A new application of Taylor expansion for approximate solution of systems of Fredholm integral equations

Mohsen Didgar; Farzan Ekhlasi

doi:10.2478/ijmce-2026-0008

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A new application of Taylor expansion for approximate solution of systems of Fredholm integral equations

International Journal of Mathematics and Computer in Engineering

Volume 4 (2026): Issue 1 (June 2026)

By: Mohsen Didgar and Farzan Ekhlasi

Open Access

|May 2026

Figures & Tables

The exact solutions (solid) versus the approximate solutions (dashed) for m = 1.

The exact solutions (solid) versus the approximate solutions (dashed) for m = 2.

The exact solutions (solid) versus the approximate solutions (dashed) for m = 3.

Variations of the errors between several approximations of ψ1(x) and the corresponding exact value.

Variations of the errors between several approximations of ψ2(x) and the corresponding exact value.

Absolute errors of Application 5 by rationalized Haar functions method in [28] for (ψ1(x), ψ2(x))_

x	M=16	M=32
0.1	(5.98 × 10⁻³, 2.40 × 10⁻⁴)	(9.44 × 10⁻³, 1.07 × 10⁻³)
0.2	(1.89 × 10⁻², 4.12 × 10⁻³)	(3.18 × 10⁻³, 7.00 × 10⁻⁴)
0.3	(1.85 × 10⁻², 5.12 × 10⁻³)	(3.08 × 10⁻³, 8.60 × 10⁻⁴)
0.4	(6.41 × 10⁻³, 2.62 × 10⁻³)	(9.34 × 10⁻³, 3.56 × 10⁻³)
0.5	(3.11 × 10⁻², 1.44 × 10⁻²)	(1.56 × 10⁻², 7.36 × 10⁻³)
0.6	(6.18 × 10⁻³, 3.53 × 10⁻³)	(9.39 × 10⁻³, 5.32 × 10⁻³)
0.7	(1.87 × 10⁻², 1.20 × 10⁻²)	(3.13 × 10⁻³, 1.98 × 10⁻³)
0.8	(1.87 × 10⁻², 1.35 × 10⁻²)	(3.13 × 10⁻³, 2.31 × 10⁻³)
0.9	(6.17 × 10⁻³, 4.45 × 10⁻³)	(9.40 × 10⁻³, 7.42 × 10⁻³)
1.0	(3.13 × 10⁻², 2.65 × 10⁻²)	(1.56 × 10⁻², 1.32 × 10⁻²)

Absolute errors of Application 1 for ψ1(x)_

x	Method in [31] (m=10)	Suggested method	Suggested method	Suggested method
	\|ψ_Exact − ψ_Taylor\|	m=1	m=2	m=3
0.1	2.548 × 10⁻⁴	2.89365 × 10⁻¹	8.76658 × 10⁻²	0
0.2	1.055 × 10⁻³	1.69063 × 10⁻¹	5.32303 × 10⁻²	0
0.3	1.558 × 10⁻³	9.81397 × 10⁻²	2.99342 × 10⁻²	0
0.4	1.513 × 10⁻³	5.38480 × 10⁻²	1.52024 × 10⁻²	0
0.5	1.467 × 10⁻³	2.82321 × 10⁻²	6.67616 × 10⁻³	0
0.6	2.634 × 10⁻³	1.73289 × 10⁻²	2.29812 × 10⁻³	0
0.7	6.293 × 10⁻³	1.93255 × 10⁻²	3.73043 × 10⁻⁴	0
0.8	1.248 × 10⁻²	3.43349 × 10⁻²	4.01271 × 10⁻⁴	0
0.9	1.655 × 10⁻²	6.45784 × 10⁻²	9.36226 × 10⁻⁴	0
1.0	4.620 × 10⁻³	1.14521 × 10⁻¹	1.79048 × 10⁻³	0

