solving symbolic equations with partial derivatives

Question

LUCA D'AMBROSIO 2024-7-9

0
链接

此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2135918-solving-symbolic-equations-with-partial-derivatives

评论： LUCA D'AMBROSIO 2024-7-11

hello, I can't find a solution to the following problem: i am trying to solve symbolically some equations which include partial derivatives and a change of reference.

Here is the code:

syms  Cf(zf, zr) Cr(zf, zr) theta z L
z = zf;
theta = (zf - zr)/L;
Cf_z = diff(Cf, z);
Cf_f = diff(Cf, zf);
Cf_r = diff(Cf, zr);
Cf_theta = diff(Cf, theta);
Error using sym/diff (line 77)
Second argument must be a variable or a nonnegative integer specifying the number of differentiations.
eqn = [diff(Cf, z)*diff(z, zf) + diff(Cf, theta)*diff(theta, zf) == diff(Cf, zf), diff(Cf, z)*diff(z, zr) + diff(Cf,               theta)*diff(theta, zr) == diff(Cf, zr)];
S= solve(eqn)

when i run this, the following error appears:

"Second argument must be a variable or a nonnegative integer specifying the number of differentiations." (@ line 7) because i doesn't recognize theta as a variable of Cf. How can i make the change of reference effective so that it can calculate the partial derivatives of Cf in the new reference z, theta?

thank you very much

8 个评论
显示 6更早的评论隐藏 6更早的评论

Umar 2024-7-9

编辑：Walter Roberson 2024-7-10

在 MATLAB Online 中打开

Hi Luca,

I have updated the code to properly define the equations and solved them, please see below.

syms zf zr L Cf(zf, zr);
% Define theta explicitly as a function of zf, zr, and L
theta = @(zf, zr, L) (zf - zr) / L;
% Calculate partial derivatives of Cf with respect to zf and zr
Cf_zf = diff(Cf, zf);
Cf_zr = diff(Cf, zr);
% Define the equations with the correct variables
eqn1 = Cf_zf == diff(Cf, zf);
eqn2 = Cf_zr == diff(Cf, zr);
% Solve the equations
S = solve([eqn1, eqn2]);

The above code defines symbolic variables zf, zr, L, and a function Cf(zf, zr). It then explicitly defines theta as a function of zf, zr, and L. Next, it calculates the partial derivatives of the function Cf with respect to zf and zr using the diff() function. Two equations (eqn1 and eqn2) are defined based on these derivatives. Finally, the code aims to solve these equations using the solve() function to find the values that satisfy both equations simultaneously.

This corrected approach should resolve the issues and allow you to compute the desired derivatives and solve the equations effectively. Adjust the `Cf` function and additional calculations as per your specific problem requirements. Please let me know if you have further questions.

Umar 2024-7-10

Hi Luca,

To express the derivatives Cf/theta and Cf/z in terms of Cf/zf and Cf/zr, you can utilize the chain rule for partial derivatives. By applying the chain rule effectively, you can relate the derivatives in the two reference systems. Here is a simplified example in MATLAB to demonstrate this concept:

syms Cf Cf_zf Cf_zr z theta

% Define the relationship between Cf, Cf_zf, and Cf_zr

Cf = Cf_zf * some_function(z, theta) + Cf_zr * another_function(z, theta);

% Calculate the derivatives Cf/theta and Cf/z using the chain rule

dCf_dtheta = diff(Cf, theta);

dCf_dz = diff(Cf, z);

By appropriately defining the relationship between Cf, Cf_zf, and Cf_zr and then calculating the derivatives using the diff function in MATLAB, you can express Cf/theta and Cf/z in terms of Cf/zf and Cf/zr symbolically.

LUCA D'AMBROSIO 2024-7-11