5
	Qe_n(s)(11)	Mue_n(s)[18]
0.0	2.304e-6	0
0.2	4.121e-7	1.263e-6
0.4	2.096e-7	2.555e-6
0.6	1.903e-7	3.879e-5
0.8	4.451e-7	5.506e-5
1.0	2.710e-6	7.751e-5

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Table 9. Comparison of error terms $Qe_{n} \left(s\right)$ and $Mue_{n} \left(s\right)$ for Example 5

s	n=9
	Qe_n(s)(11)	Mue_n(s)[18]
0.0	9.194e-10	0
0.1111	2.775e-10	9.133e-13
0.2222	2.343e-10	1.842e-12
0.3333	1.951e-10	2.753e-12
0.4444	7.917e-10	3.678e-12
0.5556	7.663e-10	4.638e-12
0.6667	1.775e-10	5.685e-12
0.7778	2.001e-10	6.871e-12
0.8889	2.224e-10	8.292e-12
1.0	6.907e-10	1.005e-11

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