A novel approach for the numerical solution of nonlinear Fredholm integral equations using the Hosoya polynomial method

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Comparison of exact, method [7] and HPM with Abs_ Error of application 2_

x	Exact solution	Method [7]	Abs. Error of Method [7]	HPM at n = 6	Abs. Error of HPM at n = 6
0.1	0.1652988882	0.1625177090	2.78E-03	0.1608554812	4.44E-03
0.2	0.2018965180	0.1921647474	9.73E-03	0.1942241951	7.67E-03
0.3	0.2465969639	0.2290236855	1.76E-02	0.2531871269	6.59E-03
0.4	0.3011942119	0.2744773506	2.67E-02	0.3047983984	3.60E-03
0.5	0.3678794412	0.3240321944	4.38E-02	0.3658308083	2.04E-03
0.6	0.2578794412	0.2134699868	4.44E-02	0.2619872746	4.11E-03
0.7	0.1278794412	0.0853296542	4.25E-02	0.1371122768	9.23E-03
0.8	−0.0221205588	−0.0603813505	3.83E-02	−0.0375967015	1.55E-02
0.9	−0.1921205588	−0.2236779332	3.16E-02	−0.2043777302	1.23E-02

Comparison with exact, method [36] and HPM with Abs_ Error of application 4_

x	Exact solution	Method [36]	Abs. Error of Method [36]	HPM at n = 6	Abs. Error of HPM at n = 6
0	0.07542668890494	0.02559478612236	4.98E-02	0.07975849286936	4.33E-03
0.1	0.38075203836056	0.33335908249980	4.74E-02	0.38049498956402	2.57E-04
0.2	0.64880672544600	0.60849186923286	4.03E-02	0.64856295681349	2.44E-04
0.3	0.85335168974252	0.82406123219325	2.93E-02	0.85347811951489	1.26E-04
0.4	0.97436464499621	0.95896574017436	1.54E-02	0.97439793249943	3.33E-05
0.5	1.00000000000000	1.00000000000000	0.00E+00	0.99990360585242	9.64E-05
0.6	0.92774838759410	0.94314729241595	1.54E-02	0.92778213023346	3.37E-05
0.7	0.76468229900737	0.79397275655664	2.93E-02	0.76480830219652	1.26E-04
0.8	0.52676377913895	0.56707863535209	4.03E-02	0.52652674951005	2.37E-04
0.9	0.23728195038934	0.28467490625009	4.74E-02	0.23703395647723	2.48E-04
1	-0.07542668890494	-0.02559478612236	4.98E-02	-0.07123971074411	4.19E-03

Comparison with exact, method [6] and HPM with Abs_ Error of application 3_

x	Exact solution	Method [6] at M = 4,k = 2	Abs. Error of Method [6]	HPM at n = 6	Abs. Error of HPM at n = 6
0	1	0.999956	4.40E-05	1.000000	7.47E-06
0.2	1.221403	1.221391	1.18E-05	1.221403	6.57E-07
0.4	1.491825	1.491845	2.00E-05	1.491825	4.13E-07
0.6	1.822119	1.822157	3.80E-05	1.822119	6.47E-07
0.8	2.225541	2.225517	2.39E-05	2.225542	1.40E-06
1	2.718282	2.718217	6.48E-05	2.718274	7.44E-06

Comparison with exact, HPM and existing method with error analysis of application 5_

x	Exact solution	Method [7]	Abs. Error of Method [7]	Abs. Error of Method [39]	HPM at n = 6	Abs. Error of HPM at n = 6
0.1	1.105170918	1.107217811	2.05E-03	2.00E-04	1.105195722	2.48E-05
0.2	1.221402757	1.218102916	3.30E-03	9.00E-03	1.221455388	5.26E-05
0.3	1.349858806	1.341165462	8.69E-03	1.00E-03	1.349857068	1.74E-06
0.4	1.491824696	1.474918603	1.69E-02	1.00E-03	1.491797307	2.74E-05
0.5	1.648721268	1.667402633	1.87E-02	1.00E-03	1.648712515	8.76E-06
0.6	1.822118797	1.833861053	1.17E-02	1.00E-03	1.822138610	1.98E-05
0.7	2.013752703	2.016679830	2.93E-03	1.00E-03	2.013770656	1.79E-05
0.8	2.225540923	2.217456630	8.08E-03	1.00E-03	2.225522500	1.84E-05
0.9	2.459603104	2.437978177	2.16E-02	1.00E-03	2.459586419	1.67E-05

Comparison with exact, method [37] and HPM with Abs_ Error of application 3_

x	Exact solution	Method at k=8 [37]	Abs. Error of Method [37]	HPM at n = 6	Abs. Error of HPM at n = 6
0.1	1.105171	1.0658	3.94E-02	1.105171	5.60E-07
0.2	1.221403	1.2091	1.23E-02	1.221403	6.57E-07
0.3	1.349859	1.3712	2.13E-02	1.349859	1.32E-07
0.4	1.491825	1.5547	6.29E-02	1.491825	4.13E-07
0.5	1.648721	1.7225	7.38E-02	1.648722	7.63E-07
0.6	1.822119	1.7625	5.96E-02	1.822119	6.47E-07
0.7	2.013753	1.9978	1.60E-02	2.013753	5.91E-07
0.8	2.225541	2.2641	3.86E-02	2.225542	1.40E-06
0.9	2.459603	2.5258	6.62E-02	2.459605	1.56E-06

Numerical solution of present method(HPM) with Abs_ Error of application 5_

x	Exact solution	HPM at n = 3	Abs. Error at n = 3	HPM at n = 6	Abs. Error at n = 6
0.1	1.105170918	1.110184967	5.01E-03	1.105195722	2.48E-05
0.2	1.221402758	1.219896572	1.51E-03	1.221455388	5.26E-05
0.3	1.349858808	1.346250257	3.61E-03	1.349857068	1.74E-06
0.4	1.491824698	1.489246022	2.58E-03	1.491797307	2.74E-05
0.5	1.648721271	1.648883866	1.63E-04	1.648712515	8.76E-06
0.6	1.822118800	1.825163790	3.04E-03	1.822138610	1.98E-05
0.7	2.013752707	2.018085793	4.33E-03	2.013770656	1.79E-05
0.8	2.225540928	2.227649877	2.11E-03	2.225522500	1.84E-05
0.9	2.459603111	2.453856039	5.75E-03	2.459586419	1.67E-05