Absolute errors of Application 2 for ψ2(x)_

x	Method in [26]	Suggested method	Suggested method
	ψ_Exact − ψ_ADM	m=1	m=2
0.1	3.45 × 10⁻³	6.25429 × 10⁻²	0
0.2	6.90 × 10⁻³	1.00914 × 10⁻¹	0
0.3	1.03 × 10⁻²	1.18371 × 10⁻¹	0
0.4	1.38 × 10⁻²	1.18171 × 10⁻¹	0
0.5	1.72 × 10⁻²	1.03571 × 10⁻¹	0
0.6	2.07 × 10⁻²	7.78286 × 10⁻²	0
0.7	2.41 × 10⁻²	4.42000 × 10⁻²	0
0.8	2.76 × 10⁻²	5.94286 × 10⁻³	0
0.9	3.10 × 10⁻²	3.36857 × 10⁻²	0
1.0	3.45 × 10⁻²	7.14286 × 10⁻²	0

Absolute errors of Application 4 for ψ1(x)_

x	m=1	m=2	m=3	m=4
0.1	1.42576 × 10⁻¹	1.87892 × 10⁻¹	1.18936 × 10⁻²	1.07546 × 10⁻²
0.2	1.02909 × 10⁻¹	1.50166 × 10⁻¹	8.40862 × 10⁻³	4.51723 × 10⁻³
0.3	7.28585 × 10⁻²	1.16840 × 10⁻¹	5.75061 × 10⁻³	3.13103 × 10⁻³
0.4	5.11274 × 10⁻²	8.77018 × 10⁻²	3.77465 × 10⁻³	2.06896 × 10⁻³
0.5	3.66628 × 10⁻²	6.25978 × 10⁻²	2.34919 × 10⁻³	1.82803 × 10⁻³
0.6	2.87143 × 10⁻²	4.14305 × 10⁻²	1.35764 × 10⁻³	7.21594 × 10⁻⁴
0.7	2.69156 × 10⁻²	2.41633 × 10⁻²	6.98557 × 10⁻⁴	3.52211 × 10⁻⁴
0.8	3.13962 × 10⁻²	1.08267 × 10⁻²	2.86319 × 10⁻⁴	1.33281 × 10⁻⁴
0.9	4.29419 × 10⁻²	1.52918 × 10⁻³	5.19763 × 10⁻⁵	2.75368 × 10⁻⁵
1.0	6.32297 × 10⁻²	3.52550 × 10⁻³	5.55572 × 10⁻⁵	6.06983 × 10⁻⁷

The maximum of the absolute errors in [59] by collocation method_

x	cos x
1.91 × 10⁻²	1.02 × 10⁻²

Absolute errors of Application 3 by B-spline wavelet method in [40] for (ψ1(x), ψ2(x))_

x	M=2	M=4
0.1	(2.07979E − 6, 1.04205E − 4)	(8.12417E − 9, 6.51057E − 6)
0.2	(1.94288E − 6, 1.14576E − 3)	(7.58937E − 9, 7.16143E − 5)
0.3	(1.80597E − 6, 1.14572E − 3)	(7.05456E − 9, 7.16142E − 5)
0.4	(1.66906E − 6, 1.04308E − 4)	(6.51976E − 9, 6.51097E − 6)
0.5	(1.53215E − 6, 2.60434E − 3)	(5.98496E − 9, 1.62761E − 4)
0.6	(1.39524E − 6, 1.04367E − 4)	(5.45016E − 9, 6.51120E − 6)
0.7	(1.25833E − 6, 1.14561E − 3)	(4.91536E − 9, 7.16137E − 5)
0.8	(1.12142E − 6, 1.14558E − 3)	(4.38055E − 9, 7.16136E − 5)
0.9	(9.84513E − 7, 1.04438E − 4)	(3.84575E − 9, 6.51148E − 6)
1.0	(8.47603E − 7, 2.60446E − 3)	(3.31095E − 9, 1.62762E − 4)

Absolute errors of Application 5 for (ψ1(x), ψ2(x))_

x	m=1	m=2
0.1	(6.39030 × 10⁻³, 4.89613 × 10⁻²)	(5.03104 × 10⁻⁴, 3.35710 × 10⁻³)
0.2	(9.88092 × 10⁻⁴, 4.28668 × 10⁻²)	(1.21393 × 10⁻⁴, 3.29583 × 10⁻³)
0.3	(2.53419 × 10⁻³, 3.56531 × 10⁻²)	(1.191079 × 10⁻⁴, 3.02207 × 10⁻³)
0.4	(4.35938 × 10⁻³, 2.79192 × 10⁻²)	(3.97546 × 10⁻⁴, 2.57961 × 10⁻³)
0.5	(4.69526 × 10⁻³, 2.02420 × 10⁻²)	(4.84346 × 10⁻⁴, 2.02815 × 10⁻³)
0.6	(3.76983 × 10⁻³, 1.31603 × 10⁻²)	(4.59428 × 10⁻⁴, 1.43648 × 10⁻³)
0.7	(1.82537 × 10⁻³, 7.15873 × 10⁻³)	(3.50314 × 10⁻⁴, 8.75759 × 10⁻⁴)
0.8	(8.88641 × 10⁻⁴, 2.65294 × 10⁻³)	(2.01529 × 10⁻⁴, 4.12923 × 10⁻⁴)
0.9	(4.12378 × 10⁻³, 2.38319 × 10⁻⁵)	(7.16129 × 10⁻⁵, 1.04535 × 10⁻⁴)
1.0	(7.64147 × 10⁻³, 6.32640 × 10⁻⁴)	(2.97710 × 10⁻⁵, 8.93103 × 10⁻⁶)

Absolute errors of Application 1 for ψ2(x)_

x	Method in [31] (m=10)	Suggested method	Suggested method	Suggested method
	\|ψ_Exact − ψ_Taylor\|	m=1	m=2	m=3
0.1	2.229 × 10⁻³	2.17774 × 10⁻¹	6.77644 × 10⁻²	0
0.2	2.271 × 10⁻²	1.12737 × 10⁻¹	3.64407 × 10⁻²	0
0.3	1.835 × 10⁻²	5.73750 × 10⁻²	1.80167 × 10⁻²	0
0.4	7.450 × 10⁻⁴	2.69276 × 10⁻²	8.04810 × 10⁻³	0
0.5	2.317 × 10⁻²	1.03606 × 10⁻²	3.20315 × 10⁻³	0
0.6	3.458 × 10⁻²	2.43533 × 10⁻⁴	1.17962 × 10⁻³	0
0.7	9.332 × 10⁻³	9.73055 × 10⁻³	5.69291 × 10⁻⁴	0
0.8	1.111 × 10⁻¹	2.60431 × 10⁻²	6.88201 × 10⁻⁴	0
0.9	6.109 × 10⁻²	5.64250 × 10⁻²	1.39386 × 10⁻³	0
1.0	1.101 × 10⁻¹	1.10788 × 10⁻¹	2.91265 × 10⁻³	0

Absolute errors of Application 2 by B-spline wavelet method in [40] for (ψ1(x), ψ2(x))_

x	M=2	M=4
0.1	(2.22045E − 16, 1.04167E − 4)	(4.44089E − 16, 6.51042E − 6)
0.2	(0.00000000000, 1.14583E − 3)	(2.22045E − 16, 7.16146E − 5)
0.3	(2.22045E − 16, 1.14583E − 3)	(2.22045E − 16, 7.16146E − 5)
0.4	(2.22045E − 16, 1.04167E − 4)	(4.44089E − 16, 6.51042E − 6)
0.5	(4.44089E − 16, 2.60417E − 3)	(1.33227E − 16, 1.62760E − 4)
0.6	(2.22045E − 16, 1.04167E − 4)	(6.66134E − 16, 6.51042E − 6)
0.7	(6.66134E − 16, 1.14583E − 3)	(4.44089E − 16, 7.16146E − 5)
0.8	(4.44089E − 16, 1.14583E − 3)	(4.44089E − 16, 7.16146E − 5)
0.9	(4.44089E − 16, 1.04167E − 4)	(6.66134E − 16, 6.51042E − 6)
1.0	(6.66134E − 16, 2.60417E − 3)	(8.88178E − 16, 1.62760E − 4)

