北航研究生数值分析上机作业 四(报告+所有程序大全)

数值分析上机作业4——数值方法求解常微

分方程及数值积分

1. 算法设计

i. 由,i i t y ，(,)t y ϕ插值求解q

ii. 将q ，y 带入Newton 迭代法求解非线性方程组 iii. 由,,,,x y p q z 得到(,)i i f t y iv.

计算1k

同理以上过程再进行3次得到234,,k k k ，由经典R-K 方法公式：

112341213243(22)

6

(,)

(0.5,0.5)(0.5,0.5)(,)

n n n n n n n n n n h

y y k k k k k f t y k f t h y hk k f t h y hk k f t h y hk +=++++==++=++=++

可得,0,1,...50i y i =。

再由复化Simpson 积分公式：1

21211

()[()()4()2()]3m m b

i i a i i h

f x dx f a f b f x f x --==≈+++∑∑⎰可得积分值I 。

2. 打印输出结果

5位有效数字格式：

t(i) y(i) x(i) z(i) p(i) q(i)

0.0 2.0000e+000 9.1445e-001 1.6333e+000 2.8181e+000 8.6000e+000 0.2 1.9533e+000 8.9816e-001 1.6605e+000 2.8477e+000 8.5396e+000 0.4 1.9108e+000 8.8411e-001 1.6844e+000 2.8765e+000 8.4951e+000 0.6 1.8722e+000 8.7209e-001 1.7048e+000 2.9043e+000 8.4660e+000 0.8 1.8373e+000 8.6194e-001 1.7216e+000 2.9307e+000 8.4516e+000 1.0 1.8056e+000 8.5353e-001 1.7347e+000 2.9558e+000 8.4516e+000 1.2 1.7770e+000 8.4675e-001 1.7440e+000 2.9792e+000 8.4656e+000

1.4 1.7513e+000 8.4151e-001 1.7495e+000 3.0008e+000 8.4932e+000

1.6 1.7283e+000 8.3772e-001 1.7511e+000 3.0205e+000 8.5342e+000

1.8 1.7078e+000 8.3533e-001 1.7489e+000 3.0382e+000 8.5883e+000

2.0 1.6897e+000 8.3432e-001 1.7430e+000

3.0537e+000 8.6552e+000

2.2 1.6739e+000 8.3465e-001 1.7334e+000

3.0669e+000 8.7349e+000

2.4 1.6603e+000 8.3632e-001 1.7203e+000

3.0779e+000 8.8272e+000

2.6 1.6489e+000 8.3934e-001 1.7039e+000

3.0866e+000 8.9320e+000

2.8 1.6395e+000 8.4374e-001 1.6844e+000

3.0929e+000 9.0491e+000

3.0 1.6320e+000 8.6317e-001 1.6204e+000 3.0924e+000 9.3945e+000

3.2 1.6263e+000 8.6990e-001 1.5984e+000 3.0947e+000 9.5260e+000

3.4 1.6226e+000 8.7709e-001 1.5769e+000 3.0952e+000 9.6532e+000

3.6 1.6210e+000 8.8477e-001 1.5559e+000 3.0940e+000 9.7760e+000

3.8 1.6216e+000 8.9294e-001 1.5354e+000 3.0910e+000 9.8945e+000

4.0 1.6244e+000 9.0160e-001 1.5153e+000 3.0865e+000 1.0008e+001

4.2 1.6294e+000 9.1072e-001 1.4958e+000 3.0804e+000 1.0117e+001

4.4 1.6366e+000 9.2027e-001 1.4770e+000 3.0728e+000 1.0219e+001

4.6 1.6462e+000 9.3020e-001 1.4590e+000 3.0639e+000 1.0315e+001

4.8 1.6581e+000 9.4045e-001 1.4418e+000 3.0537e+000 1.0403e+001

5.0 1.6724e+000 9.0334e-001 1.5304e+000 3.0511e+000 9.8131e+000

5.2 1.6891e+000 9.1466e-001 1.5099e+000 3.0369e+000 9.9016e+000

5.4 1.7082e+000 9.2998e-001 1.4823e+000 3.0211e+000 1.0034e+001

5.6 1.7299e+000 9.4964e-001 1.4481e+000 3.0042e+000 1.0212e+001

5.8 1.7544e+000 9.7407e-001 1.4082e+000 2.9865e+000 1.0438e+001

6.0 1.7817e+000 1.0038e+000 1.3632e+000 2.9686e+000 1.0714e+001

6.2 1.8122e+000 1.0395e+000 1.3140e+000 2.9510e+000 1.1043e+001

6.4 1.8462e+000 1.0817e+000 1.2615e+000 2.9342e+000 1.1428e+001

6.6 1.8840e+000 1.1312e+000 1.2068e+000 2.9186e+000 1.1871e+001

6.8 1.9261e+000 1.1889e+000 1.1507e+000 2.9049e+000 1.2375e+001

7.0 1.9732e+000 1.3243e+000 1.0353e+000 2.9047e+000 1.3693e+001

7.2 2.0274e+000 1.4003e+000 9.8689e-001 2.8963e+000 1.4300e+001

7.4 2.0875e+000 1.4798e+000 9.4271e-001 2.8890e+000 1.4907e+001

7.6 2.1536e+000 1.5626e+000 9.0237e-001 2.8826e+000 1.5514e+001

7.8 2.2262e+000 1.6485e+000 8.6549e-001 2.8770e+000 1.6118e+001

8.0 2.3055e+000 1.7373e+000 8.3174e-001 2.8720e+000 1.6717e+001

8.2 2.3917e+000 1.8286e+000 8.0082e-001 2.8674e+000 1.7309e+001

8.4 2.4850e+000 1.9224e+000 7.7242e-001 2.8631e+000 1.7892e+001

8.6 2.5853e+000 2.0103e+000 7.4900e-001 2.8571e+000 1.8391e+001

8.8 2.6930e+000 2.1088e+000 7.2455e-001 2.8532e+000 1.8959e+001

9.0 2.8084e+000 2.2125e+000 7.0107e-001 2.8501e+000 1.9546e+001

9.2 2.9316e+000 2.3222e+000 6.7835e-001 2.8480e+000 2.0157e+001

9.4 3.0628e+000 2.4387e+000 6.5623e-001 2.8470e+000 2.0798e+001

9.6 3.2024e+000 2.5627e+000 6.3465e-001 2.8472e+000 2.1474e+001

9.8 3.3505e+000 2.6945e+000 6.1362e-001 2.8486e+000 2.2188e+001

10.0 3.5072e+000 0.0000e+000 0.0000e+000 0.0000e+000 0.0000e+000