微分方程数值解法课程设计许.doc

微分方程数值解法课程设计-A组 计算022班3号 许*1. 用复化梯形计算积分解：算法思想：根据复化梯形公式:取步长，有，写出程序，如下所示：N=100;a=0;b=10;h=(b-a)/N;T=zeros(0,N);for j=1:N T(j)=0.5*h*exp(-(a+h*(j-1).2)+exp(-(a+h*j).2) t=sum(T)end运行结果：t = 0.88622. 用Euler、改进的Euler法、梯形法、R-K法解 解：根据算法思想：（1）Euler法 （2）改进的Euler法（3）梯形法（4）R-K法我们可以写出程序如下所示：format long;N=100;h=1/N;%步长U=zeros(1,N+1);%eulerT=zeros(1,N+1);%真解X=zeros(1,N+1);R=zeros(4,N+1);%误差G=zeros(1,N+1);%改进euleO=zeros(1,N+1);%梯形法K=zeros(1,N+1)%R-K法c=zeros(1,4);a=0;for j=2:N+1; X(j)=h*(j-1);endU(1)=1T(1)=1;G(1)=1;X(1)=0;O(1)=1;K(1)=1;for j=1:N U(j+1)=U(j)+h*(U(j)+X(j); %euler法公式 a=G(j)+h*(G(j)+X(j); G(j+1)=G(j)+h/2*(G(j)+X(j)+a+X(j+1); %改进的euler法公式 b=O(j)+h*(O(j)+X(j); for i=1:10 O(j+1)=O(j)+h/2*(G(j)+X(j)+b+X(j+1); %梯形法公式 b=O(j+1);endc(1)=K(j)+X(j);c(2)=K(j)+h/2*c(1)+(X(j)+h/2);c(3)=K(j)+h/2*c(2)+(X(j)+h/2);c(4)=K(j)+h*c(3)+(X(j)+h);K(j+1)=K(j)+h/6*(c(1)+2*c(2)+2*c(3)+c(4);%四阶R-K法公式 T(j+1)=-X(j+1)-1+2*exp(X(j+1); %真实值 R(1,j+1)=U(j+1)-T(j+1); %EULER法误差 R(2,j+1)=G(j+1)-T(j+1); %改进的EULER法误差 R(3,j+1)=O(j+1)-T(j+1); %梯形法误差 R(4,j+1)=K(j+1)-T(j+1); %四阶R-K法误差end真实值T：11.01011.02041.03091.04161.05251.06371.0751.08661.09831.11031.12261.1351.14771.16051.17371.1871.20061.21441.22851.24281.25741.27221.28721.30251.31811.33391.34991.36631.38291.39971.41691.43431.45191.46991.48811.50671.52551.54461.5641.58361.60361.62391.64451.66541.68661.70811.731.75211.77461.79741.82061.84411.86791.8921.91651.94131.96651.99212.0182.04422.07092.09792.12522.1532.18112.20962.23852.26782.29742.32752.3582.38892.42022.45192.4842.51662.54952.58292.61682.65112.68582.7212.75662.79272.82932.86632.90382.94182.98033.01923.05863.09863.1393.183.22143.26343.30593.34893.39253.4366欧拉法U值：11.011.02021.03061.04121.0521.0631.07431.08571.09741.10921.12131.13371.14621.15891.17191.18521.19861.21231.22621.24041.25481.26941.28431.29951.31491.33051.34641.36261.3791.39571.41271.42991.44741.46521.48321.50151.52021.53911.55821.57771.59751.61761.6381.65861.67961.70091.72251.74451.76671.78931.81221.83541.85891.88281.9071.93161.95651.98182.00742.03342.05972.08642.11352.14092.16872.19692.22552.25442.28382.31352.34372.37422.40512.43652.46832.50042.5332.56612.59952.63342.66782.70252.73782.77342.80962.84622.88322.92082.95882.99733.03623.07573.11573.15613.19713.23853.28053.3233.36613.4096改进的欧拉法G值：11.01011.02041.03091.04161.05251.06371.0751.08661.09831.11031.12261.1351.14771.16051.17371.1871.20061.21441.22851.24281.25731.27211.28721.30251.3181.33381.34991.36621.38281.39971.41681.43421.45191.46991.48811.50661.52551.54461.56391.58361.60361.62391.64451.66541.68661.70811.731.75211.