Solving non-linear equations (0)

Dear all,

I solved non-linear equations using Matlab, but all values are close to 0.

I think this solution is not incorrect, and I would like to know my coding is correct or not.

In addition, I would like to get the value of Me1. Please let me know how to get this value.

Thank you very much in advance.

Sincerely yours,

J1

function F = costate2(x)

t = 0;
tlast = 30; 
phi = 3.14;

ep = 0.1;
L1 = 157.5;
L2 = 799.5;
GHGs = 2954;

delta1 = 0.005;
gam1 = 0.10957;
eta11 = 0.40022;
eta21 = 0.99931;

ag1 = 20.49921;

alpha1 = 15.93413;
rhoq1 = 0.00710;
beta11 = 0.35;
beta21 = 0.63;
beta31 = 0.02;

g1 = 2.6005;
chi1 = 0.01;
xi1 = 0.02;
rhom1 = 0.0041773;

delta2 = 0.00581;
gam2 = 0.14990;
eta12 = 0.08;
eta22 = 0.59998;

ag2 = 21.49967;

alpha2 = 3.25702;
rhoq2 = 0.005;
beta12 = 0.60;
beta22 = 0.38;
beta32 = 0.02;

g2 = 2.8643;
chi2 = 0.01;
xi2 = 0.02;
rhom2 = -0.00800;
d = 0.05;
psi = 0.1;

K1 = 3486;
K2 = 1951;
Kg1 = 3.49;
Kg2 = 1.95;
Pw = 0.1;
Pg = 0.05;
GHG = 3162;
Q1 = 1496;
Q2 = 604;
E1 = 2216;
E2 = 2417;

a1 = 0.1;
a2 = 0.5;
b1 = 0.5;
b2 = 0.5;

Gbar = 12000;
P1 = 309;
P2 = 1338;

gam3 = 0.0492288;
gam4 = 0.0243931;
d1 = 1.2521405;
delta3 = 0.0250461;

lamda1 = x(1);
qg1 = x(2);
qp1 = x(3);
qe1 = x(4);
qc1 = x(5);
qk1 = x(6);

Gbar1 = (P1/(P1+P2))*Gbar;
Gbar2 = (P2/(P1+P2))*Gbar;
M1 = (g1*exp(-rhom1*t)-chi1*(Kg1/(K1+Kg1))^xi1)*E1;
W1 = (Pw-E1-Pg*(P1./(P1+P2))*Gbar)^2+4*beta31*Pw*Q1*M1*Pg;
J1 = (Pg*(P1./(P1+P2))*Gbar-Pw*E1)^2+4*beta31*Pw*Q1*M1*Pg;
Pe1 = ((Pw*E1-Pg*Gbar1) + sqrt(W1))/(2*E1);
Me1 = ((Pg*(P1./(P1+P2))*Gbar-Pw*E1)+sqrt(J1))/(2*Pg);
Me2 = (g2*exp(-rhom2*t)-chi2*(Kg2./(K2+Kg2))^xi2)*beta32*Q2;
Y1 = E1^2*Pe1^2+beta31*Pw*Q1*M1*Pg;
QQe1 = alpha1*exp(rhoq1*t)*(K1+Kg1)^beta11*L1^beta21*E1^beta31;
Op = Me1 - (P1./(P1+P2))*Gbar;
PP = P1*P2;

F(1) = x(1)*( phi/(2*tlast) )
-x(3)*( psi*(P1+P2)*Pg/(sqrt(PP)*Gbar) )*( (chi1*xi1*beta31*Pw*Q1*E1^2*Pe1*Kg1^xi1)/(Y1*(K1+Kg1)^(xi1+1)) )
-x(4)*chi1*xi1*E1*(Kg1^xi1)/(K1+Kg1)^(xi1+1)
-x(5)*a1*beta11*QQe1/(K1+Kg1)
-x(6)*b1*(beta31*Pw*Q1*Pg/(E1*Pe1^2))*( (chi1*xi1*beta31*Pw*Q1*E1^2*Pe1*Kg1^xi1)/(Y1*(K1+Kg1)^(xi1+1)) );

