Numerical solutions to systems of fractional Voltera Integro differential equations, using Chebyshev wavelet method

Figures & data

Table 1. The numerical results of Example 5.1.

$x_i$	$y_1$	$y_1$ (CWM)	$y_2$	$y_2$ (CWM)	Error $y_1$	Error $y_2$
0.0	0.000000	0.00000000000000000001	0.000	0.00000000000000000000	8.86E−21	5.22E−22
0.1	0.105000	0.10500000000000017529	0.095	0.09500000000000009781	1.75E−16	9.78E−17
0.2	0.220000	0.22000000000000017686	0.180	0.18000000000000009876	1.76E−16	9.87E−17
0.3	0.345000	0.34500000000000017641	0.255	0.25500000000000010000	1.76E−16	1.00E−16
0.4	0.480000	0.48000000000000017144	0.320	0.32000000000000010673	1.71E−16	1.06E−16
0.5	0.625000	0.62500000000000014947	0.375	0.37500000000000013167	1.49E−16	1.31E−16
0.6	0.780000	0.78000000000000008278	0.420	0.42000000000000020303	8.28E−17	2.03E−16
0.7	0.945000	0.94499999999999991864	0.455	0.45500000000000037359	8.12E−17	3.73E−16
0.8	1.120000	1.11999999999999956790	0.480	0.48000000000000073240	4.32E−16	7.32E−16
0.9	1.305000	1.30499999999999889160	0.495	0.49500000000000141871	1.10E−15	1.41E−15
1.0	1.500000	1.49999999999999769060	0.500	0.50000000000000264463	2.30E−15	2.64E−15

Table 2. Numerical results of Example 5.1, for different fractional order $0<\alpha \le 1$.

$x_i$	$y_1\left(0.75\right)$	$y_2\left(0.75\right)$	$y_1\left(0.99\right)$	$y_2$ (0.99)	$y_1$ (0.99999)	$y_2$ (0.99999)
0.0	0.000000000	0.0000000	0.000000	0.000000	0.00000000	0.00000000
0.1	0.176199660	0.15340263	0.106888	0.102015	0.10500185	0.09500781
0.2	0.339855138	0.25283204	0.223632	0.258342	0.22000359	0.18008987
0.3	0.496712043	0.28935057	0.350566	0.592042	0.34500559	0.25538802
0.4	0.653952604	0.24973061	0.490544	1.245713	0.48001105	0.32106665
0.5	0.812194283	0.11645073	0.651549	2.355534	0.62502893	0.37728292
0.6	0.968110842	−0.1264186	0.851227	4.042822	0.78007915	0.42417707
0.7	1.113946717	−0.4917918	1.122739	6.413404	0.94519920	0.46187177
0.8	1.236233568	−0.9885288	1.522087	9.555548	1.12045253	0.49046985
0.9	1.314090673	−1.6188345	2.136990	13.52723	1.30593864	0.51003968
1.0	1.317517557	−2.3725543	3.096760	18.32412	1.50180445	0.52057835

Figure 1. The solution graph of Example 5.1, for $y_1\left(x\right)$ at different fractional order.

Figure 2. The solution graph of Example 5.1, for $y_2\left(x\right)$ at different fractional order.

Table 3. Errors of Example 5.1, for different fractional order $0<\alpha \le 1$.

$x_i$	Error $y_1\left(0.75\right)$	Error $y_2\left(0.75\right)$	Error $y_1\left(0.99\right)$	Error $y_2$(0.99)	Error $y_1$(0.99999)	Error $y_2$(0.99999)
0.0	1.34E−20	6.79E−21	3.30E−20	4.10E−19	2.14E−21	5.34E−21
0.1	7.11E−02	5.84E−02	1.88E−03	7.01E−03	1.85E−06	7.81E−06
0.2	1.19E−01	7.28E−02	3.63E−03	7.83E−02	3.59E−06	8.98E−05
0.3	1.51E−01	3.43E−02	5.56E−03	3.37E−01	5.59E−06	3.88E−04
0.4	1.73E−01	7.02E−02	1.05E−02	9.25E−01	1.10E−05	1.06E−03
0.5	1.87E−01	2.58E−01	2.65E−02	1.9805	2.89E−05	2.28E−03
0.6	1.88E−01	5.46E−01	7.12E−02	3.6228	7.91E−05	4.17E−03
0.7	1.68E−01	9.46E−01	1.77E−01	5.9584	1.99E−04	6.87E−03
0.8	1.16E−01	1.4685	4.02E−01	9.0755	4.52E−04	1.04E−02
0.9	9.09E−03	2.1138	8.31E−01	13.032	9.38E−04	1.50E−02
1.0	1.82E−01	2.8725	1.596760	17.824	1.80E−03	2.05E−02

Table 4. The numerical solutions of Example 5.2.

$x_i$	$y_1$ (exact)	$y_1$ (CWM)	$y_2$ (exact)	$y_2$ (CWM)	$y_3$ (exact)	$y_3$ (CWM)
0.0	0.0000	0.0000000000000	0.0000	0.000000000000000	0.0000	0.00000000000000000000
0.1	0.1000	0.0999999999999	0.0100	0.010000000000013	0.0010	0.00100000000003054812
0.2	0.2000	0.1999999999990	0.0400	0.040000000000029	0.0080	0.00800000000092897704
0.3	0.3000	0.2999999999864	0.0900	0.090000000000049	0.0270	0.02700000001354235333
0.4	0.4000	0.3999999999091	0.1600	0.160000000000118	0.0640	0.06400000009091300146
0.5	0.5000	0.4999999996089	0.2500	0.250000000000588	0.1250	0.12500000039130475950
0.6	0.6000	0.5999999987301	0.3600	0.360000000003600	0.2160	0.21600000127162549054
0.7	0.7000	0.6999999966051	0.4900	0.490000000022462	0.3430	0.34300000340941206265
0.8	0.8000	0.7999999921595	0.6400	0.640000000141051	0.5120	0.51200000795371335706
0.9	0.9000	0.8999999840488	0.8100	0.810000000839143	0.7290	0.72900001670629419916
1.0	1.0000	0.9999999719332	1.0000	1.000000004487292	1.0000	1.00000003232985799840

Table 5. The errors of Example 5.2 for $\alpha =2$.

	Error $y_1$	Error $y_2$	Error $y_3$
$x_i$	$\alpha =2$	$\alpha =2$	$\alpha =2$
0.0	1.85E−21	3.34E−21	2.91E−21
0.1	2.22E−14	1.30E−14	3.05E−14
0.2	9.10E−13	2.95E−14	9.28E−14
0.3	1.35E−11	4.95E−14	1.35E−11
0.4	9.08E−11	1.18E−13	9.09E−11
0.5	3.91E−10	5.88E−13	3.91E−10
0.6	1.26E−09	3.60E−12	1.27E−09
0.7	3.39E−09	2.24E−11	3.40E−09
0.8	7.84E−09	1.41E−10	7.95E−09
0.9	1.59E−08	8.39E−10	1.67E−08
1.0	2.80E−08	4.48E−09	3.23E−08

Table 6. Numerical results of Example 5.2 for different orders $1<\alpha \le 2$.

