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湖南工程学院微分方程数值解法实验报告专业班级姓名学号组别信息与计算科学1001邓鹤201010010215实验日期2013年5 月9日第 4次实验指导老师杨继明评分实验名称用差分格式求双曲型方程的边值问题实验目的熟悉掌握双曲型方程边值问题的差分格式并程序实现实验原理与步骤：利用差分格式求下面波动方程混合边值问题的解： (精确解)波动方程混合边值问题求解。数值格式为：数值求解流程(图)：输入时间步长、空间步长、网格数，给边界条件赋值，利用式(3.1)经过多次迭代，进行计算。利用Matlab计算，在窗口命令中输入fwave(10,0.1,10,0.05,0,2)，便可得到在该条件下的结果。采用Matlab程序设计语言编程实现该问题的数值求解。在Matlab命令窗口中输入命令：T=0.5时：fwave(10,0.1,10,0.05,0,2)T=1.0时：fwave(10,0.1,20,0.05,0,2)T=1.5时：fwave(10,0.1,30,0.05,0,2)T=2.0时：fwave(10,0.1,40,0.05,0,2)运行程序得到实验结果为：T=0.5时：fwave(10,0.1,10,0.05,0,2)ans = Columns 1 through 5 0 0.6180 1.1756 1.6180 1.9021 0 0.6180 1.1756 1.6180 1.9021 0 0.5575 1.0605 1.4596 1.7159 0 0.4425 0.8416 1.1584 1.3618 0 0.2841 0.5404 0.7437 0.8743 0 0.0979 0.1862 0.2563 0.3013 0 -0.0979 -0.1862 -0.2563 -0.3013 0 -0.2841 -0.5404 -0.7437 -0.8743 0 -0.4425 -0.8416 -1.1584 -1.3618 0 -0.5575 -1.0605 -1.4596 -1.7159 0 -0.6180 -1.1756 -1.6180 -1.9021 Columns 6 through 10 2.0000 1.9021 1.6180 1.1756 0.6180 2.0000 1.9021 1.6180 1.1756 0.6180 1.8042 1.7159 1.4596 1.0605 0.5575 1.4318 1.3618 1.1584 0.8416 0.4425 0.9193 0.8743 0.7437 0.5404 0.2841 0.3168 0.3013 0.2563 0.1862 0.0979 -0.3168 -0.3013 -0.2563 -0.1862 -0.0979 -0.9193 -0.8743 -0.7437 -0.5404 -0.2841 -1.4318 -1.3618 -1.1584 -0.8416 -0.4425 -1.8042 -1.7159 -1.4596 -1.0605 -0.5575 -2.0000 -1.9021 -1.6180 -1.1756 -0.6180 Column 11 0.0000 0.0000 -0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0 0T=1.0时：fwave(10,0.1,20,0.05,0,2)ans = Columns 1 through 5 0 0.6180 1.1756 1.6180 1.9021 0 0.6180 1.1756 1.6180 1.9021 0 0.5575 1.0605 1.4596 1.7159 0 0.4425 0.8416 1.1584 1.3618 0 0.2841 0.5404 0.7437 0.8743 0 0.0979 0.1862 0.2563 0.3013 0 -0.0979 -0.1862 -0.2563 -0.3013 0 -0.2841 -0.5404 -0.7437 -0.8743 0 -0.4425 -0.8416 -1.1584 -1.3618 0 -0.5575 -1.0605 -1.4596 -1.7159 0 -0.6180 -1.1756 -1.6180 -1.9021 0 -0.6180 -1.1756 -1.6180 -1.9021 0 -0.5575 -1.0605 -1.4596 -1.7159 0 -0.4425 -0.8416 -1.1584 -1.3618 0 -0.2841 -0.5404 -0.7437 -0.8743 0 -0.0979 -0.1862 -0.2563 -0.3013 0 0.0979 0.1862 0.2563 0.3013 0 0.2841 0.5404 0.7437 0.8743 0 0.4425 0.8416 1.1584 1.3618 0 0.5575 1.0605 1.4596 1.7159 0 0.6180 1.1756 1.6180 1.9021 Columns 6 through 10 2.0000 1.9021 1.6180 1.1756 0.6180 2.0000 1.9021 