F(2) = x(2)*( phi/(2*tlast) + 1/ag1 ) 
-x(2).^2/(x(1)*2*ag1)
-x(1)*( 1/(2*ag1) - Pg*( (chi1*xi1*beta31*Pw*Q1*E1^2*Pe1*Kg1^xi1)*(1/(K1+Kg1)-1/Kg1)/(Y1*(K1+Kg1)^xi1) ) )
-x(3)*psi*(P1+P2)*Pg*(1/(sqrt(PP)*Gbar))* ( (chi1*xi1*beta31*Pw*Q1*E1^2*Pe1*Kg1^xi1)*(1/(K1+Kg1)-1/Kg1)/(Y1*(K1+Kg1)^xi1) )
-x(4)*chi1*xi1*E1*((Kg1/(K1+Kg1))^xi1)*(1/(K1+Kg1)-1/Kg1)
-x(5)*a1*beta11*QQe1/(K1+Kg1)
+x(6)*b1*beta31*Pw*Q1*Pg*(1/(E1*Pe1^2))*( (chi1*xi1*beta31*Pw*Q1*E1^2*Pe1*Kg1^xi1)*(1/(K1+Kg1)-1/Kg1)/(Y1*(K1+Kg1)^xi1) )
;

F(3) = x(3)*( phi/(2*tlast) )
+x(1)*( (Me1-Gbar1) - Pg*beta31*Pw*Q1*M1*(Me1-Gbar1)/Y1 )
+x(6)*b1*beta31*Pw*Q1*M1*(Me1-Gbar1)/Y1;

F(4) = x(4)*( phi/(2*tlast) )
+x(1).^((eta11*gam1+gam1)/(eta11*gam1+gam1-1)) * eta21*(ep/eta11)^(eta11*gam1/(eta11+gam1-1)) * GHGs^(-eta21*gam1/(eta11*gam1+gam1-1)) * GHG^(((eta21-eta11)*gam1-gam1+1)/(eta11*gam1+gam1-1));

F(5) = x(5)*( phi/(2*tlast) )
-x(1)*(1-Pg*beta31*Pw+M1*E1*Pe1/Y1)
-x(3)*psi*(P1+P2)*Pg*(1/(sqrt(PP)*Gbar))*beta31*Pw+M1*E1*Pe1/Y1
-x(6)*b1*beta31*Pw+M1*E1*Pe1/Y1;

F(6) = x(6)*( phi/(2*tlast) )
+x(1)*Pg*M1*(1/E1)*beta31*Pw*Q1*Pg*(Me1-Gbar1)/Y1
-x(3)*psi*(P1+P2)*Pg*(1/(sqrt(PP)*Gbar))*M1*(1/E1)*beta31*Pw*Q1*Pg*(Me1-Gbar1)/Y1
-x(4)*M1/E1
-x(5)*a1*beta31*QQe1/E1;

fun = @costate2;
x0 = [1,1,1,1,1,1];
x = fsolve(fun,x0)

[Image Analyst]

http://www.mathworks.com/matlabcentral/answers/13205#answer_18099

[Walter Roberson]

You did not present the equations in symbolic form such as an image of the equations, so as far as we as onlookers are concerned, the equations are defined by your current code, and so must be correct. We would need some other representation of the equations in order to be able to check the code against the equations.

[Star Strider]

Since ‘Me1’ does not appear to be a function of ‘x’, to display its value, put this line after the ‘Me1’ assignment:

fprintf(1, '\n\t\tMe1 = %23.15E\n', Me1)

in context:

Me1 = ((Pg*(P1./(P1+P2))*Gbar-Pw*E1)+sqrt(J1))/(2*Pg);
fprintf(1, '\n\t\tMe1 = %23.15E\n', Me1)

It will display the value of ‘Me1’ (several times) in the Command Window.

The result will be:

    Me1 =   1.476154199966165E+02

Answer 1

[Redwood]

Dear all,

Thank you very much for your help.

I revised my code to solve non-linear equations.

I got some errors, and I would like to know how to fix these errors.

Thank you very much for your help.

