nm_ch_0501.m

(有源代码)数值分析作业,本文主要包括两个部分,第一部分是常微分方程(ODE)的三个实验题,第二部分是有关的拓展讨论,包括高阶常微分的求解和边值问题的求解(BVP).文中的算法和算例都是基于Matla

function [xx,yy,exact] = nm_ch_0501(a,h)
% This is the function for Numerical analysis Chapter 5.1 to solve the ODE
% equations with the Runge-Kutta Method 

% [xx,yy] = nm_ch_0501(a,h)

% a is the  coef of  dy/dx
% h  is the  length of step
%  xx is  x, yy is y, exact is exact solution

func = inline(' 1 + (y - x) .* a ');
%
% func =
%      Inline function:
%      func(a,x,y) = 1 + (y - x) .* a


% y' = a*y - a* x + 1 , 0<x<1, 
% y(0)=1;

% -50 < a <50 .
% the exact solution is 
% y(x) = e^(a*x) +x ;



if ((h>=1) | (h<=0))
    error(' the step :h , must be in interval (0,1) 1');
%     break;
    return;
end

x = 0 :h :1 ;
y = x;
k = length(x);
y(1) =1;
% y(1) is the  first element of y ,and in fact is  equal to  the inital
% value 
    
for n=1:k-1
   k1 = func(a,x(n),y(n));
   k2 = func(a,x(n)+0.5*h,y(n)+0.5*h*k1);
   k3 = func(a,x(n)+0.5*h,y(n)+0.5*h*k2);
   k4 = func(a,x(n)+h,y(n)+h*k3);
   y(n+1) = y(n) + h*(k1+2.0*k2+2.0*k3+k4)/6.0;
%    xn = xn+h; x(k+1)=xn; y(k+1)=y(n);
end
xx=x;
yy=y;
exact=exp(a*x) +x;