How to solve a five equations systems to get the five unknown

I have my code that i wrote to solve a system of 5 equations in which there is five unknown. I'm trying to get an expression for five of those variables. The functions in the equation are quite complex and i don't think its doable by hand. However when i try to run my code the output is not what i expect to get

Here is my code :

clc; clear all; close all;

syms alpha_inf phi sigma lambda V;

syms T_1_exp T_2_exp T_3_exp T_4_exp T_5_exp;

syms theta_1 theta_2 theta_3 theta_4 theta_5;

syms E_0(theta);

syms coeff_transmission(theta);

syms gamma;

omega = 2*pi*1e5;

syms l;

c_0 = 340;

syms nu;

rho_0 = 1.2;

P_r = 0.07;

delta = sqrt(nu/(omega*rho_0));

A = alpha_inf*(1/lambda)*(1 + (gamma - 1)/(3*sqrt(P_r))) ;

B_1 = alpha_inf*((gamma - 1)/(3*lambda*P_r) +4*(gamma - 1)/(3*sigma^2*sqrt(P_r)) + 3/sigma) ;

B_2 = 4*alpha_inf^2*(1/lambda)*(1 + (gamma - 1)/(3*sqrt(P_r))) ;

C_1 = alpha_inf*((15/(4*V))*((gamma-1)/(27*P_r) +1) + (6*(gamma-1)/(3*sigma*lambda))*(1/(3*P_r) + 1/sqrt(P_r)));

C_2 = alpha_inf^2*(1/lambda)*(1 + (gamma - 1)/(3*sqrt(P_r)))*((gamma - 1)/(3*lambda*P_r) ...

+4*(gamma - 1)/(3*sigma^2*sqrt(P_r)) + 3/sigma);

C_3 = alpha_inf^3*((1/lambda)*(1 + (gamma - 1)/(3*sqrt(P_r))))^3;

C_4 = C_2;

E_0(theta) = phi*sqrt(alpha_inf - sin(theta)^2)/(cos(theta)*alpha_inf);

coeff_transmission(theta) = abs(4*E_0(theta)/(1 + E_0(theta))^2)*exp(-(omega*l/(2*c_0))*sqrt(2)*delta*(A/sqrt(alpha_inf - sin(theta)^2)) ...

- nu*l/(2*c_0*rho_0)*(B_1/sqrt(alpha_inf - sin(theta)^2) ...

- B_2/(8*sqrt(alpha_inf - sin(theta)^2)^3)) ...

- (omega*l*sqrt(2)/(2*c_0))*delta^3*(C_1/(2*sqrt(alpha_inf - sin(theta)^2)) ...

- C_2/(12*sqrt(alpha_inf - sin(theta)^2)^3) ...

+ C_3/(4*sqrt(alpha_inf - sin(theta)^2)^5) ...

- C_4/(24*sqrt(alpha_inf - sin(theta)^2)^5)));

eqns = [T_1_exp == coeff_transmission(theta_1), T_2_exp == coeff_transmission(theta_2), T_3_exp == coeff_transmission(theta_3), T_4_exp == coeff_transmission(theta_4), T_5_exp == coeff_transmission(theta_5)];

assume(alpha_inf>0 & phi>=0 & phi<=1 )

s = solve(eqns, [alpha_inf, phi, lambda, sigma, V],'IgnoreAnalyticConstraints', true, 'ReturnConditions',true);

disp(s)

And here is the output :

alpha_inf: z3

phi: z4

lambda: z5

sigma: z6

V: z7

parameters: [z3 z4 z5 z6 z7]

conditions: T_1_exp - (4*exp(- (l*nu*((z3*(3/z6 + (100*(gamma - 1))/(21*z5) + (10*7^(1/2)*…

Does anyone know why it doesn't give me an expression for my unknown and if it is even possible to do

[Torsten]

I don't understand the symbolic variables theta_1,...,theta_5,T_1_exp,...,T_5_exp. Do you want a solution for alpha_inf, phi, lambda, sigma, V that is parametric in theta_1,...,theta_5,T_1_exp,...,T_5_exp and gamma ?

[Romain]

Well basically i have an equation for a coefficient T which depends on theta the angle of incidence of a wave on a porous material. So in order to find my parameters i just vary the angle to get get different values of T which i can evaluate experimentally on a graph which here are the T_1_exp. My goal is to be able to evaluate alpah phi lambda sigma and V independently with this equation

[Star Strider]

syms alpha_inf phi sigma lambda V;
syms T_1_exp T_2_exp T_3_exp T_4_exp T_5_exp;
syms theta_1 theta_2 theta_3 theta_4 theta_5;
syms E_0(theta);
syms coeff_transmission(theta);
syms gamma;
omega = 2*pi*1e5;
syms l;
c_0 = 340;
syms nu;
rho_0 = 1.2;
P_r = 0.07;
delta = sqrt(nu/(omega*rho_0));
A = alpha_inf*(1/lambda)*(1 + (gamma - 1)/(3*sqrt(P_r))) ;
B_1 = alpha_inf*((gamma - 1)/(3*lambda*P_r) +4*(gamma - 1)/(3*sigma^2*sqrt(P_r)) + 3/sigma) ;
B_2 = 4*alpha_inf^2*(1/lambda)*(1 + (gamma - 1)/(3*sqrt(P_r))) ;
C_1 = alpha_inf*((15/(4*V))*((gamma-1)/(27*P_r) +1) + (6*(gamma-1)/(3*sigma*lambda))*(1/(3*P_r) + 1/sqrt(P_r)));
C_2 = alpha_inf^2*(1/lambda)*(1 + (gamma - 1)/(3*sqrt(P_r)))*((gamma - 1)/(3*lambda*P_r) ...
    +4*(gamma - 1)/(3*sigma^2*sqrt(P_r)) + 3/sigma);
C_3 = alpha_inf^3*((1/lambda)*(1 + (gamma - 1)/(3*sqrt(P_r))))^3;
C_4 = C_2;
E_0(theta) = phi*sqrt(alpha_inf - sin(theta)^2)/(cos(theta)*alpha_inf);
coeff_transmission(theta) = abs(4*E_0(theta)/(1 + E_0(theta))^2)*exp(-(omega*l/(2*c_0))*sqrt(2)*delta*(A/sqrt(alpha_inf - sin(theta)^2)) ...
        - nu*l/(2*c_0*rho_0)*(B_1/sqrt(alpha_inf - sin(theta)^2) ...
        - B_2/(8*sqrt(alpha_inf - sin(theta)^2)^3)) ...
        - (omega*l*sqrt(2)/(2*c_0))*delta^3*(C_1/(2*sqrt(alpha_inf - sin(theta)^2)) ...
        - C_2/(12*sqrt(alpha_inf - sin(theta)^2)^3) ...
        + C_3/(4*sqrt(alpha_inf - sin(theta)^2)^5) ...
        - C_4/(24*sqrt(alpha_inf - sin(theta)^2)^5)));
eqns = [T_1_exp == coeff_transmission(theta_1), T_2_exp == coeff_transmission(theta_2), T_3_exp == coeff_transmission(theta_3), T_4_exp == coeff_transmission(theta_4), T_5_exp == coeff_transmission(theta_5)];
assume(alpha_inf>0 & phi>=0 & phi<=1 )
s = solve(eqns, [alpha_inf, phi, lambda, sigma, V],'IgnoreAnalyticConstraints', true, 'ReturnConditions',true);
% disp(s)
disp("Conditions: ")

Conditions: 

pretty(vpa(simplify(s.conditions, 500), 5))

z3 cos(theta_2) + z4 sqrt(#16) ~= 0.0 and z3 cos(theta_3) + z4 sqrt(#17) ~= 0.0 and z3 cos(theta_4) + z4 sqrt(#18) ~= 0.0 and z3 cos(theta_5) + z4 sqrt(#19) ~= 0.0

                           2
   and z3 (sin(0.5 theta_1)  2.0 - 1.0) ~= z4 sqrt(#15) and not 0.31831 theta_1 - 0.5 in integer and not 0.31831 theta_2 - 0.5 in integer and not 0.31831 theta_3 - 0.5 in

  integer and not 0.31831 theta_4 - 0.5 in integer and not 0.31831 theta_5 - 0.5 in integer and 0.0 < z3

                                              2               /        /   z3 #13    #7 \                  /     #1          #6       #3       #4    \               #5      \
   and z4 in [0.0, 1.0] and sqrt(|sin(theta_1)  - 1.0 z3|) exp| - l nu | --------- - -- | 0.0012255 - l #2 | --------- + ---------- - -- - --------- | 1306.7 - ------------ |
                                                              |        \ sqrt(#15)   #8 /                  | sqrt(#15)     3    5/2   #8         5/2 |          z5 sqrt(#15) |
                                                              \                                            \             z5  #15           z5 #15    /                       /

                        |   z4 sqrt(#15)        | 2                                           2               /        /   z3 #13    #7 \
   |z4| 4.0 == T_1_exp  | --------------- + 1.0 |   |cos(theta_1)| |z3| and sqrt(|sin(theta_2)  - 1.0 z3|) exp| - l nu | --------- - -- |
                        | z3 cos(theta_1)       |                                                             |        \ sqrt(#16)   #9 /
                                                                                                              \