1.6180 1.1756 0.6180 1.8042 1.7159 1.4596 1.0605 0.5575 1.4318 1.3618 1.1584 0.8416 0.4425 0.9193 0.8743 0.7437 0.5404 0.2841 0.3168 0.3013 0.2563 0.1862 0.0979 -0.3168 -0.3013 -0.2563 -0.1862 -0.0979 -0.9193 -0.8743 -0.7437 -0.5404 -0.2841 -1.4318 -1.3618 -1.1584 -0.8416 -0.4425 -1.8042 -1.7159 -1.4596 -1.0605 -0.5575 -2.0000 -1.9021 -1.6180 -1.1756 -0.6180 -2.0000 -1.9021 -1.6180 -1.1756 -0.6180 -1.8042 -1.7159 -1.4596 -1.0605 -0.5575 -1.4318 -1.3618 -1.1584 -0.8416 -0.4425 -0.9193 -0.8743 -0.7437 -0.5404 -0.2841 -0.3168 -0.3013 -0.2563 -0.1862 -0.0979 0.3168 0.3013 0.2563 0.1862 0.0979 0.9193 0.8743 0.7437 0.5404 0.2841 1.4318 1.3618 1.1584 0.8416 0.4425 1.8042 1.7159 1.4596 1.0605 0.5575 2.0000 1.9021 1.6180 1.1756 0.6180 Column 11 0.0000 0.0000 -0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0 0 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0000T=1.5时： fwave(10,0.1,30,0.05,0,2)ans = Columns 1 through 5 0 0.6180 1.1756 1.6180 1.9021 0 0.6180 1.1756 1.6180 1.9021 0 0.5575 1.0605 1.4596 1.7159 0 0.4425 0.8416 1.1584 1.3618 0 0.2841 0.5404 0.7437 0.8743 0 0.0979 0.1862 0.2563 0.3013 0 -0.0979 -0.1862 -0.2563 -0.3013 0 -0.2841 -0.5404 -0.7437 -0.8743 0 -0.4425 -0.8416 -1.1584 -1.3618 0 -0.5575 -1.0605 -1.4596 -1.7159 0 -0.6180 -1.1756 -1.6180 -1.9021 0 -0.6180 -1.1756 -1.6180 -1.9021 0 -0.5575 -1.0605 -1.4596 -1.7159 0 -0.4425 -0.8416 -1.1584 -1.3618 0 -0.2841 -0.5404 -0.7437 -0.8743 0 -0.0979 -0.1862 -0.2563 -0.3013 0 0.0979 0.1862 0.2563 0.3013 0 0.2841 0.5404 0.7437 0.8743 0 0.4425 0.8416 1.1584 1.3618 0 0.5575 1.0605 1.4596 1.7159 0 0.6180 1.1756 1.6180 1.9021 0 0.6180 1.1756 1.6180 1.9021 0 0.5575 1.0605 1.4596 1.7159 0 0.4425 0.8416 1.1584 1.3618 0 0.2841 0.5404 0.7437 0.8743 0 0.0979 0.1862 0.2563 0.3013 0 -0.0979 -0.1862 -0.2563 -0.3013 0 -0.2841 -0.5404 -0.7437 -0.8743 0 -0.4425 -0.8416 -1.1584 -1.3618 0 -0.5575 -1.0605 -1.4596 -1.7159 0 -0.6180 -1.1756 -1.6180 -1.9021 Columns 6 through 10 2.0000 1.9021 1.6180 1.1756 0.6180 2.0000 1.9021 1.6180 1.1756 0.6180 1.8042 1.7159 1.4596 1.0605 0.5575 1.4318 1.3618 1.1584 0.8416 0.4425 0.9193 0.8743 0.7437 0.5404 0.2841 0.3168 0.3013 0.2563 0.1862 0.0979 -0.3168 -0.3013 -0.2563 -0.1862 -0.0979 -0.9193 -0.8743 -0.7437 -0.5404 -0.2841 -1.4318 -1.3618 -1.1584 -0.8416 -0.4425 -1.8042 -1.7159 -1.4596 -1.0605 -0.5575 -2.0000 -1.9021 -1.6180 -1.1756 -0.6180 -2.0000 -1.9021 -1.6180 -1.1756 -0.6180 -1.8042 -1.7159 -1.4596 -1.0605 -0.5575 -1.4318 -1.3618 -1.1584 -0.8416 -0.4425 -0.9193 -0.8743 -0.7437 -0.5404 -0.2841 -0.3168 -0.3013 -0.2563 -0.1862 -0.0979 0.3168 0.3013 0.2563 0.1862 0.0979 0.9193 0.8743 0.7437 0.5404 0.2841 1.4318 1.3618 1.1584 0.8416 0.4425 1.8042 1.7159 1.4596 1.0605 0.5575 2.0000 1.9021 1.6180 1.1756 0.6180 2.0000 1.9021 1.6180 1.1756 0.6180 1.8042 1.7159 1.4596 1.0605 0.5575 1.4318 1.3618 1.1584 0.8416 0.4425 0.9193 0.8743 0.7437 0.5404 0.2841 0.3168 0.3013 0.2563 0.1862 0.0979 -0.3168 -0.3013 -0.2563 -0.1862 -0.0979 -0.9193 -0.8743 -0.7437 -0.5404 -0.2841 -1.4318 -1.3618 -1.1584 -0.8416 -0.4425 -1.8042 -1.7159 -1.4596 -1.0605 -0.5575 -2.0000 -1.9021 -1.6180 -1.1756 -0.6180 Column 11 0.0000 0.0000 -0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0 0 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 -0.0000 0.0000 0 -0.0000 -0.0000 -0.0000 0.0000 0 0T=2.0时：fwave(10,0.1,40,0.05,0,2)ans = Columns 1 through 5 0 0.6180 1.1756 1.6180 1.9021 0 0.6180 1.1756 1.6180 1.9021 0 0.5575 1.0605 1.4596 1.7159 0 0.4425 0.8416 1.1584 1.3618 0 0.2841 0.5404 0.7437 0.8743 0 0.0979 0.1862 0.2563 0.3013 0 -0.0979 -0.1862 -0.2563 -0.3013 0 -0.2841 -0.5404 -0.7437 -0.8743 0 -0.4425 -0.8416 -1.1584 -1.3618 0 -0.5575 -1.0605 -1.4596 -1.7159 0 -0.6180 -1.1756 -1.6180 -1.9021 0 -0.6180 -1.1756 -1.6180 -1.9021 0 -0.5575 -1.0605 -1.4596 -1.7159 0 -0.4425 -0.8416 -1.1584 -1.3618 0 -0.2841 -0.5404 -0.7437 -0.8743 0 -0.0979 -0.1862 -0.2563 -0.3013 0 0.0979 0.1862 0.2563 0.3013 0 0.2841 0.5404 0.7437 0.8743 0 0.4425 0.8416 1.1584 1.3618 0 0.5575 1.0605 1.4596 1.7159 0 0.6180 1.1756 1.6180 1.9021 0 0.6180 1.1756 1.6180 1.9021 0 0.5575 1.0605 1.4596 1.7159 0 0.4425 0.8416 1.1584 1.3618 0 0.2841 0.5404 0.7437 0.8743 0 0.0979 0.1862 0.2563 0.3013 0 -0.0979 -0.1862 -0.2563 -0.3013 0 -0.2841 -0.5404 -0.7437 -0.8743 0 -0.4425 -0.8416 -1.1584 -1.3618 0 -0.5575 -1.0605 -1.4596 -1.7159 0 -0.6180 -1.1756 -1.6180 -1.9021 0 -0.6180 -1.1756 -1.6180 -1.9021 0 -0.5575 -1.0605 -1.4596 -1.7159 0 -0.4425 -0.8416 -1.1584 -1.3618 0 -0.2841 -0.5404 -0.7437 -0.8743 0 -0.0979 -0.1862 -0.2563 -0.3013 0 0.0979 0.1862 0.2563 0.3013 0 0.2841 0.5404 0.7437 0.8743 0 0.4425 0.8416 1.1584 1.3618 0 0.5575 1.0605 1.4596 1.7159 0 0.6180 1.1756 1.6180 1.9021 Columns 6 through 10 2.0000 1.9021 1.6180 1.1756 0.6180 2.0000 1.9021 1.6180 1.1756 0.6180 1.8042 1.7159 1.4596 1.0605 0.5575 1.4318 1.3618 1.1584 0.8416 0.4425 0.9193 0.8743 0.7437 0.5404 0.2841 0.3168 0.3013 0.2563 0.1862 0.0979 -0.3168 -0.3013 -0.2563 -0.1862 -0.0979 -0.9193 -0.8743 -0.7437 -0.5404 -0.2841 -1.4318 -1.3618 -1.1584 -0.8416 -0.4425 -1.8042 -1.7159 -1.4596 -1.0605 -0.5575 -2.0000 -1.9021 -1.6180 -1.1756 -0.6180 -2.0000 -1.9021 -1.6180 -1.1756 -0.6180 -1.8042 -1.7159 -1.4596 -1.0605 -0.5575 -1.4318 -1.3618 -1.1584 -0.8416 -0.4425 -0.9193 -0.8743 -0.7437 -0.5404 -0.2841 -0.3168 -0.3013 -0.2563 -0.1862 -0.0979 0.3168 0.3013 0.2563 0.1862 0.0979 0.9193 0.8743 0.7437 0.5404 0.2841 1.4318 1.3618 1.1584 0.8416 0.4425 1.8042 1.7159 1.4596 1.0605 0.5575 2.0000 1.9021 1.6180 1.1756 0.6180 2.0000 1.9021 1.6180 1.1756 0.6180 1.8042 1.7159 1.4596 1.0605 0.5575 1.4318 1.3618 1.1584 0.8416 0.4425 0.9193 0.8743 0.7437 0.5404 0.2841 0.3168 0.3013 0.2563 0.1862 0.0979 -0.3168 -0.3013 -0.2563 -0.1862 -0.0979 -0.9193 -0.8743 -0.7437 -0.5404 -0.2841 -1.4318 -1.3618 -1.1584 -0.8416 -0.4425 -1.8042 -1.7159 -1.4596 -1.0605 -0.5575 -2.0000 -1.9021 -1.6180 -1.1756 -0.6180 -2.0000 -1.9021 -1.6180 -1.1756 -0.6180 -1.8042 -1.7159 -1.4596 -1.0605 -0.5575 -1.4318 -1.3618 -1.1584 -0.8416 -0.4425 -0.9193 -0.8743 -0.7437 -0.5404 -0.2841 -0.3168 -0.3013 -0.2563 -0.1862 -0.0979 0.3168 0.3013 0.2563 0.1862 0.0979 0.9193 0.8743 0.7437 0.5404 0.2841 1.4318 1.3618 1.1584 0.8416 0.4425 1.8042 1.7159 1.4596 1.0605 0.5575 2.0000 1.9021 1.6180 1.1756 0.6180 Column 11 0.0000 0.0000 -0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0 0 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 -0.0000 -0.0000 0.0000 -0.0000 -0.0000 -0.0000 0.0000 0 -0.0000 -0.0000 -0.0000 0.0000 0 0 0.0000 0.0000 -0.0000 0.0000 0.0000 0.0000 -0.0000 -0.0000 -0.0000 0.0000实验结果分析： 当步长取的越大时，其值越精确，与精确值产生的误差就会越小。当步长一定时，t的值越大，其分段数就越大，计算的值就接近精确值。实验小结： 运用适合的差分格式和方法解答波动方程混合边值问题的解，有利于提高解题效率。同时多次调试和修改程序，使之能正确运行，得到我们所要的结果。附Matlab程序代码：function u=fwave(nx,hx,nt,ht,lowb
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