微分方程數(shù)值解課程設(shè)計(jì)

1、利用五點(diǎn)差分格式求解拉普拉斯方程的混合邊值問題 要求：給出問題，計(jì)算格式，格式的矩陣形式，線性代數(shù)方程組求解的數(shù)值 方法，穩(wěn)定性，格式的截?cái)嗾`差；程序流程圖；沿 y=1 畫出計(jì)算結(jié)果與精確 解的比較圖形以及誤差圖形(在一個(gè)圖上)；分析兩個(gè)迭代法的收斂速度。 E = -1.1000 0.5000 0 0 0 0 0 0 0 0 0.5000 -2.1000 0.5000 0 0 0 0 0 0 0 0 0.5000 -2.1000 0.5000 0 0 0 0 0 0 0 0 0.5000 -2.1000 0.5000 0 0 0 0 0 0 0 0 0.5000 -2.1000 0.5000

2、0 0 0 0 0 0 0 0 0.5000 -2.1000 0.5000 0 0 0 0 0 0 0 0 0.5000 -2.1000 0.5000 0 0 0 0 0 0 0 0 0.5000 -2.1000 0.5000 0 0 0 0 0 0 0 0 0.5000 -2.1000 0.5000 0 0 0 0 0 0 0 0 0.5000 -2.1000h=0.1Em = -2.1000 1.0000 0 0 0 0 0 0 0 0 1.0000 -4.0000 1.0000 0 0 0 0 0 0 0 0 1.0000 -4.0000 1.0000 0 0 0 0 0 0 0 0 1

3、.0000 -4.0000 1.0000 0 0 0 0 0 0 0 0 1.0000 -4.0000 1.0000 0 0 0 0 0 0 0 0 1.0000 -4.0000 1.0000 0 0 0 0 0 0 0 0 1.0000 -4.0000 1.0000 0 0 0 0 0 0 0 0 1.0000 -4.0000 1.0000 0 0 0 0 0 0 0 0 1.0000 -4.0000 1.0000 0 0 0 0 0 0 0 0 1.0000 -4.0000K = 0.5000 0 0 0 0 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 0 0 0 0 1

4、.0000 0 0 0 0 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 0 0 0 0 1.0000A=E K zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) K Em K zeros(10) zer

5、os(10) zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) K Em K zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) K Em K zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) K Em K zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) ze

6、ros(10) zeros(10) zeros(10) K Em K zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) K Em K zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) K Em K zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) K Em K zero

7、s(10) zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) zeros(10) K Emg = 0 -0.0100 -0.0400 -0.0900 -0.1600 -0.2500 -0.3600 -0.4900 -0.6400 -0.8100 0.0100 0 0 0 0 0 0 0 0 -9.9000 0.0400 0 0 0 0 0 0 0 0 -9.6000 0.0900 0 0 0 0 0 0 0 0 -9.1000 0.1600 0 0 0 0 0 0 0 0 -8.4000 0.2500 0 0 0 0 0 0

8、 0 0 -7.5000 0.3600 0 0 0 0 0 0 0 0 -6.4000 0.4900 0 0 0 0 0 0 0 0 -5.1000 0.6400 0 0 0 0 0 0 0 0 -3.6000 5.8100 9.9000 9.6000 9.1000 8.4000 7.5000 6.4000 5.1000 3.60001.9000U=zeros(100，1)nm=100; w=10-2;jacobi(A,h*g,U,nm,w)迭代次數(shù)為n = 44方程組的解為x = 0 0.0087 0.0350 0.0795 0.1437 0.2290 0.3368 0.4680 0.622

9、8 0.8006 -0.0087 0 0.0260 0.0702 0.1338 0.2187 0.3261 0.4572 0.6121 0.7902 -0.0350 -0.0260 0 0.0439 0.1072 0.1915 0.2984 0.4289 0.5832 0.7607 -0.0795 -0.0702 -0.0439 -0.0000 0.0631 0.1469 0.2531 0.3826 0.5358 0.7120 -0.1437 -0.1338 -0.1072 -0.0631 -0.0000 0.0835 0.1891 0.3176 0.4693 0.6439 -0.2290 -

10、0.2187 -0.1915 -0.1469 -0.0835 0 0.1051 0.2327 0.3831 0.5558 -0.3368 -0.3261 -0.2984 -0.2531 -0.1891 -0.1051 0 0.1271 0.2764 0.4476 -0.4680 -0.4572 -0.4289 -0.3826 -0.3176 -0.2327 -0.1271 0 0.1487 0.3189 -0.6228 -0.6121 -0.5832 -0.5358 -0.4693 -0.3831 -0.2764 -0.1487 0 0.1697 -0.8006 -0.7902 -0.7607

11、 -0.7120 -0.6439 -0.5558 -0.4476 -0.3189 -0.1697 0ans = 44gaussseidel(A,h*g,U,nm,w)迭代次數(shù)為n = 29方程組的解為x = -0.0045 0.0033 0.0282 0.0706 0.1309 0.2087 0.3020 0.4036 0.4908 0.4780 -0.0139 -0.0062 0.0185 0.0607 0.1209 0.1990 0.2941 0.4041 0.5271 0.6779 -0.0412 -0.0334 -0.0086 0.0339 0.0949 0.1747 0.2740 0

