Solving system of integro differential equations using discrete adomian decomposition method

Figures & data

Figure 1. Curves of the exact solution $u\left(x\right)$ and the approximate solution using DADM based on the Trapezoidal rule.

Table 1. The comparison between exact solution $u\left(x\right)$ and the approximate solution using DADM based on the Trapezoidal rule.

x	Exact	DADM	Absolute Error
0	1.00000000	1.00000000	0
0.20	1.42140276	1.42140311	3.51966153e−07
0.40	1.89182470	1.89183726	1.25586920e−05
0.60	2.42211880	2.42222676	1.07957623e−04
0.80	3.02554093	3.02606413	5.23203886e−04
1	3.71828183	3.72014498	1.86315626e−03

Figure 2. Curves of the exact solution $v\left(x\right)$ and the approximate solution using DADM based on the Trapezoidal rule.

Table 2. The comparison between exact solution $v\left(x\right)$ and the approximate solution using DADM based on the Trapezoidal rule.

x	Exact	DADM	Absolute Error
0	−1.00000000	−1.00000000	0
0.20	−1.02140276	−1.02140264	1.17998654e−07
0.40	−1.09182470	−1.09182052	4.17977601e−06
0.60	−1.22211880	−1.22208370	3.50985477e−05
0.80	−1.42554093	−1.42537777	1.63155502e−04
1	−1.71828183	−1.71773462	5.47210550e−04

Figure 3. Curves of the exact solution $u\left(x\right)$ and the approximate solution using DADM based on the Simpson's rule.

Table 3. The comparison between the exact solution $u\left(x\right)$ and the approximate solution using DADM based on the Simpson's rule.

x	Exact	DADM	Absolute Error
0	1.00000000	1.00000000	0
0.10	1.20517092	1.20517092	1.42545776e−15
0.20	1.42140276	1.42140276	1.86990969e−13
0.30	1.64985881	1.64985881	2.00290725e−12
0.40	1.89182470	1.89182470	3.23009992e−11
0.50	2.14872127	2.14872127	9.09490688e−10

Figure 4. Curves of the exact solution $v\left(x\right)$ and the approximate solution using DADM based on the Simpson's rule.

Table 4. The comparison between the exact solution $v\left(x\right)$ and the approximate using DADM based on the Simpson's rule.

x	Exact	DADM	Absolute Error
0	−1.00000000	−1.00000000	0
0.10	−1.00517092	−1.00517092	3.46668166e−16
0.20	−1.02140276	−1.02140276	4.65531657e−14
0.30	−1.04985881	−1.04985881	8.38358300e−13
0.40	−1.09182470	−1.09182470	6.79761258e−12
0.50	−1.14872127	−1.14872127	3.83945477e−11

Figure 5. Curves of the exact solution $u\left(x\right)$ and the approximate solution using DADM based on the Trapezoidal rule.

Table 5. The comparison between the exact solution $u\left(x\right)$ and the approximate solution using DADM based on the Trapezoidal rule.

x	Exact	DADM	Absolute Error
0	1.00000000	1.00000000	0
0.20	1.04000000	1.03996370	3.62969667e−05
0.40	1.16000000	1.15970962	2.90375734e−04
0.60	1.36000000	1.35901998	9.80018101e−04
0.80	1.64000000	1.63767699	2.32300587e−03
1	2.00000000	1.99546288	4.53712084e−03

Figure 6. Curves of the exact solution $v\left(x\right)$ and the approximate solution using DADM based on the Trapezoidal rule.

Table 6. The comparison between the exact solution $v\left(x\right)$ and the approximate solution using DADM based on the Trapezoidal rule.

x	Exact	DADM	Absolute Error
0	1.00000000	1.00000000	0
0.20	0.96000000	0.95996436	3.56392238e−05
0.40	0.84000000	0.83971489	2.85113790e−04
0.60	0.64000000	0.63903774	9.62259042e−04
0.80	0.36000000	0.35771909	2.28091032e−03
1	0	−0.00445490	4.45490297e−03

Figure 7. Curves of the exact solution $u\left(x\right)$ and the approximate solution using DADM based on the Simpson's rule.

Table 7. The comparison between the exact solution $u\left(x\right)$ and the approximate solution using DADM based on the Simpson's rule.

x	Exact	DADM	Absolute Error
0	1.00000000	1.00000000	0
0.10	1.01000000	1.00999945	5.47032873e−07
0.20	1.04000000	1.03999562	4.37626298e−06
0.30	1.09000000	1.08998523	1.47698876e−05
0.40	1.16000000	1.15996499	3.50101039e−05
0.50	1.25000000	1.24993162	6.83791091e−05

Figure 8. Curves of the exact solution $v\left(x\right)$ and the approximate solution using DADM based on the Simpson's rule.

Table 8. The comparison between the exact solution $v\left(x\right)$ and the approximate solution using DADM based on the Simpson's rule.

x	Exact	DADM	Absolute Error
0	1.00000000	1.00000000	0
0.1	0.99000000	0.98999967	3.28891242e−07
0.20	0.96000000	0.95999737	2.63112993e−06
0.30	0.91000000	0.90999112	8.88006352e−06
0.40	0.84000000	0.83997895	2.10490395e−05
0.50	0.75000000	0.74995889	4.11114052e−05