	$y_1$	$y_1$	$y_1$	$y_2$	$y_2$	$y_2$	$y_3$	$y_3$	$y_3$
$x_i$	$\alpha ={1.9}^4$	$\alpha ={1.9}^7$	$\alpha ={1.9}^{12}$	$\alpha ={1.9}^4$	$\alpha ={1.9}^7$	$\alpha ={1.9}^{12}$	$\alpha ={1.9}^4$	$\alpha ={1.9}^7$	$\alpha ={1.9}^{12}$
0.0	0.000000	0.000000	0.0000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000
0.1	0.099999	0.099999	0.0999999	0.008722	0.009998	0.009999	0.001000	0.001000	0.0010000
0.2	0.199989	0.199999	0.1999999	0.022193	0.039982	0.039999	0.008013	0.008000	0.0080000
0.3	0.299831	0.299999	0.2999999	0.011349	0.089921	0.089999	0.027175	0.027000	0.0270001
0.4	0.398859	0.399998	0.3999999	−0.05789	0.159782	0.159999	0.065154	0.064001	0.0640011
0.5	0.495066	0.499995	0.4999999	−0.21936	0.249532	0.249999	0.129958	0.125004	0.1250049
0.6	0.583904	0.599983	0.5999999	−0.50564	0.359137	0.359999	0.232116	0.216016	0.2160160
0.7	0.656706	0.699956	0.6999999	−0.94931	0.488565	0.489998	0.386155	0.343043	0.3430430
0.8	0.698732	0.799899	0.7999999	−1.58401	0.637783	0.639998	0.611985	0.512099	0.5120996
0.9	0.686822	0.899787	0.8999998	−2.44344	0.806757	0.809997	0.935064	0.729205	0.7292053
1.0	0.586575	0.999588	0.9999996	−3.55600	0.995458	0.999996	1.382024	1.000380	1.0003806

Figure 3. Solution graph of Example 5.2 for $y_1\left(x\right)$ at different fractional order.

Figure 4. Solution graph of Example 5.2 for $y_2\left(x\right)$ at different fractional order.

Figure 5. Solution graph of Example 5.2 for $y_3\left(x\right)$ at different fractional order.

Table 7. The error analysis of Example 5.2 for different order$\alpha ={1.9}^4,{1.9}^7,{1.9}^{12}$.

	Error $y_1$	Error $y_1$	Error $y_1$	Error $y_2$	Error $y_2$	Error $y_2$	Error $y_3$	Error $y_3$	Error $y_3$
$x_i$	$\alpha ={1.9}^4$	$\alpha ={1.9}^7$	$\alpha ={1.9}^{12}$	$\alpha ={1.9}^4$	$\alpha ={1.9}^7$	$\alpha ={1.9}^{12}$	$\alpha ={1.9}^4$	$\alpha ={1.9}^7$	$\alpha ={1.9}^{12}$
0.0	4.71E−21	5.84E−21	7.49E−21	7.41E−20	4.28E−21	1.84E−21	4.40E−21	3.11E−21	1.68E−21
0.1	9.47E−08	9.44E−11	8.37E−14	1.27E−03	1.27E−06	1.11E−09	4.65E−07	4.65E−11	8.44E−14
0.2	1.09E−05	1.08E−08	9.48E−12	1.78E−02	1.77E−05	1.54E−08	1.32E−05	1.32E−08	9.50E−12
0.3	1.68E−04	1.67E−07	1.46E−10	7.86E−02	7.83E−05	6.82E−08	1.75E−04	1.74E−07	9.89E−10
0.4	1.14E−03	1.13E−06	9.89E−10	2.17E−01	2.17E−04	1.89E−07	1.15E−03	1.15E−06	9.90E−10
0.5	4.93E−03	4.91E−06	4.28E−09	4.69E−01	4.67E−04	4.07E−07	4.95E−03	4.94E−06	4.28E−09
0.6	1.60E−02	1.60E−05	1.39E−08	8.65E−01	8.62E−04	7.51E−07	1.61E−02	1.60E−05	1.39E−08
0.7	4.32E−02	4.31E−05	3.75E−08	1.4393	1.43E−03	1.24E−06	4.31E−02	4.30E−05	3.73E−08
0.8	1.01E−01	1.00E−04	8.78E−08	2.2240	2.21E−03	1.93E−06	9.99E−02	9.96E−05	8.66E−08
0.9	2.31E−01	2.12E−04	1.84E−07	3.2534	3.24E−03	2.82E−06	2.06E−01	2.05E−04	1.78E−07
1.0	4.13E−01	4.11E−04	3.58E−07	4.5560	4.54E−03	3.95E−06	3.82E−01	3.80E−04	3.31E−07

Table 8. Numerical results for Example 5.3.