                    /     #1          #6       #3       #4    \               #5      \
   0.0012255 - l #2 | --------- + ---------- - -- - --------- | 1306.7 - ------------ | |z4| 4.0
                    | sqrt(#16)     3    5/2   #9         5/2 |          z5 sqrt(#16) |
                    \             z5  #16           z5 #16    /                       /

               |   z4 sqrt(#16)        | 2                                           2               /        /   z3 #13     #7 \
   == T_2_exp  | --------------- + 1.0 |   |cos(theta_2)| |z3| and sqrt(|sin(theta_3)  - 1.0 z3|) exp| - l nu | --------- - --- |
               | z3 cos(theta_2)       |                                                             |        \ sqrt(#17)   #10 /
                                                                                                     \

                    /     #1          #6        #3       #4    \               #5      \
   0.0012255 - l #2 | --------- + ---------- - --- - --------- | 1306.7 - ------------ | |z4| 4.0
                    | sqrt(#17)     3    5/2   #10         5/2 |          z5 sqrt(#17) |
                    \             z5  #17            z5 #17    /                       /

               |   z4 sqrt(#17)        | 2                                           2               /        /   z3 #13     #7 \
   == T_3_exp  | --------------- + 1.0 |   |cos(theta_3)| |z3| and sqrt(|sin(theta_4)  - 1.0 z3|) exp| - l nu | --------- - --- |
               | z3 cos(theta_3)       |                                                             |        \ sqrt(#18)   #11 /
                                                                                                     \

                    /     #1          #6        #3       #4    \               #5      \
   0.0012255 - l #2 | --------- + ---------- - --- - --------- | 1306.7 - ------------ | |z4| 4.0
                    | sqrt(#18)     3    5/2   #11         5/2 |          z5 sqrt(#18) |
                    \             z5  #18            z5 #18    /                       /

               |   z4 sqrt(#18)        | 2                                           2               /        /   z3 #13     #7 \
   == T_4_exp  | --------------- + 1.0 |   |cos(theta_4)| |z3| and sqrt(|sin(theta_5)  - 1.0 z3|) exp| - l nu | --------- - --- |
               | z3 cos(theta_4)       |                                                             |        \ sqrt(#19)   #12 /
                                                                                                     \

                    /     #1          #6        #3       #4    \               #5      \                      |   z4 sqrt(#19)        | 2
   0.0012255 - l #2 | --------- + ---------- - --- - --------- | 1306.7 - ------------ | |z4| 4.0 == T_5_exp  | --------------- + 1.0 |   |cos(theta_5)| |z3|
                    | sqrt(#19)     3    5/2   #12         5/2 |          z5 sqrt(#19) |                      | z3 cos(theta_5)       |
                    \             z5  #19            z5 #19    /                       /

where

            / (0.5291 gamma + 0.4709) 3.75   (6.0 gamma - 6.0) 2.8472 \
   #1 == z3 | ---------------------------- + ------------------------ | 0.5
            \              z7                          z5 z6          /

                       3/2
   #2 == (1.3263e-6 nu)

           2
   #3 == z3  #14 #13 0.083333

           2
   #4 == z3  #14 #13 0.041667

   #5 == l z3 sqrt(1.3263e-6 nu) #14 1306.7

           3    3
   #6 == z3  #14  0.25

           2
   #7 == z3  #14 0.5

               3/2
   #8 == z5 #15

               3/2
   #9 == z5 #16

                3/2
   #10 == z5 #17

                3/2
   #11 == z5 #18

                3/2
   #12 == z5 #19

          4.7619 (gamma - 1.0)   3.0   (4.0 gamma - 4.0) 1.2599
   #13 == -------------------- + --- + ------------------------
                   z5             z6                2
                                                  z6

   #14 == 1.2599 gamma - 0.25988

                               2
   #15 == z3 - 1.0 sin(theta_1)

                               2
   #16 == z3 - 1.0 sin(theta_2)

                               2
   #17 == z3 - 1.0 sin(theta_3)

                               2
   #18 == z3 - 1.0 sin(theta_4)

                               2
   #19 == z3 - 1.0 sin(theta_5)

.

Answer 1

[Torsten]

Directly use the five experimental values (T_exp,theta) as numerical data and "lsqcurvefit" or "fmincon" to solve for the five unknowns.

But note that the measurement data must be very precise to get physically senseful values for the unknown parameters alpha_inf, phi, lambda, sigma, V. It would be better if you had more than five measurements.