Sincerely yours,

J1

t = 0;
tlast = 30; 
phi = 3.14;

ep = 0.1;
L1 = 157.5;
L2 = 799.5;
GHGs = 2954;

delta1 = 0.005;
gam1 = 0.10957;
eta11 = 0.40022;
eta21 = 0.99931;

ag1 = 20.49921;

alpha1 = 15.93413;
rhoq1 = 0.00710;
beta11 = 0.35;
beta21 = 0.63;
beta31 = 0.02;

g1 = 2.6005;
chi1 = 0.01;
xi1 = 0.02;
rhom1 = 0.0041773;

delta2 = 0.00581;
gam2 = 0.14990;
eta12 = 0.08;
eta22 = 0.59998;

ag2 = 21.49967;

alpha2 = 3.25702;
rhoq2 = 0.005;
beta12 = 0.60;
beta22 = 0.38;
beta32 = 0.02;

g2 = 2.8643;
chi2 = 0.01;
xi2 = 0.02;
rhom2 = -0.00800;
d = 0.05;
psi = 0.1;

K1 = 3486;
K2 = 1951;
Kg1 = 3.49;
Kg2 = 1.95;
Pw = 0.1;
Pg = 0.05;
GHG = 3162;
Q1 = 1496;
Q2 = 604;
E1 = 2216;
E2 = 2417;

a1 = 0.1;
a2 = 0.5;
b1 = 0.5;
b2 = 0.5;

Gbar = 12000;
P1 = 309;
P2 = 1338;

gam3 = 0.0492288;
gam4 = 0.0243931;
d1 = 1.2521405;
delta3 = 0.0250461;

Gbar1 = (P1/(P1+P2))*Gbar;
Gbar2 = (P2/(P1+P2))*Gbar;
M1 = (g1*exp(-rhom1*t)-chi1*(Kg1/(K1+Kg1))^xi1)*E1;
W1 = (Pw-E1-Pg*(P1./(P1+P2))*Gbar)^2+4*beta31*Pw*Q1*M1*Pg;
J1 = (Pg*(P1./(P1+P2))*Gbar-Pw*E1)^2+4*beta31*Pw*Q1*M1*Pg;
Pe1 = ((Pw*E1-Pg*Gbar1) + sqrt(W1))/(2*E1);

Me1 = ((Pg*(P1./(P1+P2))*Gbar-Pw*E1)+sqrt(J1))/(2*Pg ); 
Me2 = (g2*exp(-rhom2*t)-chi2*(Kg2./(K2+Kg2))^xi2)*beta32*Q2;
Y1 = E1^2*Pe1^2+beta31*Pw*Q1*M1*Pg;
QQe1 = alpha1*exp(rhoq1*t)*(K1+Kg1)^beta11*L1^beta21*E1^beta31;
Op = Me1 - (P1./(P1+P2))*Gbar;
PP = P1*P2;

syms lamda1 qg1 qp1 qn1 qc1 qk1

eq1 = lamda1*( phi/(2*tlast) )
-qp1*( psi*(P1+P2)*Pg/(sqrt(PP)*Gbar) )*( (chi1*xi1*beta31*Pw*Q1*E1^2*Pe1*Kg1^xi1)/(Y1*(K1+Kg1)^(xi1+1)) )
-qn1*chi1*xi1*E1*(Kg1^xi1)/(K1+Kg1)^(xi1+1)
-qc1*a1*beta11*QQe1/(K1+Kg1)
-qk1*b1*(beta31*Pw*Q1*Pg/(E1*Pe1^2))*( (chi1*xi1*beta31*Pw*Q1*E1^2*Pe1*Kg1^xi1)/(Y1*(K1+Kg1)^(xi1+1)) );

eq2 = qg1*( phi/(2*tlast) + 1/ag1 ) 
-qg1^2/(lamda1*2*ag1)
-lamda1*( 1/(2*ag1) - Pg*( (chi1*xi1*beta31*Pw*Q1*E1^2*Pe1*Kg1^xi1)*(1/(K1+Kg1)-1/Kg1)/(Y1*(K1+Kg1)^xi1) ) )
-qp1*psi*(P1+P2)*Pg*(1/(sqrt(PP)*Gbar))* ( (chi1*xi1*beta31*Pw*Q1*E1^2*Pe1*Kg1^xi1)*(1/(K1+Kg1)-1/Kg1)/(Y1*(K1+Kg1)^xi1) )
-qn1*chi1*xi1*E1*((Kg1/(K1+Kg1))^xi1)*(1/(K1+Kg1)-1/Kg1)
-qc1*a1*beta11*QQe1/(K1+Kg1)
+qk1*b1*beta31*Pw*Q1*Pg*(1/(E1*Pe1^2))*( (chi1*xi1*beta31*Pw*Q1*E1^2*Pe1*Kg1^xi1)*(1/(K1+Kg1)-1/Kg1)/(Y1*(K1+Kg1)^xi1) );