$x_i$	$y_1$ (exact)	$y_1$ (CWM)	Error $y_1$	$y_2$ (exact)	$y_2$(CWM)	Error $y_2$
0.0	1.000	0.99999999999999999996	4.0E−20	1.000	0.99999999999999999997	3.0E−20
0.1	1.110	1.11000000000000000000	0.0000	0.910	0.91000000000000000001	1.0E−20
0.2	1.240	1.24000000000000000000	0.0000	0.840	0.84000000000000000023	2.3E−19
0.3	1.390	1.39000000000000000010	1.0E−19	0.790	0.79000000000000000063	6.3E−19
0.4	1.560	1.56000000000000000020	2.0E−19	0.760	0.76000000000000000126	1.2E−18
0.5	1.750	1.75000000000000000030	3.0E−19	0.750	0.75000000000000000212	2.1E−18
0.6	1.960	1.96000000000000000050	5.0E−19	0.760	0.76000000000000000316	3.1E−18
0.7	2.190	2.19000000000000000070	7.0E−19	0.790	0.79000000000000000446	4.4E−18
0.8	2.440	2.44000000000000000100	1.0E−18	0.840	0.84000000000000000597	5.9E−18
0.9	2.710	2.71000000000000000140	1.4E−18	0.910	0.91000000000000000767	7.6E−18
1.0	3.000	3.00000000000000000170	1.7E−18	1.000	1.00000000000000000960	9.6E−18

Figure 6. The graph of $y_1\left(exact\right)$ and $y_1\left(CWM\right)$ of Example 5.3.

Figure 7. The solution graph of $y_2\left(exact\right)$ and $y_2\left(CWM\right)$.

Figure 8. The error graph of $Error{y}_1\left(x\right)$ and for other fractional orders.

Figure 9. The error graph of $y_2\left(x\right)$ for different fractional order $\alpha$, where $0\le \alpha \le 2$.

Table 9. The numerical results of Example 5.3, for fractional orders,$2<\alpha \le 3$_.

	Error $y_1$	Error $y_1$	Error $y_1$	Error$y_2$	$y_2$	$y_2$
$x_i$	$\alpha =2.25$	$\alpha =2.50$	$\alpha =2.75$	$\alpha =2.25$	$\alpha =2.50$	$\alpha =2.75$
0.0	3.00E−20	1.00E−19	0.00000	0.000000	1.00E−20	0.000000
0.1	0.00000	1.00E−19	0.0000	0.000000	2.00E−20	3.00E−20
0.2	1.00E−19	2.00E−19	0.0000	0.000000	1.80E−19	2.00E−19
0.3	2.00E−19	2.00E−19	1.00E−19	2.00E−19	4.30E−19	5.10E−19
0.4	3.00E−19	3.00E−19	1.00E−19	6.70E−19	7.40E−19	9.50E−19
0.5	5.00E−19	5.00E−19	2.00E−19	9.70E−19	1.15E−18	1.52E−18
0.6	6.00E−19	6.00E−19	5.00E−19	1.28E−18	1.59E−18	2.20E−18
0.7	8.00E−19	8.00E−19	7.00E−19	1.59E−18	2.09E−18	2.96E−18
0.8	1.10E−18	9.00E−19	1.00E−18	1.91E−18	2.61E−18	3.85E−18
0.9	1.10E−18	1.20E−18	1.40E−18	2.26E−18	3.17E−18	4.80E−18
1.0	1.40E−18	1.40E−18	1.30E−18	2.60E−18	3.80E−18	5.90E−18