微分方程数值解欧拉法

微分方程数值解欧拉法 dy,,x,,xe,y1.1、求解初值问 题 快递公司问题件快递公司问题件货款处理关于圆的周长面积重点题型关于解方程组的题及答案关于南海问题 ，已知精确解为 ,dx ,，，y0,1, 12,x，，，，yx,x，2x 2 当h=0.1时，解为: ，， yx,y，，xyyxnnnnn 0 1 1 0 0.1 0.900000 0.909362 9.3616E-03 0.2 0.819048 0.835105 1.6057E-02 0.3 0.753518 0.774155 2.0637E-02 0.4 0.700391 0.723946 2.3555E-02 0.5 0.657165 0.682347 2.5182E-02 0.6 0.621775 0.647598 2.5823E-02 0.7 0.592526 0.618249 2.5723E-02 0.8 0.568034 0.593114 2.5080E-02 0.9 0.547177 0.571230 2.4053E-02 1.0 0.529051 0.551819 2.2768E-02 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.500.10.20.30.40.50.60.70.80.91 当h=0.05时，解为: ，，yx,y，，xyyx nnnnn 0 1 1 0 0.05 0.950000 0.952418 2.4185E-03 0.10 0.904878 0.909362 4.4835E-03 0.15 0.864158 0.870391 6.2326E-03 0.20 0.827406 0.835105 7.6996E-03 0.25 0.794223 0.803138 8.9155E-03 0.30 0.764247 0.774155 9.9084E-03 0.35 0.737147 0.747850 1.0704E-02 0.40 0.712621 0.723946 1.1324E-02 0.45 0.690397 0.702188 1.1791E-02 0.50 0.670223 0.682347 1.2124E-02 0.55 0.651876 0.664213 1.2338E-02 0.60 0.635148 0.647598 1.2450E-02 0.65 0.619855 0.632328 1.2473E-02 0.70 0.605829 0.618249 1.2420E-02 0.75 0.592918 0.605220 1.2302E-02 0.80 0.580985 0.593114 1.2129E-02 0.85 0.569909 0.581819 1.1909E-02 0.90 0.559579 0.571230 1.1651E-02 0.95 0.549896 0.561258 1.1362E-02 1.00 0.540771 0.551819 1.1048E-02 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.500.10.20.30.40.50.60.70.80.91 h=50时，解为: ，，yx,y，，xyyx nnnnn 0 1 1 0 0.02 0.980000 0.980395 3.9471E-04 0.04 0.960792 0.961558 7.6599E-04 0.06 0.942345 0.943460 1.1148E-03 0.08 0.924628 0.926070 1.4422E-03 0.10 0.907613 0.909362 1.7491E-03 0.12 0.891270 0.893306 2.0363E-03 0.14 0.875573 0.877878 2.3048E-03 0.16 0.860496 0.863051 2.5553E-03 0.18 0.846013 0.848802 2.7888E-03 0.20 0.832100 0.835105 3.0058E-03 0.22 0.818732 0.821940 3.2073E-03 0.24 0.805889 0.809283 3.3938E-03 0.26 0.793547 0.797113 3.5662E-03 0.28 0.781685 0.785410 3.7250E-03 0.30 0.770284 0.774155 3.8709E-03 0.32 0.759323 0.763328 4.0045E-03 0.34 0.748784 0.752911 4.1264E-03 0.36 0.738649 0.742886 4.2371E-03 0.38 0.728899 0.733236 4.3373E-03 0.40 0.719518 0.723946 4.4274E-03 0.42 0.710490 0.714998 4.5079E-03 0.44 0.701800 0.706379 4.5793E-03 0.46 0.693431 0.698073 4.6421E-03 0.48 0.685371 0.690067 4.6967E-03 0.50 0.677603 0.682347 4.7435E-03 0.52 0.670117 0.674900 4.7830E-03 0.54 0.662897 0.667713 4.8156E-03 0.56 0.655933 0.660775 4.8415E-03 0.58 0.649212 0.654073 4.8613E-03 0.60 0.642723 0.647598 4.8751E-03 0.62 0.636454 0.641337 4.8835E-03 0.64 0.630395 0.635282 4.8866E-03 0.66 0.624537 0.629422 4.8848E-03 0.68 0.618868 0.623747 4.8784E-03 0.70 0.613381 0.618249 4.8676E-03 0.72 0.608066 0.612918 4.8528E-03 0.74 0.602914 0.607748 4.8341E-03 0.76 0.597917 0.602728 4.8119E-03 0.78 0.593067 0.597853 4.7863E-03 0.80 0.588357 0.593114 4.7577E-03 0.82 0.583779 0.588505 4.7261E-03 0.84 0.579326 0.584018 4.6918E-03 0.86 0.574992 0.579647 4.6550E-03 0.88 0.570771 0.575387 4.6159E-03 0.90 0.566656 0.571230 4.5746E-03 0.92 0.562641 0.567172 4.5314E-03 0.94 0.558721 0.563207 4.4864E-03 0.96 0.554890 0.559330 4.4397E-03 0.98 0.551144 0.555535 4.3916E-03 1.00 0.547477 0.551819 4.3420E-03 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.500.10.20.30.40.50.60.70.80.91 有图像看出，当步长越小，计算得到的解越逼近精确解。 dy,,,y，x，1, dx,1.2、求解初值问题，已知精确解为: ,y,1x,0, ,xy,x，e ，h=0.1 改进的欧拉预报-校正 ,x欧拉法 xy,x，e 法 格式 pdf格式笔记格式下载页码格式下载公文格式下载简报格式下载 0 1 1 1 1 0.1 1.00483742 1.00000000 1.00500000 1.00476190 0.2 1.01873075 1.01000000 1.01925000 1.01859410 0.3 1.04081822 1.02900000 1.04183750 1.04063276 0.4 1.07032005 1.05610000 1.07194063 1.07009631 0.5 1.10653066 1.09049000 1.10881909 1.10627761 0.6 1.14881164 1.13144100 1.15180609 1.14853689 0.7 1.19658530 1.17829690 1.20030094 1.19629528 0.8 1.24932896 1.23046721 1.25376253 1.24902906 0.9 1.30656966 1.28742049 1.31170338 1.30626439 1.0 1.36787944 1.34867844 1.37368429 1.36757254 欧拉法图像: 1.4 1.35 1.3 1.25 1.2 1.15 1.1 1.05 100.10.20.30.40.50.60.70.80.91 改进欧拉法图像: 1.4 1.35 1.3 1.25 1.2 1.15 1.1 1.05 100.10.20.30.40.50.60.70.80.91 预测-校正法图像: 1.4 1.35 1.3 1.25 1.2 1.15 1.1 1.05 100.10.20.30.40.50.60.70.80.91 附录:源代码 1.1、 clear clc X0=0; X1=1; n=10;%更改分点数 h=1/n;%步长 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)); end x=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 on plot(X,Y) X=X0:h:X1; Y=0.5.*(X.^2+2).*exp(-X); Y=vpa(Y',6) yy=abs(y-Y) 1.2、 欧拉法: clear clc 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); end y=vpa(y',9) plot(x,y,'*') hold on plot(XX,YY) 改进的欧拉法: clear clc 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); end for i=1:n y(i+1)=y(i)+0.5*h*((-y(i)+x(i)+1)+(-y(i+1)+x(i+1)+1)); end y=vpa(y',9) plot(x,y,'*') hold on plot(XX,YY) 预报-校正格式: clear clc 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); end for 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)); end end y=vpa(y',9) plot(x,y,'*') hold on plot(XX,YY)