数值分析课程设计实验六文档格式.docx
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数值分析课程设计实验六文档格式.docx
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x=%3.1f,y=%9.7f'
x(i),y(i))
end
disp('
四阶龙格-库塔法求得的解为:
k1=feval(f,x(i-1),y(i-1));
k2=feval(f,x(i-1)+h/2,y(i-1)+k1*h/2);
k3=feval(f,x(i-1)+h/2,y(i-1)+k2*h/2);
k4=feval(f,x(i-1)+h,y(i-1)+k3*h);
y(i)=y(i-1)+h/6*(k1+2*k2+2*k3+k4);
运行结果为:
x=0.1,y=1.0900000
x=0.2,y=1.1717500
x=0.3,y=1.2470768
x=0.4,y=1.3172249
x=0.5,y=1.3830861
x=0.6,y=1.4453250
x=0.7,y=1.5044519
x=0.8,y=1.5608688
x=0.9,y=1.6148989
x=1.0,y=1.6668064
x=0.1,y=1.0855051
x=0.2,y=1.1634777
x=0.3,y=1.2355334
x=0.4,y=1.3027862
x=0.5,y=1.3660420
x=0.6,y=1.4259056
x=0.7,y=1.4828446
x=0.8,y=1.5372290
x=0.9,y=1.5893579
x=1.0,y=1.6394763
x
euler公式求得的解
四阶龙格-库塔法求得的解
x=0.1
y=1.0900000
y=1.0855051
x=0.2
y=1.1717500
y=1.1634777
x=0.3
y=1.2470768
y=1.2355334
x=0.4
y=1.3172249
y=1.3027862
x=0.5
y=1.3830861
y=1.3660420
x=0.6
y=1.4453250
y=1.4259056
x=0.7
y=1.5044519
y=1.4828446
x=0.8
y=1.5608688
y=1.5372290
x=0.9
y=1.6148989
y=1.5893579
x=1.0
y=1.6668064
y=1.6394763
(2)f12.m
alpha=2;
-x*y/(1+x^2)'
x=0.1,y=2.0000000
x=0.2,y=1.9801980
x=0.3,y=1.9421173
x=0.4,y=1.8886645
x=0.5,y=1.8235382
x=0.6,y=1.7505966
x=0.7,y=1.6733644
x=0.8,y=1.5947500
x=0.9,y=1.5169573
x=1.0,y=1.4415285
x=0.1,y=1.9900743
x=0.2,y=1.9611612
x=0.3,y=1.9156523
x=0.4,y=1.8569530
x=0.5,y=1.7888539
x=0.6,y=1.7149854
x=0.7,y=1.6384634
x=0.8,y=1.5617372
x=0.9,y=1.4865879
x=1.0,y=1.4142132
y=2.0000000
y=1.9900743
y=1.9801980
y=1.9611612
y=1.9421173
y=1.9156523
y=1.8886645
y=1.8569530
y=1.8235382
y=1.7888539
y=1.7505966
y=1.7149854
y=1.6733644
y=1.6384634
y=1.5947500
y=1.5617372
y=1.5169573
y=1.4865879
y=1.4415285
y=1.4142132
四、精度分析:
Euler公式的精度阶为一,而四阶龙格-库塔法的精度阶为四。
所以四阶龙格-库塔法的精度更高。
6.2
用求解常微分方程初值问题的方法计算积分上限函数
的值,实际上是将上面表达式两端求导化为常微分方程形式,并用初值条件
。
试用MATLAB中的指令ode23解决定积分
计算问题
因为
,其中
则
该问题就转化成了求解常微分方程初值问题,下面调用指令ode23求解
三、程序及运行结果
f2.m
调用edo23函数求得的微分值为:
[x,y]=ode23(@eq1,[45],0)
计算结果为:
X的值
对应的积分值
4.000000000000000
4.000000932807829
0.000000800000280
4.000005596846973
0.000004800010074
4.000028917042694
0.000024800268931
4.000145518021295
0.000124806810519
4.000728522914301
0.000624970727467
4.003643547379329
0.003129073833207
4.018218669704470
0.015732090904789
.0910********
0.080862383169153
4.191094281330174
0.176********7794
4.291094281330174
0.279248598876569
4.391094281330173
0.390424305048003
4.491094281330173
0.510526123353964
4.591094281330173
0.640336524230162
4.691094281330172
0.780708375359451
4.791094281330172
0.932571457279386
4.891094281330171
1.096939588890575
5.000000000000000
1.291468550321271
本题运用的ode23方法计算的,精度阶最多只能达到3阶
6.3
将下列高阶常微分方程初值问题化为一阶常微分方程组的初值问题,然后用MATEAB指令ode23求数值解。
将下列高阶常微分方程初值问题化为一阶常微分方程组的初值问题为:
f3.m
[x,Y]=ode23(@cdq1,[0,10],[0000])
x=
0
0.000001000000000
0.000006000000000
0.000031000000000
0.000156000000000
0.000781000000000
0.003906000000000
0.009275457558261
0.016064903682848
.022*********
.029*********
0.036619445053325
.0435********
0.050653408315816
0.058966909999049
0.068685571310153
0.080061842525860
.0933********
0.108724974090130
0.123365186352687
0.137********3554
0.151********0112
0.167434693326125
0.184********2580
0.204161647712462
0.225555546391267
0.249285210101193
0.275626369042573
0.304892019015618
0.337438572072545
0.373673319418414
0.414063582875753
0.459148079979070
0.509551255756144
0.566001695301314
0.629356323624332
0.700633109650712
0.781056791019875
0.872125505977080
0.975712912052453
1.094234669978660
1.230941682969375
1.390491280121142
1.580226733493732
1.813732499511545
.0953********
2.339582919898136
2.536607179323829
2.765646634712564
3.015633336439359
3.280468423137612
3.482470855269472
3.684473287401333
3.909496779464162
4.156********2576
4.418821603169769
4.692736604010163
4.926773195981463
5.132********4824
5.367286762739621
5.555084960747961
5.742883158756301
5.957170800499516
6.197527338844070
6.362211208287101
6.526895077730131
6.717157254797291
6.936189921342121
7.130********5802
7.358430222510848
7.607585545660934
7.871471735860521
.0574********
8.243377367049810
8.444033723466909
8.676059270618181
8.865908911250951
.0557********
9.257513138959393
9.455810335202022
9.674046550803718
9.918864595053151
10.000000000000000
Y=
1.0e+002*
0000