77461.79741.82061.8441.86781.8921.91651.94131.96651.9922.01792.04422.07082.09782.12522.15292.1812.20952.23842.26772.29742.32752.35792.38882.42012.45182.48392.51652.54952.58292.61672.6512.68582.72092.75662.79272.82922.86632.90382.94172.98023.01913.05863.09853.13893.17993.22133.26333.30583.34883.39243.4365梯形法O值：11.01011.02041.03091.04161.05251.06371.0751.08661.09841.11031.12261.1351.14771.16061.17371.1871.20061.21441.22851.24281.25741.27221.28721.30251.31811.33391.34991.36631.38291.39971.41691.43431.45191.46991.48811.50671.52551.54461.5641.58371.60361.62391.64451.66541.68661.70821.731.75221.77461.79751.82061.84411.86791.8921.91651.94141.96651.99212.0182.04422.07092.09792.12522.1532.18112.20962.23852.26782.29742.32752.3582.38892.42022.45192.4842.51662.54952.5832.61682.65112.68582.7212.75672.79272.82932.86632.90382.94182.98033.01923.05873.09863.1393.183.22143.26343.30593.34893.39253.4366R-K法K值：11.01011.02041.03091.04161.05251.06371.0751.08661.09831.11031.12261.1351.14771.16051.17371.1871.20061.21441.22851.24281.25741.27221.28721.30251.31811.33391.34991.36631.38291.39971.41691.43431.45191.46991.48811.50671.52551.54461.5641.58361.60361.62391.64451.66541.68661.70811.731.75211.77461.79741.82061.84411.86791.8921.91651.94131.96651.99212.0182.04422.07092.09792.12522.1532.18112.20962.23852.26782.29742.32752.3582.38892.42022.45192.4842.51662.54952.58292.61682.65112.68582.7212.75662.79272.82932.86632.90382.94182.98033.01923.05863.09863.1393.183.22143.26343.30593.34893.39253.4366误差R值：第一组0-0.00010033-0.00020268-0.00030707-0.00041353-0.00052209-0.00063279-0.00074566-0.00086072-0.00097802-0.0010976-0.0012194-0.0013436-0.0014702-0.0015992-0.0017306-0.0018645-0.0020008-0.0021398-0.0022813-0.0024254-0.0025722-0.0027217-0.002874-0.003029-0.0031868-0.0033475-0.0035111-0.0036777-0.0038472-0.0040198-0.0041954-0.0043742-0.0045561-0.0047412-0.0049296-0.0051213-0.0053163-0.0055147-0.0057166-0.0059219-0.0061308-0.0063433-0.0065595-0.0067793-0.0070029-0.0072303-0.0074615-0.0076966-0.0079358-0.0081789-0.0084261-0.0086775-0.008933-0.0091928-0.0094569-0.0097254-0.0099983-0.010276-0.010558-0.010844-0.011135-0.011431-0.011732-0.012038-0.012349-0.012664-0.012985-0.013311-0.013642-0.013979-0.014321-0.014668-0.015021-0.015379-0.015743-0.016113-0.016489-0.01687-0.017258-0.017651-0.018051-0.018457-0.01887-0.019288-0.019714-0.020146-0.020584-0.02103-0.021482-0.021941-0.022407-0.02288-0.023361-0.023849-0.024344-0.024847-0.025358-0.025876-0.026402-0.026936第二组0-3.3417e-007-6.7505e-007-1.0228e-006-1.3774e-006-1.739e-006-2.1078e-006-2.4838e-006-2.8672e-006-