eq3 = qp1*( phi/(2*tlast) )
+lamda1*( (Me1-Gbar1) - Pg*beta31*Pw*Q1*M1*(Me1-Gbar1)/Y1 )
+qk1*b1*beta31*Pw*Q1*M1*(Me1-Gbar1)/Y1;

eq4 = qn1*( phi/(2*tlast) )
+lamda1^((eta11*gam1+gam1)/(eta11*gam1+gam1-1)) * eta21*(ep/eta11)^(eta11*gam1/(eta11+gam1-1)) * GHGs^(-eta21*gam1/(eta11*gam1+gam1-1)) * GHG^(((eta21-eta11)*gam1-gam1+1)/(eta11*gam1+gam1-1));

eq5 = qc1*( phi/(2*tlast) )
-lamda1*(1-Pg*beta31*Pw+M1*E1*Pe1/Y1)
-qp1*psi*(P1+P2)*Pg*(1/(sqrt(PP)*Gbar))*beta31*Pw+M1*E1*Pe1/Y1
-qk1*b1*beta31*Pw+M1*E1*Pe1/Y1;

eq6 = qk1*( phi/(2*tlast) )
+lamda1*Pg*M1*(1/E1)*beta31*Pw*Q1*Pg*(Me1-Gbar1)/Y1
-qp1*psi*(P1+P2)*Pg*(1/(sqrt(PP)*Gbar))*M1*(1/E1)*beta31*Pw*Q1*Pg*(Me1-Gbar1)/Y1
-qn1*M1/E1
-qc1*a1*beta31*QQe1/E1;

sol = solve( eq1,eq2, eq3. eq4, eq5, eq6 );
sol. lamda1;
sol. qg1;
sol. qp1;
sol. qn1;
sol. qc1;
sol. qk1;

[Walter Roberson]

You are missing continuation marks for your eq* variables.

eq1 = lamda1*( phi/(2*tlast) )
-qp1*( psi*(P1+P2)*Pg/(sqrt(PP)*Gbar) )*( (chi1*xi1*beta31*Pw*Q1*E1^2*Pe1*Kg1^xi1)/(Y1*(K1+Kg1)^(xi1+1)) )

is not considered all one long expression: it is considered an assignment, followed by the calculation and display (but no assignment) of the expression on the second line.

You would need

eq1 = lamda1*( phi/(2*tlast) ) ...
-qp1*( psi*(P1+P2)*Pg/(sqrt(PP)*Gbar) )*( (chi1*xi1*beta31*Pw*Q1*E1^2*Pe1*Kg1^xi1)/(Y1*(K1+Kg1)^(xi1+1)) )

Answer 2

[Redwood]

Dear Walter,

Thank you very much for your comment.

I revised my coding, and I still got some errors.

Please let me know how to fix this problem.

Thank you very much in advance.

Sincerely yours,

J1

t = 0;
tlast = 30; 
phi = 3.14;

ep = 0.1;
L1 = 157.5;
L2 = 799.5;
GHGs = 2954;

delta1 = 0.005;
gam1 = 0.10957;
eta11 = 0.40022;
eta21 = 0.99931;

ag1 = 20.49921;

alpha1 = 15.93413;
rhoq1 = 0.00710;
beta11 = 0.35;
beta21 = 0.63;
beta31 = 0.02;

g1 = 2.6005;
chi1 = 0.01;
xi1 = 0.02;
rhom1 = 0.0041773;

delta2 = 0.00581;
gam2 = 0.14990;
eta12 = 0.08;
eta22 = 0.59998;

ag2 = 21.49967;

alpha2 = 3.25702;
rhoq2 = 0.005;
beta12 = 0.60;
beta22 = 0.38;
beta32 = 0.02;

g2 = 2.8643;
chi2 = 0.01;
xi2 = 0.02;
rhom2 = -0.00800;
d = 0.05;
psi = 0.1;

