Question

I am trying to implement the Taylor method for ODEs in MatLab:

My code (so far) looks like this...

function [x,y] = TaylorEDO(f, a, b, n, y0)
% syms t
% x = sym('x(t)'); % x(t)
% f = (t^2)*x+x*(1-x);

h = (b - a)/n;

fprime = diff(f);
f2prime = diff(fprime);

y(0) = y0,
for i=1:n
    T((i-1)*h, y(i-1), n) = double(f((i-1)*h, y(i-1)))+(h/2)*fprime((i-1)*h, y(i-1))
    y(i+1) = w(i) + h*T(t(i), y(i), n);

I was trying to use symbolic variables, but I don´t know if/when I have to use double.

I also tried this other code, which is from a Matlab function, but I do not understand how f should enter the code and how this df is calculated.

http://www.mathworks.com/matlabcentral/fileexchange/2181-numerical-methods-using-matlab-2e/content/edition2/matlab/chap_9/taylor.m

As error using the function from this link, I got:

>> taylor('f',0,2,0,20)
Error using feval
Undefined function 'df' for input arguments of type 'double'.

Error in TaylorEDO (line 28)
  D = feval('df',tj,yj)

The f I used here was

 syms t
 x = sym('x(t)'); % x(t)
 f = (t^2)*x+x*(1-x);

Answer 1

这是一种数值方法，因此需要数值函数。然而，它们中的一些是根据函数f的导数计算的。为此，你需要象征性的区分。

相关的Matlab命令是symfun（创建符号函数）和matlabFunction（将符号函数转换为数字）。

到目前为止，您所拥有的代码似乎并不具备可抢救性。你需要从更接近基础的地方开始，例如，“Matlab索引从1开始”。所以我将填补您链接的代码中的差距（df的计算）。评论应该解释发生了什么。

function [T,Y] = taylor(f,a,b,ya,m)
syms t y
dfs(1) = symfun(f, [t y]);                         % make sure that the function has 2 arguments, even if the user passes in something like 2*y
for k=1:3
    dfs(k+1) = diff(dfs(k),t)+f*diff(dfs(k),y);    % the idea of Taylor method: calculate the higher derivatives of solution from the ODE
end
df = matlabFunction(symfun(dfs,[t y]));            % convert to numerical function; again, make sure it has two variables

h = (b - a)/m;                                     % the rest is unchanged except one line
T = zeros(1,m+1);
Y = zeros(1,m+1);
T(1) = a;
Y(1) = ya;
for j=1:m
  tj = T(j);
  yj = Y(j);
  D = df(tj,yj);                                   % syntax change here; feval is unnecessary with the above approach to df
  Y(j+1) = yj + h*(D(1)+h*(D(2)/2+h*(D(3)/6+h*D(4)/24)));
  T(j+1) = a + h*j;
end
end

使用示例：

syms t y
[T, Y] = taylor(t*y, 0, 1, 2, 100); 
plot(T,Y)

Taylor Method ODE

1 个答案: