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matlabsymbolic-math
Maverick
Maverick

Reputation: 440

Partial differential equation in Matlab (incorrect solution)

Let's define function x(t), its time derivative xdot(t), and expression T that is dependent on them:

syms t x(t)
xdot(t) = diff(x,t);
T = (xdot + x)^2;

We can all agree that partial derivative of T with respect to x is ∂T/∂x = 2*(xdot+x). However, if I do this in Matlab I get wrong answer:

dT_dx  = subs( diff( subs(T,x,'x'), 'x' ), 'x', x);
>> dT_dx  = 2 x(t)

Note that it returns the correct answer for ∂T/∂xdot:

dT_dxdot = subs( diff( subs(T,xdot,'x1'), 'x1' ), 'x1', xdot);
>> dT_dxdot = 2*x(t) + 2*diff(x(t), t)

It looks like Matlab ignores the product 2*x*xdot, when calculating derivatives in terms of lower order variables (x), but it doesn't ignore this product when calculating derivative in terms of higher order variables (xdot). If we redefined the expression T as T = (100 + x)^2, we would get ∂T/∂x:

>> ans = 2 x(t) + 200

Thus, after having swapped xdot with a constant we now obtain correct answer.

Comments:

Question

How can one reliably calculate partial and absolute derivatives of T, especially ∂T/∂x?

Is subs( diff(subs() ) ) a good approach, or is there a better way, and, if so, what is it?

Upvotes: 1

Views: 624

Answers (1)

horchler
horchler

Reputation: 18484

As you've noticed, taking the derivative of an abstract symfun like x(t) is not the same as taking the derivative of a symbolic variable like x (assuming x(t) hasn't already been declared in the scope) – see my answer here for more. One needs to be very careful substituting like you're doing. The problem arrises because x(t) gets substituted for 'x' inside of diff(x,t) (xdot(t)) too, i.e., the first substitution in your code, subs(T,x,'x'), already returns an incorrect result relative to what you expect.

You can try this:

syms x(t)
xdot(t) = diff(x,t);
T = (xdot + x)^2;
x1 = {x,xdot};
x2 = {'x','xdot'};
dT_dx = subs(diff(subs(T,x1,x2),'x'),x2,x1)

which returns 2*x(t) + 2*diff(x(t), t).

Upvotes: 1

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