MATLAB改进欧拉法与四阶龙格-库塔求解一阶常微分方程

姓名：樊元君 学号：2012200902 日期：2012.11.06一、实验目的掌握MATLAB语言、C/C++语言编写计算程序的方法、掌握改进欧拉法与四阶龙格-库塔求解一阶常微分方程的初值问题。掌握使用MATLAB程序求解常微分方程问题的方法。、实验内容1、分别写出改进欧拉法与四阶龙格-库塔求解的算法，编写程序上机调试出结果,要求所编程序适用于任何一阶常微分方程的数值解问题，即能解决这一类问题，而不是某一个冋题。实验中以下列数据验证程序的正确性。&&-*~ /，步长h=0.252、实验注意事项yr=一科亠応—二 -必-汇的精确解为- ，通过调整步长，观察结果的精度的变化三、程序流程图:•改进欧拉格式流程图:•四阶龙格库塔流程图:开始初!|点CxO,yO);(K.Y)xl=xO+h;kl=f(xO,yO)；k2=f(x0+h/2ty0+h/2*kl)；k3=f(xO+h/2.y0+h/2*k2)；k4=f(xl,yO+h*k3)；yl=yO+h/6*(k!+2*k2+2*k3+k4).i=i+l；xO=xl；yO=y1四、源程序:•改进后欧拉格式程序源代码:function[]=GJOL(h,x0,y0,X,Y)formatlongh=input('h=');x0=input('x0=');y0=input('y0=');disp('输入的范围是：');X=input('X=');Y=input('Y=');n=round((Y-X)/h);i=1;x1=0;yp=0;yc=0;fori=1:1:nx1=x0+h;yp=yO+h*(-xO*(yOF2);%yp=yO+h*(yO-2*xO/yO);%yc=y0+h*(-x1*(ypf2);%yc=y0+h*(yp-2*x1/yp);%y1=(yp+yc)/2;x0=x1;y0=y1;y=2/(1+x0A2);%y=sqrt(1+2*x0);%fprintf('结果=%.3f,%.8f,%.8f\n',x1,y1,y);endend•四阶龙格库塔程序源代码：function[]=LGKT(h,x0,y0,X,Y)formatlongh=input('h=');x0=input('x0=');y0=input('y0=');disp('输入的范围是：');X=input('X=');Y=input('Y=');n=round((Y-X)/h);i=1;x1=0;k1=0;k2=0;k3=0;k4=0;fori=1:1:nx1=x0+h;k1=-xO*yOA2;%k1=yO-2*xO/yO;%k2=(-(x0+h/2)*(y0+h/2*k1)A2);%k2=(y0+h/2*k1)-2*(x0+h/2)/(y0+h/2*k1);%k3=(-(x0+h/2)*(y0+h/2*k2)A2);%k3=(y0+h/2*k2)-2*(x0+h/2)/(y0+h/2*k2);%k4=(-(x1)*(y0+h*k3f2);%k4=(y0+h*k3)-2*(x1)/(y0+h*k3);%y1=y0+h/6*(k1+2*k2+2*k3+k4);%y1=y0+h/6*(k1+2*k2+2*k3+k4);%x0=x1;y0=y1;y=2/(1+x0A2);%y=sqrt(1+2*x0);%fprintf('结果=%.3f,%.7f,%.7f\n',x1,y1,y);endend五、运行结果：改进欧拉格式结果：»GJOLh=0»25xO=Oy0=2输入的范国是：X=0Y=5结果刊・250,k87600000,1.88235294结果二0.500,1.59389108,1.60000000结果二0.750,L2823900S,1-28000000结果叫.000,1,00962125,L00000000结果=1.250,0.79318809,0.73048780结果“・500,0.62915123,0.61538462结果=1.750,0-50372S54,0.4923076S结果二N000,0.40966655,0.40000000结果二2.250,0*33736499,0.3298&691结果二近500,0.2823574S,0.275S6207结果W・750,0.23885673,0.23357564结果=王000,0.20429990,0.20000000结果250,0.17648S88,0.17297297结果二500,0.15383629,0.15094340结果二3.「50,61351747S,0.13278008结果二4・000,0.11964242,0.11764706结果二4・250,0.10659158,0.10-191803结果二4・500,0.09553028.0*09411765结果匕.为山（L03608040,0.O848SO64结果二弓.000,0.077948Q7J0.07692308四阶龙格库塔结果:»LGKTh=0,25i0=0y0=2输入的萄Hl是：X=0Y=5结^0,250,1.8823030,1.8823529结果=0.500,1.5998962,1.6000000结果=0.750,1.2799478,1.2800000结果=1.000,L0000271,1-0000000结果=1.250,0-7805556,0.7804378结果二1.500j0.6154594j0.6153846Slltl.750,0.4923742：,0.4923077结果=2.000,0.4000543,0.4000000结果-2.250,0.3299396,0.3298969结果=2.500,0.2758952,0.2758621结果=2.750,0.2336023,0.2335766^^=3,000,0,2000200,0,2000000结果=3・25Q卫17298S6.0,1728730结果=1500,0.1509558,0-1509434结果=3-750,0.1327899,0.1327801结果=4.000,0.1176550,0.1176471结果=4.250,0.1049245,0.1049180^^4.500,0.0941229,0,0941176结果=4.750,0.0848850,0.0848806结果=5.000,0.0769267,0.0769231

»LGKTX1xO=Oy€=2輸入的范園是；1=0Y=5结果100,1.98019?8,1,9801980结果=0.200,1.9230763,1.923076&结果刃*300,1.8348612」.8348624结果刃.400,L7241364,L724137&结果二Ck500#1.5999^84,1.6000000结果=0,60^1.4705870,1*4705802结果=0.700,1.3422312,L342281&结果-0.800fL2195122f1.2195122结果刃*900,1.1049731,L1049724结果000,1.0000012,1,0000000结果=1-100,0.9049790,0.9049774结果=1,200,0,8196739,0.8196721结果=】*300,0.7434963,0,7434944结果打*400,0.6756776,0.6756757结果=1*500,0.6153865,山6153346结果600,0.5617996,认5617978结果=1.F00?0.5141405,0.5141388结果=1.800,0,4716997,0,4716981结果=】*900,0.4338409,0,4338395结赛2・000,Q.400001%1.4000000结果=2.100,0.3696870,0-3696858结果200,0.3424668,仇3424658

结果哑.300,0.3179660,0.3179650结果二工400,0.2958589,0.2958580结果=2.500,0,2*58629,0.2758621结果弍*600,0.2577327,0.2577320结果=監700,0.2412552,0.^12545结果=2.800,0.2262449,0.2262443结果=2.900,0.2125404,0.2125399结果=J.：：\LI.2000uu57'：.：：'U|jijij0结果-3.100^0.1835018,0.1885014结果=3.200,0.1779363,0.1779359结果=3・300,0.16S2OS9,0.1682086结果=3.400,0.1592360,0,1592357结果二3.500」.1609437,0-1509434结果=3,600,0.143266^0.1432665结果书*700,0,1361473,0,136U70结果=3.800,0.1295330,0.1295337结果=1900,0.1233808,0,1233806结果=4.000,0.1176472,0.1176471拮UM・M0.1122966,0,1122965结果X*200,0.1072963,0.1072961结果=4.300,0.1026169,0.1026167结果=4.400,0.0982320,(L0982313结果=4.500,0.09411F8?O.0941176结果二4*600,0..0902628,0,0902527结果二4.7»,0.0866177,0,0866176絃肚4・800,0.0831948,0.0831947^$=4.900,0.0799681,0.0799680结果翡.000,0.0769232,0,0769231步长分别为:0.25和0.1时，不冋结果显示验证了步长减少，对于精度的提高起到很大作用，有效数字位数明显增加。六、 实验小结：通过这次实验学习，首先第一点对改进欧拉格式和四阶龙格库塔的原理推导有了深入的理解，改进欧拉格式采用（预报+校正）模式得到较精确的原函数数值解；而四阶龙格库塔则采用多预报几个点的斜率值，采用加权平均作为平均斜率的近似值的思想达到更高精度的数值解，二阶龙格库塔的特例就是改进后的欧拉格式。七、 思考题