00.0000000000000000.0000000000004000.000000799999200
0.0000000000000000.0000000000000000.0000000000144000.000004799971200
0.0000000000000000.0000000000000040.0000000003843920.000024799231216
0.0000000000000000.0000000000005060.0000000097333880.000124780533225
0.0000000000000070.0000000000635030.0000002438573950.000624312285196
0.0000000000045720.0000000079366160.0000060868611640.003112626268527
0.0000000002152420.0000001059702840.0000342017040030.007351962207821
0.0000000021065030.0000005485934350.0001021352532830.012647648220721
0.0000000088923630.0000015803176900.0002063328200300.017895501922103
0.0000000255218630.0000034549783420.0003467970648160.023095673874587
0.0000000587712500.0000064299299570.0005235327921290.028248372491969
0.0000001173022260.0000107661216090.0007365464349640.033353803719360
0.0000002146243330.0000168993661960.0009925040289100.038537287181025
0.0000003931058300.0000265515158460.0013377262562660.044497284646587
0.0000007211344510.0000417626312260.0018035378923430.051339929087025
0.0000013255420630.0000657734540610.0024323697483570.059182062183504
0.0000024425233770.0001037479422300.0032817720394930.068150841272184
0.0000044588472160.0001624455539760.0044033604602300.078164243323076
0.0000073498624150.0002356248437540.0056159695974130.087445327838037
0.0000112728044210.0003238268508390.0069109032097990.096131247349977
0.0000166243860090.0004321482033260.0083391572035740.104643494985282
0.0000245275476530.0005767548355290.0100585594836440.113780751489420
0.0000362050609350.0007698055571330.0121269192805620.123560346195156
0.0000534706923150.0010275531882630.0146131898991760.133********5438
0.0000790169586430.0013717154968040.0175993286385670.145084133276851
0.0001168454749440.0018313189808630.0211825343401600.156********4747
0.0001729110706970.0024451679960660.0254777907569770.169190000785565
0.0002560878907660.0032651531776420.0306207238901820.182********0814
0.0003796218986680.0043606874099070.0367707618986380.195623945286940
0.0005633206421260.0058246584606660.0441145530261880.209539853661262
0.0008368649384160.0077814248725100.0528695417095080.223767010239295
0.0012448358362470.0103975701261450.0632875115284620.238136281057342
0.0018543788800060.0138963903652310.0756577518263190.252421953263467
0.0027669516350900.0185774544956930.0903092504565390.266325071826012
0.0041364488479570.0248430913736340.1076108824759790.279450106143691
0.0061974020832590.0332344082023020.1279678138606620.291271332217253
0.0093093335838730.0444805706493270.151********45830.301083043883125
0.0140275453962450.0595668544055220.1795742291237060.307923791298525
.021*********
0.0322527077017180.1070949969786650.2482931552989030.306782739340534
0.0493441290692150.1438795619566630.2894658146959450.294104550994398
0.0761834713936860.1937222635262550.3344754656417250.268146157335427
0.1192651918384970.2617830981703980.3812268907088480.221945800835107
0.1912572067710890.3562441454980440.4242791677276130.142866909165689
0.3089391296287230.4799450424453670.4474833538162030.016822473735432
0.4395431733788320.5885159239769470.435707696648763-0.116302684599302
0.5638036088926780.6713591730333040.401086614094405-0.236540717639941
0.7276179834972840.7557509939343260.329880779846736-0.386072134344135
0.9258481237345450.8244166926107140.212314742061070-0.553779431208195
1.1499138157646920.8591594094078220.042523080891770-0.725225092982078
1.3233373800039910.852104076703162-0.116178187898725-0.842940535611336
1.4919361329971740.810693583120805-0.296868670107743-0.941697943003061
1.6650568618859220.719233005297356-0.518436754835483-1.020466319732361
1.824402*********0.559289856527625-0.776522734550902-1.057425466615932
1.9411773192707200.319257795055316-1.051587115277669-1.026331601655872
1.985661354669631-0.006355226903674-1.318233919246126-0.902966534385858
1.946142927160647-0.338246993316069-1.509284284117149-0.716326656205778
1.843601907956106-0.662449106607701-1.634178951331275-0.487861564255529
1.641934909829494-1.056930002261673-1.711025369515181-0.154105449610795
1.413102884646365-1.379230706347288-1.7106461481560540.165213060696099
1.124102545487569-1.695570116933253-1.6463224973601610.52563664035238
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