K1 = 3486;
K2 = 1951;
Kg1 = 3.49;
Kg2 = 1.95;
Pw = 0.1;
Pg = 0.05;
GHG = 3162;
Q1 = 1496;
Q2 = 604;
E1 = 2216;
E2 = 2417;

a1 = 0.1;
a2 = 0.5;
b1 = 0.5;
b2 = 0.5;

Gbar = 12000;
P1 = 309;
P2 = 1338;

gam3 = 0.0492288;
gam4 = 0.0243931;
d1 = 1.2521405;
delta3 = 0.0250461;

Gbar1 = (P1/(P1+P2))*Gbar;
Gbar2 = (P2/(P1+P2))*Gbar;
M1 = (g1*exp(-rhom1*t)-chi1*(Kg1/(K1+Kg1))^xi1)*E1;
W1 = (Pw-E1-Pg*(P1./(P1+P2))*Gbar)^2+4*beta31*Pw*Q1*M1*Pg;
J1 = (Pg*(P1./(P1+P2))*Gbar-Pw*E1)^2+4*beta31*Pw*Q1*M1*Pg;
Pe1 = ((Pw*E1-Pg*Gbar1) + sqrt(W1))/(2*E1);

Me1 = ((Pg*(P1./(P1+P2))*Gbar-Pw*E1)+sqrt(J1))/(2*Pg ); 
Me2 = (g2*exp(-rhom2*t)-chi2*(Kg2./(K2+Kg2))^xi2)*beta32*Q2;
Y1 = E1^2*Pe1^2+beta31*Pw*Q1*M1*Pg;
QQe1 = alpha1*exp(rhoq1*t)*(K1+Kg1)^beta11*L1^beta21*E1^beta31;
Op = Me1 - (P1./(P1+P2))*Gbar;
PP = P1*P2;

syms lamda1 qg1 qp1 qn1 qc1 qk1

eq1 = lamda1*( phi/(2*tlast) )...
-qp1*( psi*(P1+P2)*Pg/(sqrt(PP)*Gbar) )*( (chi1*xi1*beta31*Pw*Q1*E1^2*Pe1*Kg1^xi1)/(Y1*(K1+Kg1)^(xi1+1)) ) 
-qn1*chi1*xi1*E1*(Kg1^xi1)/(K1+Kg1)^(xi1+1) 
-qc1*a1*beta11*QQe1/(K1+Kg1) 
-qk1*b1*(beta31*Pw*Q1*Pg/(E1*Pe1^2))*( (chi1*xi1*beta31*Pw*Q1*E1^2*Pe1*Kg1^xi1)/(Y1*(K1+Kg1)^(xi1+1)) );

eq2 = qg1*( phi/(2*tlast) + 1/ag1 )... 
-qg1^2/(lamda1*2*ag1) -lamda1*( 1/(2*ag1) - Pg*( (chi1*xi1*beta31*Pw*Q1*E1^2*Pe1*Kg1^xi1)*(1/(K1+Kg1)-1/Kg1)/(Y1*(K1+Kg1)^xi1) ) ) 
-qp1*psi*(P1+P2)*Pg*(1/(sqrt(PP)*Gbar))* ( (chi1*xi1*beta31*Pw*Q1*E1^2*Pe1*Kg1^xi1)*(1/(K1+Kg1)-1/Kg1)/(Y1*(K1+Kg1)^xi1) ) 
-qn1*chi1*xi1*E1*((Kg1/(K1+Kg1))^xi1)*(1/(K1+Kg1)-1/Kg1) 
-qc1*a1*beta11*QQe1/(K1+Kg1) 
+qk1*b1*beta31*Pw*Q1*Pg*(1/(E1*Pe1^2))*( (chi1*xi1*beta31*Pw*Q1*E1^2*Pe1*Kg1^xi1)*(1/(K1+Kg1)-1/Kg1)/(Y1*(K1+Kg1)^xi1) );

eq3 = qp1*( phi/(2*tlast) )...
+lamda1*( (Me1-Gbar1) - Pg*beta31*Pw*Q1*M1*(Me1-Gbar1)/Y1 ) 
+qk1*b1*beta31*Pw*Q1*M1*(Me1-Gbar1)/Y1;

