数值方法-课后答案-第9章

第九章常微分方程file:///F|/khdaw/大三/0909152348a84ab552cc3c549f/数值方法+第2版+课后习题完整解答/ch9.htm[2009-9-1613:38:20]                  第九章 常微分方程习题9-11. 就初值问题 分别导出显式欧拉法和改进欧拉法的近似表达式，并与精确解  相比较。 2．用梯形法解初值问题第九章常微分方程file:///F|/khdaw/大三/0909152348a84ab552cc3c549f/数值方法+第2版+课后习题完整解答/ch9.htm[2009-9-1613:38:20] 解：精确解： 第九章常微分方程file:///F|/khdaw/大三/0909152348a84ab552cc3c549f/数值方法+第2版+课后习题完整解答/ch9.htm[2009-9-1613:38:20]x00.20.40.60.81.0y=2ex-x-111.2428061.5836502.0442382.6510823.436564 显式欧拉法：    yi+1=yi+hf(xi, yi)     h=0.2nxnynfn= xn+ ynh fnyn+1001.01.00.21.210.21.21.40.281.4820.41.481.880.3761.85630.61.8562.4560.49122.347240.82.34723.14720.629442.9766451.02.97664    改进欧拉法（单步的预测校正系统） xiyixi + yixi+1yi+101.01.00.21.20.241.241.240.21.241.440.41.5280.33681.57681.57680.41.57681.97680.61.972160.454902.0316962.0316960.62.0316962.6316960.82.5580350.5989732.6306692.6306690.82.6306693.4306691.03.3168030.7747473.4054163.405416第九章常微分方程file:///F|/khdaw/大三/0909152348a84ab552cc3c549f/数值方法+第2版+课后习题完整解答/ch9.htm[2009-9-1613:38:20]1.03.405416       四阶龙格一库塔法   i xi yi j   插 点 kj=hfk1+2k2+2k3+k4(k1+2k2+2k3+k4)/6  yk+1 x y0    0    1  123400.10.10.21.01.11.121.2440.20.240.2440.2888  1.4568   0.2428   1.2428 1 0.2 1.2428 120.20.31.24281.38710.28860.3374      第九章常微分方程file:///F|/khdaw/大三/0909152348a84ab552cc3c549f/数值方法+第2版+课后习题完整解答/ch9.htm[2009-9-1613:38:20]      340.30.41.41151.58510.34230.39702.0450 0.3408 1.5836 2   0.4   1.5836   12340.40.50.50.61.58361.78201.81182.04600.39670.45640.46240.5292  2.7635   0.4606   2.0442 3   0.6   2.0442   12340.60.70.70.82.04422.30862.34512.65320.52880.60170.60900.6906  3.6410   0.6068   2.6510 4   0.8   2.6510   12340.80.90.91.02.65102.99613.04073.43920.69020.77920.78810.8878  4.7128   0.7855   3.4365  4.  证明下列龙格-库塔法是三阶的：第九章常微分方程file:///F|/khdaw/大三/0909152348a84ab552cc3c549f/数值方法+第2版+课后习题完整解答/ch9.htm[2009-9-1613:38:20]第九章常微分方程file:///F|/khdaw/大三/0909152348a84ab552cc3c549f/数值方法+第2版+课后习题完整解答/ch9.htm[2009-9-1613:38:20] 5．用梯形公式计算时，需用迭代法求解yi+1，问当步长h满足什么条件时迭代收敛？证明你的结论。 第九章常微分方程file:///F|/khdaw/大三/0909152348a84ab552cc3c549f/数值方法+第2版+课后习题完整解答/ch9.htm[2009-9-1613:38:20]6．试画出梯形法的绝对稳定区域。习题9-2第九章常微分方程file:///F|/khdaw/大三/0909152348a84ab552cc3c549f/数值方法+第2版+课后习题完整解答/ch9.htm[2009-9-1613:38:20]  第九章常微分方程file:///F|/khdaw/大三/0909152348a84ab552cc3c549f/数值方法+第2版+课后习题完整解答/ch9.htm[2009-9-1613:38:20]第九章常微分方程file:///F|/khdaw/大三/0909152348a84ab552cc3c549f/数值方法+第2版+课后习题完整解答/ch9.htm[2009-9-1613:38:20]称为米尔尼—哈明系统。