12、.3935 0.5370 0.7170 -0.0875 -0.0795 -0.0542 -0.0111 0.0510 0.1328 0.2355 0.3607 0.5114 0.6932 -0.1537 -0.1455 -0.1197 -0.0758 -0.0126 0.0708 0.1756 0.3032 0.4556 0.6347 -0.2411 -0.2325 -0.2062 -0.1613 -0.0971 -0.0126 0.0933 0.2216 0.3736 0.5500 -0.3501 -0.3412 -0.3142 -0.2685 -0.2033 -0.1179 -0.0116

13、 0.1165 0.2673 0.4417 -0.4809 -0.4718 -0.4441 -0.3974 -0.3312 -0.2451 -0.1386 -0.0114 0.1372 0.3094 -0.6333 -0.6239 -0.5954 -0.5477 -0.4804 -0.3934 -0.2867 -0.1608 -0.0164 0.1488 -0.8066 -0.7969 -0.7676 -0.7188 -0.6502 -0.5620 -0.4544 -0.3290 -0.1909 -0.0580ans = 29x=linspace(0,1,100)u=x.2-1X=linspa

14、ce(0,1,10)Y=(-0.8066 -0.7969 -0.7676 -0.7188 -0.6502 -0.5620 -0.4544 -0.3290 -0.1909 -0.0580)plot(x,u) hold on plot(X,Y)截?cái)嗾`差分析：h=0.2e = -1.2000 0.5000 0 0 0 0.5000 -2.2000 0.5000 0 0 0 0.5000 -2.2000 0.5000 0 0 0 0.5000 -2.2000 0.5000 0 0 0 0.5000 -2.2000k = 0.5000 0 0 0 0 0 1.0000 0 0 0 0 0 1.0000

15、0 0 0 0 0 1.0000 0 0 0 0 0 1.0000em = -2.2000 1.0000 0 0 0 1.0000 -4.0000 1.0000 0 0 0 1.0000 -4.0000 1.0000 0 0 0 1.0000 -4.0000 1.0000 0 0 0 1.0000 -4.0000k = 0.5000 0 0 0 0 0 1.0000 0 0 0 0 0 1.0000 0 0 0 0 0 1.0000 0 0 0 0 0 1.0000a=e k zeros(5) zeros(5) zeros(5) k em k zeros(5) zeros(5) zeros(5

16、) k em k zeros(5) zeros(5) zeros(5) k em k zeros(5) zeros(5) zeros(5) k emv = 0 -0.0080 -0.0320 -0.0720 -0.6280 0.0080 0 -0.9600 0.0320 0 -0.8400 0.0720 0 -0.6400 1.1280 0.9600 0.8400 0.6400 0u=zeros(25,1)nm=100; w=10-2;jacobi(a,v,u,nm,w)迭代次數(shù)為n =15方程組的解為x = -0.0141 0.0258 0.1454 0.3488 0.6330 -0.059

17、7 -0.0214 0.1030 0.3061 0.5927 -0.2023 -0.1546 -0.0246 0.1854 0.4722 -0.4618 -0.3847 -0.2323 -0.0166 0.2732 -0.9327 -0.6955 -0.5145 -0.2929 -0.0049x=linspace(0,1,100)u=x.2-1X=linspace(0,1,5)Y=-0.9327 -0.6955 -0.5145 -0.2929 -0.0049plot(x,u)hold onplot(X,Y)畫圖如下：h=0.1的時(shí)候的解的一部分： u1=0 0.0350 0.1437 0.33

18、68 0.6228 -0.0350 0 0.1072 0.2984 0.5832 -0.1437 -0.1072 -0.0000 0.1891 0.4693 -0.3368 -0.2984 -0.1891 0 0.2764 -0.6228 -0.5832 -0.4693 -0.2764 0 u2= -0.0141 0.0258 0.1454 0.3488 0.6330 -0.0597 -0.0214 0.1030 0.3061 0.5927 -0.2023 -0.1546 -0.0246 0.1854 0.4722 -0.4618 -0.3847 -0.2323 -0.0166 0.2732

19、-0.9327 -0.6955 -0.5145 -0.2929 -0.0049ur=00.040.160.360.64-0.0400.120.320.6-0.16-0.1200.20.48-0.36-0.32-0.200.28-0.64-0.6-0.48-0.280ur-u1ans = 0 0.0050 0.0163 0.0232 0.0172 -0.0050 0 0.0128 0.0216 0.0168 -0.0163 -0.0128 0 0.0109 0.0107 -0.0232 -0.0216 -0.0109 0 0.0036 -0.0172 -0.0168 -0.0107 -0.0036 0ur-u2ans = 0.0141 0.0142 0.0146 0.0112 0.0070 0.0197 0.0214 0.0170 0.0139 0.0073 0.0423 0.0346 0.0246 0.0