3.258e-006-3.6564e-006-4.0624e-006-4.4763e-006-4.8981e-006-5.3278e-006-5.7658e-006-6.212e-006-6.6665e-006-7.1296e-006-7.6014e-006-8.0818e-006-8.5712e-006-9.0696e-006-9.5772e-006-1.0094e-005-1.062e-005-1.1156e-005-1.1702e-005-1.2257e-005-1.2822e-005-1.3398e-005-1.3983e-005-1.458e-005-1.5186e-005-1.5804e-005-1.6432e-005-1.7071e-005-1.7722e-005-1.8384e-005-1.9057e-005-1.9742e-005-2.0439e-005-2.1148e-005-2.1869e-005-2.2603e-005-2.3349e-005-2.4108e-005-2.4879e-005-2.5664e-005-2.6462e-005-2.7273e-005-2.8098e-005-2.8937e-005-2.979e-005-3.0657e-005-3.1539e-005-3.2435e-005-3.3346e-005-3.4272e-005-3.5213e-005-3.617e-005-3.7142e-005-3.8131e-005-3.9135e-005-4.0156e-005-4.1193e-005-4.2247e-005-4.3318e-005-4.4407e-005-4.5513e-005-4.6636e-005-4.7778e-005-4.8938e-005-5.0116e-005-5.1313e-005-5.2529e-005-5.3765e-005-5.502e-005-5.6294e-005-5.7589e-005-5.8904e-005-6.024e-005-6.1596e-005-6.2974e-005-6.4373e-005-6.5794e-005-6.7237e-005-6.8703e-005-7.0191e-005-7.1702e-005-7.3236e-005-7.4794e-005-7.6376e-005-7.7982e-005-7.9613e-005-8.1269e-005-8.2949e-005-8.4656e-005-8.6388e-005-8.8147e-005-8.9932e-005第三组01.6834e-0073.3755e-0075.076e-0076.7848e-0078.5018e-0071.0227e-0061.196e-0061.3701e-0061.5449e-0061.7205e-0061.8968e-0062.0739e-0062.2516e-0062.43e-0062.6091e-0062.7888e-0062.9692e-0063.1501e-0063.3317e-0063.5138e-0063.6964e-0063.8796e-0064.0633e-0064.2475e-0064.4321e-0064.6172e-0064.8027e-0064.9885e-0065.1747e-0065.3613e-0065.5481e-0065.7353e-0065.9227e-0066.1103e-0066.2981e-0066.486e-0066.6741e-0066.8623e-0067.0506e-0067.2389e-0067.4272e-0067.6155e-0067.8037e-0067.9918e-0068.1798e-0068.3676e-0068.5552e-0068.7425e-0068.9295e-0069.1162e-0069.3025e-0069.4884e-0069.6738e-0069.8587e-0061.0043e-0051.0227e-0051.041e-0051.0592e-0051.0774e-0051.0955e-0051.1135e-0051.1314e-0051.1492e-0051.1669e-0051.1846e-0051.2021e-0051.2194e-0051.2367e-0051.2538e-0051.2708e-0051.2877e-0051.3044e-0051.3209e-0051.3373e-0051.3535e-0051.3696e-0051.3854e-0051.4011e-0051.4166e-0051.4319e-0051.4469e-0051.4618e-0051.4764e-0051.4908e-0051.5049e-0051.5188e-0051.5324e-0051.5458e-0051.5588e-0051.5716e-0051.5841e-0051.5963e-0051.6082e-0051.6198e-0051.631e-0051.6419e-0051.6524e-0051.6626e-0051.6724e-0051.6818e-005第四组0-1.6693e-012-3.3724e-012-5.1097e-012-6.8812e-012-8.6882e-012-1.053e-011-1.2409e-011-1.4324e-011-1.6277e-011-1.8267e-011-2.0295e-011-2.2363e-011-2.447e-011-2.6617e-011-2.8805e-011-3.1034e-011-3.3305e-011-3.5619e-011-3.7976e-011-4.0376e-011-4.2821e-011-4.5311e-011-4.7847e-011-5.0429e-011-5.3058e-011-5.5735e-011-5.846e-011-6.1235e-011-6.4059e-011-6.6934e-011-6.986