Numerical solution of quadratic Riccati differential equations

Figures & data

Table 1 Rate of convergence for some model examples with different mesh sizes.

	N	10	40	70	100	200	400
Rate of convergence	Example 2	3.9522	3.9851	3.9915	3.9941	3.9972	3.9935
	Example 3	3.9584	3.9904	3.9946	3.9964	4.0010	4.0581
	Example 4	4.1313	4.0350	4.0202	4.0144	4.0076	3.9293

Table 2 Absolute error for example 1 (mesh size $h=0.1$ OR N = 10).

t	Method [3]	The present method
0.0	0.0000000000000	0
0.1	0.000233600365141	0
0.3	0.00045422294912	2.2204e-16
0.5	9.375e-11	2.2204e-16
0.7	0.00045422275331	2.2204e-16
0.9	0.00023360043610	4.4409e-16
1.0	0.0000000000000	8.8818e-16

Table 3 Absolute error for example 2.

t	N = 10	N = 40	N = 70	N = 100	N = 200	N = 400
0.1	2.2551e-06	9.8491e-09	1.0669e-09	2.8533e-10	1.6233e-11	1.0184e-12
0.2	4.7763e-06	2.0641e-08	2.2327e-09	5.3915e-10	3.3923e-11	2.1275e-12
0.3	7.3083e-06	3.1235e-08	3.3731e-09	8.1402e-10	5.1180e-11	3.2087e-12
0.4	9.5635e-06	4.0441e-08	4.3607e-09	1.0517e-09	6.6078e-11	4.1415e-12
0.5	1.1301e-05	4.7374e-08	5.1021e-09	1.2299e-09	7.7230e-11	4.8390e-12
0.6	1.2408e-05	5.1724e-08	5.5661e-09	1.3414e-09	8.4199e-11	5.2707e-12
0.7	1.2940e-05	5.3815e-08	5.7892e-09	1.3949e-09	8.7546e-11	5.4756e-12
0.8	1.3100e-05	5.4419e-08	5.8528e-09	1.4101e-09	8.8489e-11	5.5316e-12
0.9	1.3141e-05	5.4381e-08	5.8450e-09	1.4079e-09	8.8322e-11	5.5178e-12
1	1.3245e-05	5.4260e-08	5.8236e-09	1.4019e-09	8.7889e-11	5.5029e-12

Table 4 Absolute errors for example 3.

t	N = 10	N = 40	N = 70	N = 100	N = 200	N = 400
0.1	1.1153e-07	4.5427e-10	4.8711e-11	1.1722e-11	7.3475e-13	4.5963e-14
0.2	2.6297e-07	1.0710e-09	1.1484e-10	2.7633e-11	1.7317e-12	1.0880e-13
0.3	4.6838e-07	1.9073e-09	2.0451e-10	4.9211e-11	3.0835e-12	1.9384e-13
0.4	7.4674e-07	3.0404e-09	3.2600e-10	7.8447e-11	4.9163e-12	3.0909e-13
0.5	1.1237e-06	4.5748e-09	4.9051e-10	1.1803e-10	7.3965e-12	4.6496e-13
0.6	1.6338e-06	6.6511e-09	7.1312e-10	1.7160e-10	1.0753e-11	6.7191e-13
0.7	2.3239e-06	9.4596e-09	1.0142e-09	2.4406e-10	1.5293e-11	9.5124e-13
0.8	3.2569e-06	1.3257e-08	1.4214e-09	3.4203e-10	2.1432e-11	1.3305e-12
0.9	4.5182e-06	1.8390e-08	1.9717e-09	4.7445e-10	2.9730e-11	1.8439e-12
1	6.2225e-06	2.5327e-08	2.7154e-09	6.5340e-10	4.0941e-11	2.5673e-12

Table 5 Absolute errors for example 4.

t	N = 10	N = 40	N = 70	N = 100	N = 200	N = 400
0.1	3.8296e-07	1.2712e-09	1.3226e-10	3.1445e-11	1.9426e-12	1.2057e-13
0.2	5.7951e-07	1.9396e-09	2.0206e-10	4.8062e-11	2.9710e-12	1.8452e-13
0.3	6.8133e-07	2.2939e-09	2.3918e-10	5.6914e-11	3.5196e-12	2.1860e-13
0.4	7.3394e-07	2.4816e-09	2.5893e-10	6.1630e-11	3.8125e-12	1.8452e-13
0.5	7.6091e-07	2.5808e-09	2.6941e-10	6.4137e-11	3.9686e-12	2.4647e-13
0.6	7.7483e-07	2.6340e-09	2.7506e-10	6.5490e-11	4.0530e-12	2.5280e-13
0.7	7.8257e-07	2.6648e-09	2.7834e-10	6.6278e-11	4.1022e-12	2.5668e-13
0.8	7.8799e-07	2.6865e-09	2.8066e-10	6.6837e-11	4.1374e-12	2.5946e-13
0.9	7.9326e-07	2.7069e-09	2.8284e-10	6.7358e-11	4.1697e-12	2.6190e-13
1	7.9961e-07	2.7304e-09	2.8533e-10	6.7954e-11	4.2070e-12	2.6240e-13

Fig. 1 The graph of numerical and exact solution of example 1 for N = 30.

Fig. 2 The graph of numerical and exact solution of example 2 for N = 30.

Fig. 3 The graph of numerical and exact solution of example 3 for N = 30.

Fig. 4 The graph of numerical and solution exact solution of example 4 for N = 30.