Absolute errors of Application 4 by Block-Pulse functions method in [30] for (ψ1(x), ψ2(x))_

x	M=16	M=32
0.1	(7.570 × 10⁻³, 5.960 × 10⁻³)	(1.124 × 10⁻², 8.260 × 10⁻³)
0.2	(2.212 × 10⁻², 1.481 × 10⁻²)	(3.560 × 10⁻³, 2.370 × 10⁻³)
0.3	(2.606 × 10⁻², 1.441 × 10⁻²)	(4.390 × 10⁻³, 2.700 × 10⁻³)
0.4	(8.320 × 10⁻³, 3.660 × 10⁻³)	(1.406 × 10⁻², 6.500 × 10⁻³)
0.5	(5.185 × 10⁻², 1.989 × 10⁻²)	(2.572 × 10⁻², 9.680 × 10⁻³)
0.6	(1.221 × 10⁻², 4.020 × 10⁻³)	(1.699 × 10⁻², 4.950 × 10⁻³)
0.7	(3.724 × 10⁻², 8.690 × 10⁻³)	(6.060 × 10⁻³, 1.390 × 10⁻³)
0.8	(4.174 × 10⁻², 9.480 × 10⁻³)	(6.440 × 10⁻³, 7.800 × 10⁻⁴)
0.9	(1.499 × 10⁻², 1.670 × 10⁻³)	(2.309 × 10⁻², 4.130 × 10⁻³)
1.0	(8.308 × 10⁻², 1.299 × 10⁻²)	(4.217 × 10⁻², 6.130 × 10⁻³)

Absolute errors of Application 3 for ψ1(x)_

x	Method in [30]	Method in [30]	Suggested method	Suggested method
	m=16	m=32	m=1	m=2
0.1	8.698 × 10⁻²	1.980 × 10⁻³	2.38200 × 10⁻¹	0
0.2	1.898 × 10⁻²	3.450 × 10⁻³	1.91404 × 10⁻¹	0
0.3	1.855 × 10⁻²	8.540 × 10⁻³	1.49116 × 10⁻¹	0
0.4	5.420 × 10⁻³	1.210 × 10⁻²	1.11397 × 10⁻¹	0
0.5	3.123 × 10⁻²	1.386 × 10⁻²	7.83878 × 10⁻²	0
0.6	8.538 × 10⁻²	6.410 × 10⁻³	5.03014 × 10⁻²	0
0.7	1.876 × 10⁻²	9.140 × 10⁻³	2.74219 × 10⁻²	0
0.8	1.877 × 10⁻²	1.314 × 10⁻²	1.01035 × 10⁻²	0
0.9	6.150 × 10⁻³	1.512 × 10⁻²	1.22547 × 10⁻³	0
1.0	3.137 × 10⁻²	6.860 × 10⁻³	6.05633 × 10⁻³	0

Absolute errors of Application 1 by B-spline wavelet method in [40] for (ψ1(x), ψ2(x))_

x	M=2	M=4
0.1	(1.05947E − 4, 3.16487E − 4)	(6.51740E − 6, 5.47084E − 6)
0.2	(1.14356E − 3, 1.75687E − 3)	(7.16057E − 5, 1.16058E − 4)
0.3	(1.14313E − 3, 2.26496E − 3)	(7.16040E − 5, 1.34611E − 4)
0.4	(1.07237E − 4, 3.70402E − 5)	(6.52242E − 6, 1.72473E − 5)
0.5	(2.60757E − 3, 6.50994E − 3)	(1.62774E − 4, 4.06899E − 4)
0.6	(1.07887E − 4, 4.83014E − 4)	(6.52497E − 6, 1.53021E − 5)
0.7	(1.14179E − 3, 3.46452E − 3)	(7.15988E − 5, 2.23463E − 4)
0.8	(1.14144E − 3, 3.97184E − 3)	(7.15974E − 5, 2.42013E − 4)
0.9	(1.08955E − 4, 2.05924E − 4)	(6.52915E − 6, 2.70877E − 5)
1.0	(2.60940E − 3, 1.01976E − 2)	(1.62781E − 4, 6.47543E − 4)

