微分方程数值解欧拉法.docx

1、微分方程数值解欧拉法1.1、求解初值问题，已知精确解为 当h=0。1时，解为：01100.10。9000000.9093629.3616E-030.20.8190480。8351051.6057E-020.30。7535180.7741552。0637E020。40。7003910。7239462.3555E020。50.6571650.6823472。5182E-020。60.6217750。6475982。5823E-020。70。5925260.6182492。5723E-020。80。5680340.5931142.5080E020。90.5471770.5712302.4053E-0

2、21.00。5290510。5518192.2768E02当h=0。05时，解为：01100。050.9500000。9524182.4185E-030。100.9048780.9093624.4835E030。150。8641580。8703916.2326E-030.200.8274060。8351057.6996E030。250.7942230。8031388。9155E-030.300.7642470.7741559。9084E-030。350.7371470。7478501.0704E-020.400.7126210.7239461.1324E-020。450。6903970.702

3、1881。1791E-020。500。6702230.6823471。2124E-020。550。6518760。6642131.2338E-020。600。6351480.6475981.2450E020.650.6198550。6323281.2473E-020。700。6058290。6182491.2420E020.750.5929180.6052201.2302E020.800.5809850.5931141.2129E-020。850。5699090.5818191.1909E-020。900.5595790.5712301.1651E-020。950。5498960。561258

4、1.1362E-021。000.5407710。5518191.1048E-02h=50时，解为：01100.020。9800000.9803953.9471E-040.040。9607920.9615587。6599E040。060。9423450。9434601.1148E-030.080。9246280.9260701。4422E-030。100.9076130。9093621.7491E030.120。8912700。8933062。0363E030.140。8755730.8778782。3048E-030.160.8604960.8630512.5553E-030。180.8460

5、130.8488022.7888E-030。200.8321000。8351053.0058E030.220.8187320.8219403.2073E-030.240.8058890.8092833。3938E030.260.7935470。7971133.5662E030.280.7816850.7854103.7250E-030.300.7702840.7741553。8709E030。320.7593230.7633284。0045E-030.340.7487840。7529114。1264E-030。360。7386490。7428864。2371E-030.380.7288990.

6、7332364。3373E030。400.7195180。7239464.4274E030。420。7104900.7149984。5079E030.440.7018000.7063794。5793E030。460。6934310。6980734。6421E-030.480。6853710。6900674.6967E-030。500.6776030。6823474。7435E-030。520.6701170。6749004.7830E030。540.6628970。6677134.8156E-030。560。6559330。6607754.8415E030.580.6492120.654073

7、4.8613E-030。600.6427230.6475984.8751E030.620.6364540。6413374。8835E030。640。6303950。6352824.8866E030。660.6245370.6294224。8848E030。680。6188680。6237474。8784E030。700.6133810。6182494.8676E030.720。6080660。6129184.8528E030。740.6029140。6077484。8341E-030.760.5979170.6027284。8119E030。780.5930670。5978534。7863E-

8、030。800.5883570.5931144。7577E-030.820。5837790.5885054。7261E-030。840。5793260.5840184.6918E-030。860.5749920.5796474.6550E030。880。5707710.5753874.6159E030。900.5666560.5712304.5746E-030。920.5626410.5671724。5314E-030。940。5587210.5632074.4864E-030。960。5548900。5593304。4397E030。980。5511440。5555354.3916E031.

9、000。5474770.5518194。3420E03有图像看出,当步长越小，计算得到的解越逼近精确解.1。2、求解初值问题，已知精确解为： ，h=0。1欧拉法改进的欧拉法预报-校正格式011110.11.004837421。000000001.005000001.004761900.21。018730751。010000001。019250001.018594100.31。040818221.029000001。041837501。040632760。41。070320051.056100001.071940631。070096310。51。106530661。090490001。10881

10、9091.106277610。61.148811641.131441001.151806091。148536890.71。196585301。178296901.200300941。196295280。81。249328961。230467211.253762531。249029060。91.306569661.287420491.311703381.306264391.01.367879441.348678441.373684291。36757254欧拉法图像：改进欧拉法图像：预测-校正法图像：附录:源代码1.1、clearclc X0=0；X1=1；n=10;%更改分点数h=1/n;%步长

11、 y(1）=1；x（1）=X0;for i=1：n x（i+1）=x(i)+h； y(i+1)=y(i)+h*（x（i）*exp（-x（i)y(i)；endx=vpa（x，6）y=vpa(y,6) X=(X0:0。001：X1);n=1/0.001；for(i=1：n+1) Y(i）=0.5(X(i)2+2)*exp（-X(i）);end plot(x，y,*）hold onplot（X，Y) X=X0：h:X1；Y=0。5。*(X。2+2).exp(X）；Y=vpa（Y，6）yy=abs(yY）1.2、欧拉法:clearclcX0=0；X1=1；n=10；h=1/n；y(1）=1;x（1）=

12、X0;X=X0：h:X1；Y=X+exp(-X）；Y=vpa(Y,9）；精确解XX=X0:0.0001：X1；YY=XX+exp(XX)；for i=1:nx(i+1)=x(i）+h；y（i+1)=y（i）+h*（-y(i)+x（i)+1）;endy=vpa（y,9)plot（x，y，*)hold onplot（XX,YY)改进的欧拉法：clearclc X0=0；X1=1;n=10;h=1/n； y（1)=1；x(1)=X0; X=X0：h:X1；Y=X+exp(-X）；Y=vpa（Y，9);精确解 XX=X0:0.0001：X1;YY=XX+exp(XX）; for i=1：n x（i+1

13、)=x(i)+h; y(i+1)=y（i)+h（-y(i)+x（i）+1）；endfor i=1：n y（i+1）=y（i)+0.5h*(y（i)+x（i)+1）+(-y(i+1）+x(i+1)+1)）;end y=vpa(y,9)plot（x，y，）hold onplot(XX，YY)预报-校正格式：clearclc X0=0；X1=1；n=10；h=1/n; y（1）=1；x（1)=X0; X=X0：h：X1;Y=X+exp（-X）;Y=vpa（Y,9)；精确解 XX=X0:0。0001：X1；YY=XX+exp(-XX）； for i=1：n x(i+1）=x(i）+h； y（i+1)=y（i）+h(y（i)+x（i)+1);endfor i=1：n for count=1：10 预报校正格式，迭代十次 y(i+1）=y(i）+0.5*h（(-y(i)+x（i）+1）+（y(i+1)+x(i+1）+1）); endend y=vpa(y,9)plot(x，y，*）hold onplot(XX,YY）