Homotopy perturbation method and Chebyshev polynomials for solving a class of singular and hypersingular integral equations

Zainidin Eshkuvatov

doi:10.3934/naco.2018022

Article Contents

2018, Volume 8, Issue 3: 337-350. Doi: 10.3934/naco.2018022

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Homotopy perturbation method and Chebyshev polynomials for solving a class of singular and hypersingular integral equations

Zainidin Eshkuvatov^1,2, ,

1.
Faculty of Science and Technology, Universiti Sains Islam Malaysia, Malaysia (USIM), Negeri Sembilan, Malaysia
2.
Institute for Mathematical Research, Universiti Putra Malaysia (UPM), Malaysia

* Corresponding author: Zainidin Eshkuvatov
* Corresponding author: Zainidin Eshkuvatov

Received: April 2017

Revised: March 2018

Published: June 2018

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Abstract

TIn this note, we review homotopy perturbation method (HPM), Discrete HPM, Chebyshev polynomials and its properties. Moreover, the convergences of HPM and error term of Chebyshev polynomials were discussed. Then, linear singular integral equations (SIEs) and hyper-singular integral equations (HSIEs) are solved by combining modified HPM together with Chebyshev polynomials. Convergences of the mixed method for the linear HSIEs are also obtained. Finally, illustrative examples and comparisons with different methods are presented.

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Mathematics Subject Classification: Primary: 45E05, 42A10; Secondary: 65D30, 35L81.

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Table 1. Comparisons with other methods

$x$	Error term in [45]	Error of HPM	Error of MHPM in [28]
-1	0	0	0
-0.5	$13*10^{-17}$	0	0
0.	0	0	0
0.5	$7.8*10^{-18}$	0	0
1	0	0	0

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Table 2. Comparisons with other methods

$x$	Error term in [45]	Error of MHPM [28]
-1.0	0.0	0
-0.8	$2.1\cdot 10^{-10}$	0
-0.4	$3.4\cdot 10^{-9}$	0
0.0	$2.0\cdot 10^{-9}$	0
0.4	$3.3\cdot 10^{-9}$	0
0.8	$2.7\cdot 10^{-9}$	0
1.0	0.0	0

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Table 3. Error terms for different value of $n$

$ x$	Exact Solution	Error MHPM for m=n=6	Error MHPM for m=n=10
-0.9999	0.14140368029	1.0851729 10?4	1.4040446 10?10
-0.901	3.94739842327	3.5501594 10?4	1.8203460 10?9
-0.436	5.75413468725	3.0648319 10?4	2.6221447 10?8
-0.015	5.03721659280	8.6784464 10?5	1.7385004 10?8
0.015	4.96222081226	1.9992451 10?5	1.8524990 10?8
0.436	3.69436233615	3.3916634 10?1	1.9700974 10?8
0.901	1.49541222584	1.2745747 10?4	1.3403888 10?8
0.9999	0.04714084491	3.3327100 10?5	3.5400390 10?10

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