Absolute errors of Application 1 by BPM and HBBPFM in [41] for (ψ1(x), ψ2(x))_

x	BPM	HBBPFM	HBBPFM
x	n = 10	n=6, M=25	n=7, M=14
0.1	(3.47 × 10⁻⁴, 9.66 × 10⁻⁴)	(1.34 × 10⁻⁴, 1.28 × 10⁻⁴)	(9.88 × 10⁻⁶, 5.60 × 10⁻⁶)
0.2	(4.78 × 10⁻⁴, 8.25 × 10⁻⁴)	(2.77 × 10⁻⁴, 2.95 × 10⁻⁴)	(7.63 × 10⁻⁶, 9.90 × 10⁻⁶)
0.3	(5.28 × 10⁻⁴, 6.35 × 10⁻⁴)	(4.13 × 10⁻⁴, 4.88 × 10⁻⁴)	(3.65 × 10⁻⁶, 9.40 × 10⁻⁶)
0.4	(7.92 × 10⁻⁴, 4.56 × 10⁻⁴)	(6.75 × 10⁻⁵, 9.67 × 10⁻⁵)	(1.33 × 10⁻⁶, 3.50 × 10⁻⁶)
0.5	(3.68 × 10⁻⁴, 1.96 × 10⁻⁴)	(6.03 × 10⁻⁵, 7.36 × 10⁻⁵)	(6.94 × 10⁻⁶, 7.00 × 10⁻⁷)
0.6	(4.73 × 10⁻⁴, 9.25 × 10⁻⁴)	(2.04 × 10⁻⁴, 1.66 × 10⁻⁴)	(3.77 × 10⁻⁶, 1.00 × 10⁻⁶)
0.7	(3.11 × 10⁻⁵, 7.41 × 10⁻⁴)	(9.25 × 10⁻⁵, 3.55 × 10⁻⁴)	(6.46 × 10⁻⁶, 3.70 × 10⁻⁶)
0.8	(9.43 × 10⁻⁵, 3.67 × 10⁻⁴)	(6.04 × 10⁻⁵, 3.56 × 10⁻⁴)	(8.80 × 10⁻⁶, 8.00 × 10⁻⁷)
0.9	(7.36 × 10⁻⁴, 8.93 × 10⁻⁴)	(3.82 × 10⁻⁵, 2.00 × 10⁻⁴)	(6.14 × 10⁻⁶, 5.09 × 10⁻²)

Absolute errors of Application 2 for ψ1(x)_

x	Method in [26]	Suggested method	Suggested method
	ψ_Exact − ψ_ADM	m=1	m=2
0.1	1.33 × 10⁻²	2.40124 × 10⁻¹	0
0.2	1.52 × 10⁻²	2.18895 × 10⁻¹	0
0.3	1.71 × 10⁻²	1.90581 × 10⁻¹	0
0.4	1.89 × 10⁻²	1.57067 × 10⁻¹	0
0.5	2.08 × 10⁻²	1.20238 × 10⁻¹	0
0.6	2.26 × 10⁻²	8.19810 × 10⁻²	0
0.7	2.45 × 10⁻²	4.41810 × 10⁻²	0
0.8	2.64 × 10⁻²	8.72381 × 10⁻³	0
0.9	2.82 × 10⁻²	2.25048 × 10⁻²	0
1.0	3.02 × 10⁻¹	4.76190 × 10⁻²	0

Absolute errors of Application 3 for ψ2(x)_

x	Method in [30]	Method in [30]	Suggested method	Suggested method
	m=16	m=32	m=1	m=2
0.1	5.600 × 10⁻⁴	2.900 × 10⁻⁴	6.62637 × 10⁻³	0
0.2	2.810 × 10⁻⁴	1.720 × 10⁻³	6.70369 × 10⁻³	0
0.3	8.230 × 10⁻³	1.340 × 10⁻³	5.58039 × 10⁻³	0
0.4	1.123 × 10⁻²	8.120 × 10⁻³	3.55018 × 10⁻⁴	0
0.5	3.136 × 10⁻²	1.409 × 10⁻²	3.43149 × 10⁻³	0
0.6	3.171 × 10⁻²	2.405 × 10⁻²	4.53355 × 10⁻³	0
0.7	2.701 × 10⁻²	9.000 × 10⁻⁵	2.99261 × 10⁻³	0
0.8	2.666 × 10⁻²	1.795 × 10⁻²	1.48519 × 10⁻³	0
0.9	9.855 × 10⁻³	4.490 × 10⁻³	8.86505 × 10⁻³	0
1.0	2.900 × 10⁻²	2.900 × 10⁻⁴	1.88099 × 10⁻²	0