e-011-7.2838e-011-7.5869e-011-7.8954e-011-8.2093e-011-8.5287e-011-8.8537e-011-9.1844e-011-9.5208e-011-9.8631e-011-1.0211e-010-1.0565e-010-1.0926e-010-1.1292e-010-1.1665e-010-1.2044e-010-1.2429e-010-1.2821e-010-1.322e-010-1.3625e-010-1.4038e-010-1.4457e-010-1.4883e-010-1.5316e-010-1.5756e-010-1.6204e-010-1.6659e-010-1.7122e-010-1.7592e-010-1.807e-010-1.8556e-010-1.905e-010-1.9551e-010-2.0061e-010-2.058e-010-2.1106e-010-2.1641e-010-2.2185e-010-2.2738e-010-2.3299e-010-2.3869e-010-2.4449e-010-2.5037e-010-2.5635e-010-2.6243e-010-2.686e-010-2.7487e-010-2.8124e-010-2.8771e-010-2.9428e-010-3.0095e-010-3.0773e-010-3.1461e-010-3.216e-010-3.287e-010-3.3591e-010-3.4323e-010-3.5067e-010-3.5822e-010-3.6588e-010-3.7367e-010-3.8157e-010-3.8959e-010-3.9774e-010-4.0601e-010-4.1441e-010-4.2293e-010-4.3159e-010-4.4037e-010-4.4929e-0103. 用差分格式计算两点边值问题解：算法思想：根据有%-u(t)+u(t)=(1/20*x4)-6*x;%u(0)=0，U(1)=21/20,U(x)=1/20*x5+x3N=100;X=zeros(1,N+1);F=zeros(N-1,1);h=1/N;A=zeros(N-1,N-1);U=zeros(N-1,1);R=zeros(N-1,1);X=0:h:1;T=zeros(N-1,1);A(1,1)=2+h2;A(1,2)=-1;A(N-1,N-1)=2+h2;A(N-1,N-2)=-1;for i=2:N-2 A(i,i-1)=-1; A(i,i)=2+h2; A(i,i+1)=-1; endfor j=1:N-1 F(j)=h2*(1/20*X(j+1)4)-6*X(j+1); T(j)=1/20*X(j+1)5+X(j+1)3; end U=inv(A)*F; R=U-T运行结果：U值：-0.0089337 -0.017862 -0.026781 -0.035684 -0.044566 -0.053424 -0.06225 -0.071041 -0.079791 -0.088494 -0.097147 -0.10574 -0.11428 -0.12275 -0.13114 -0.13946 -0.1477 -0.15585 -0.16391 -0.17187 -0.17973 -0.18748 -0.19511 -0.20263 -0.21002 -0.21729 -0.22442 -0.23141 -0.23826 -0.24495 -0.2515 -0.25788 -0.26409 -0.27013 -0.276 -0.28168 -0.28717 -0.29248 -0.29758 -0.30248 -0.30717 -0.31164 -0.31589 -0.31992 -0.32372 -0.32727 -0.33059 -0.33365 -0.33646 -0.33902 -0.3413 -0.34331 -0.34505 -0.3465 -0.34766 -0.34853 -0.3491 -0.34936 -0.34931 -0.34894 -0.34824 -0.34721 -0.34585 -0.34415 -0.34209 -0.33968 -0.33691 -0.33377 -0.33026 -0.32636 -0.32208 -0.31741 -0.31234 -0.30686 -0.30097 -0.29466 -0.28793 -0.28076 -0.27315 -0.2651 -0.2566 -0.24764 -0.23822 -0.22832 -0.21794 -0.20708 -0.19573 -0.18387 -0.17151 -0.15863 -0.14523 -0.13131 -0.11684 -0.10184 -0.086285 -0.070173 -0.053496 -0.036246 -0.018417误差R值：-0.0089347 -0.01787 -0.026808 -0.035748 -0.044692 -0.05364 -0.062593 -0.071553 -0.08052 -0.089495 -0.098479 -0.10747 -0.11648 -0.12549 -0.13452 -0.14356 -0.15262 -0.16169 -0.17078 -0.17989 -0.18901 -0.19815 -0.20731 -0.21649 -0.2257 -0.23493 -0.24418 -0.25345 -0.26275 -0.27208 -0.28143 -0.29081 -0.30022 -0.30966 -0.31913 -0.32864 -0.33817 -0.34774 -0.35735 -0.36699 -0.37667 -0.38638 -0.39614 -0.40593 -0.41576 -0.42564 -0.43556 -0.44552 -0.45553 -0.46558 -0.47568 -0.48582 -0.49602 -0.50626 -0.51656 -0.5269 -0.5373 -0.54775 -0.55826 -0.56882 -0.57944 -0.59012 -0.60086 -0.61166 -0.62252 -0.63344 -0.64442 -0.65547 -0.66658 -0.67776 -0.68901 -0.70033 -0.71172 -0.72318 -0.73471 -0.74631 -0.75799 -0.76975 -0.78158 -0.79349 -0.80548 -0.81755 -0.8297 -0.84193 -0.85425 -0.86666 -0.87915 -0.89173 -0.9044 -0.91715 -0.93 -0.94295 -0.95599 -0.96912 -0.98235 -0.99568 -1.0091 -1.0226 -1.03634. 利用向前差分格式计算一下定解问题解：根据算法思想，向前差分格式即为 ，根据题意，我们可以写出如下程序：N=10; %把X值从0-1 N等分M=20; %把Y值从0-1M等分h=1/N; %X的步长t=1/M; %Y的步长X=zeros(1,N+1);Y=zeros(1,M+1);Y=0:t:1X=0:h:1;r=2*t/(h2); U=zeros(M+1,N+1);%W=zeros(N-1,1);%S=zeros(N-1,N-1);%A=zeros(N-1,N-1);%B=zeros(N-1,N-1);%C=zeros(N-1,N-1);%I=zeros(N-1,N-1);T=zeros(M+1,N+1);R=zeros(M+1,N+1);%for j=2:N-2 %S(j,j-1)=1; %S(j,j+1)=1; % I(j,j)=1; %end%I(1,1)=1;%I(N-1,N-1)=1;%S(1,2)=1;%S(N-1,N-2)=1;%A=(1-2*r)*I+r*S;for j=1:N+1 %Y值固定时，X的变化 U(1,j)=exp(X(j)*sin(1/2); T(1,j)=exp(X(j)*sin(1/2);endfor j=2:M+1 %X值固定时，Y的变化 U(j,N+1)=exp(1)*sin(1/2-Y(j); U(j,1)=sin(1/2-Y(j); T(j,N+1)=exp(1)*sin(1/2-Y(j); T(j,1)=sin(1/2-Y(j);endfor k=1:M for i=2:N U(k+1,i)=r*U(k,i-1)+(1-2*r)*U(k,i)+r*U(k,i+1); %向前差分格式 T(k+1,i)=exp(X(i)*sin(1/2-Y(k+1); end end for j=1:M+1 for i=1:N+1 R(j,i)=T(j,i)-U(j,i); %误差公式 end end运行结果：真实值T0.479430.529850.585570.647160.715220.790440.873570.965441.0671.17921.30320.434970.480710.531270.587140.648890.717140.792560.875910.968031.06981.18240.389420.430370.475640.525660.580940.642040.709570.784190.866670.957811.05850.34290.378960.418820.462860.511540.565340.62480.690510.763130.843390.932090.295520.32660.360950.398910.440860.487230.538470.59510.657690.726860.803310.24740.273420.302180.333960.369080.40790.45080.498210.550610.608520.672510.198670.219560.242660.268180.296380.327550.3620.400070.442150.488650.540040.149440.165150.182520.201720.222940.246380.272290.300930.332580.367560.406210.0998330.110330.121940.134760.148930.16460.181910.201040.222180.245550.271380.0499790.0552360.0610450.0674650.074560.0824020.0910680.100650.111230.122930.1358600000000000-0.049979-0.055236-0.061045-0.067465-0.07456-0.082402-0.091068-0.10065-0.11123-0.12293-0.13586-0.099833-0.11033-0.12194-0.13476-0.14893-0.1646-0.18191-0.20104-0.22218-0.24555-0.27138-0.14944-0.16515-0.18252-0.20172-0.22294-0.24638-0.27229-0.30093-0.33258-0.36756-0.40621-0.19867-0.21956-0.24266-0.26818-0.29638-0.32755-0.362-0.40007-0.44215-0.48865-0.54004-0.2474-0.27342-0.30218-0.33396-0.36908-0.4079-0.4508-0.49821-0.55061-0.60852-0.67251-0.29552-0.3266-0.36095-0.39891-0.44086-0.