Absolute errors of Application 4 for ψ2(x)_

x	m=1	m=2	m=3	m=4
0.1	5.08674 × 10⁻²	9.01505 × 10⁻²	1.27633 × 10⁻³	6.45456 × 10⁻³
0.2	5.50850 × 10⁻²	6.80369 × 10⁻²	1.68664 × 10⁻⁴	2.04838 × 10⁻³
0.3	5.35456 × 10⁻²	4.97035 × 10⁻²	4.25853 × 10⁻⁴	1.32564 × 10⁻³
0.4	4.75126 × 10⁻²	3.47864 × 10⁻²	6.50643 × 10⁻⁴	8.12332 × 10⁻⁴
0.5	3.80803 × 10⁻²	2.29547 × 10⁻²	6.50718 × 10⁻⁴	4.62022 × 10⁻⁴
0.6	2.61641 × 10⁻²	1.39002 × 10⁻²	5.13929 × 10⁻⁴	2.36744 × 10⁻⁴
0.7	1.24855 × 10⁻²	7.32388 × 10⁻³	2.9481 × 10⁻⁴	1.03630 × 10⁻⁴
0.8	2.45176 × 10⁻³	2.92136 × 10⁻³	1.59256 × 10⁻⁴	3.45574 × 10⁻⁵
0.9	1.84003 × 10⁻²	3.64138 × 10⁻⁴	4.44145 × 10⁻⁵	6.15417 × 10⁻⁶
1.0	3.54318 × 10⁻²	7.23693 × 10⁻⁴	6.92641 × 10⁻⁶	9.48343 × 10⁻⁸

Absolute errors of Application 5 by Taylor-series expansion method in [31] with m = 10_

x	ψ₁(x)	ψ₂(x)
0.1	3.809 × 10⁻²	6.406 × 10⁻³
0.2	2.748 × 10⁻²	4.259 × 10⁻³
0.3	1.745 × 10⁻²	4.240 × 10⁻⁴
0.4	8.999 × 10⁻³	2.835 × 10⁻³
0.5	2.914 × 10⁻³	3.644 × 10⁻³
0.6	3.770 × 10⁻⁴	5.120 × 10⁻⁴
0.7	9.700 × 10⁻⁴	7.755 × 10⁻³
0.8	3.290 × 10⁻⁴	2.216 × 10⁻²
0.9	1.818 × 10⁻³	4.367 × 10⁻²
1.0	7.100 × 10⁻⁴	7.329 × 10⁻²

Absolute errors of Application 4 by rationalized Haar functions method in [28] for (ψ1(x), ψ2(x))_

x	M=16	M=32
0.1	(7.970 × 10⁻³, 5.950 × 10⁻³)	(1.014 × 10⁻², 8.370 × 10⁻³)
0.2	(2.203 × 10⁻², 1.490 × 10⁻²)	(3.550 × 10⁻³, 2.480 × 10⁻³)
0.3	(2.615 × 10⁻², 1.436 × 10⁻²)	(4.480 × 10⁻³, 2.410 × 10⁻³)
0.4	(8.310 × 10⁻³, 3.790 × 10⁻³)	(1.418 × 10⁻², 6.410 × 10⁻³)
0.5	(5.175 × 10⁻², 1.968 × 10⁻²)	(2.582 × 10⁻², 9.660 × 10⁻³)
0.6	(1.231 × 10⁻², 3.950 × 10⁻³)	(1.693 × 10⁻², 4.990 × 10⁻³)
0.7	(3.724 × 10⁻², 8.610 × 10⁻³)	(6.070 × 10⁻³, 1.410 × 10⁻³)
0.8	(4.214 × 10⁻², 9.200 × 10⁻²)	(7.140 × 10⁻³, 1.590 × 10⁻³)
0.9	(1.478 × 10⁻³, 1.690 × 10⁻³)	(2.492 × 10⁻², 4.030 × 10⁻³)
1.0	(8.418 × 10⁻², 1.261 × 10⁻²)	(4.227 × 10⁻², 6.030 × 10⁻³)