eq4 = qn1*( phi/(2*tlast) )...
+lamda1^((eta11*gam1+gam1)/(eta11*gam1+gam1-1)) * eta21*(ep/eta11)^(eta11*gam1/(eta11+gam1-1)) * GHGs^(-eta21*gam1/(eta11*gam1+gam1-1)) * GHG^(((eta21-eta11)*gam1-gam1+1)/(eta11*gam1+gam1-1));

eq5 = qc1*( phi/(2*tlast) )...
-lamda1*(1-Pg*beta31*Pw+M1*E1*Pe1/Y1) 
-qp1*psi*(P1+P2)*Pg*(1/(sqrt(PP)*Gbar))*beta31*Pw+M1*E1*Pe1/Y1 
-qk1*b1*beta31*Pw+M1*E1*Pe1/Y1;

eq6 = qk1*( phi/(2*tlast) )...
+lamda1*Pg*M1*(1/E1)*beta31*Pw*Q1*Pg*(Me1-Gbar1)/Y1 
-qp1*psi*(P1+P2)*Pg*(1/(sqrt(PP)*Gbar))*M1*(1/E1)*beta31*Pw*Q1*Pg*(Me1-Gbar1)/Y1 
-qn1*M1/E1 -qc1*a1*beta31*QQe1/E1;

sol = solve( eq1,eq2, eq3. eq4, eq5, eq6 );
sol. lamda1;
sol. qg1;
sol. qp1;
sol. qn1;
sol. qc1;
sol. qk1;

[Walter Roberson]

You accidentally wrote

sol = solve( eq1,eq2, eq3. eq4, eq5, eq6 );

instead of

sol = solve( eq1,eq2, eq3, eq4, eq5, eq6 );

[Redwood]

Dear Walter,

Thank you so much!!!

Sincerely yours,

J1

Answer 3

[Redwood]

Dear Walter,

I tried to put "..."

For example,

eq1 = lamda1*( phi/(2*tlast) )... 
-qp1*( psi*(P1+P2)*Pg/(sqrt(PP)*Gbar) )*( (chi1*xi1*beta31*Pw*Q1*E1^2*Pe1*Kg1^xi1)/(Y1*(K1+Kg1)^(xi1+1)) )... 
-qn1*chi1*xi1*E1*(Kg1^xi1)/(K1+Kg1)^(xi1+1)...
-qc1*a1*beta11*QQe1/(K1+Kg1)...
-qk1*b1*(beta31*Pw*Q1*Pg/(E1*Pe1^2))*( (chi1*xi1*beta31*Pw*Q1*E1^2*Pe1*Kg1^xi1)/(Y1*(K1+Kg1)^(xi1+1)) );

But I also got some errors. Please let me know how to fix this problem

Thank you very much in advance.

Sincerely yours,

J1

Answer 4

[Redwood]

Dear Walter,

I got some answers and I think my code has no errors.

But I cannot see the values of the unknown variables(lamda1, qg1, qp1, qn1, qc1, qk1), and I am not sure if I solver non-linear equations or not.

Please let me know how to get the values of the unknown variables(lamda1, qg1, qp1, qn1, qc1, qk1).

Thank you very much in advance.

Sincerely yours,

J1

t = 0; tlast = 30; phi = 3.14;

ep = 0.1; L1 = 157.5; L2 = 799.5; GHGs = 2954;

delta1 = 0.005; gam1 = 0.10957; eta11 = 0.40022; eta21 = 0.99931;

ag1 = 20.49921;

alpha1 = 15.93413; rhoq1 = 0.00710; beta11 = 0.35; beta21 = 0.63; beta31 = 0.02;

g1 = 2.6005; chi1 = 0.01; xi1 = 0.02; rhom1 = 0.0041773;

delta2 = 0.00581; gam2 = 0.14990; eta12 = 0.08; eta22 = 0.59998;

ag2 = 21.49967;

alpha2 = 3.25702; rhoq2 = 0.005; beta12 = 0.60; beta22 = 0.38; beta32 = 0.02;

g2 = 2.8643; chi2 = 0.01; xi2 = 0.02; rhom2 = -0.00800; d = 0.05; psi = 0.1;

K1 = 3486; K2 = 1951; Kg1 = 3.49; Kg2 = 1.95; Pw = 0.1; Pg = 0.05; GHG = 3162; Q1 = 1496; Q2 = 604; E1 = 2216; E2 = 2417;

a1 = 0.1; a2 = 0.5; b1 = 0.5; b2 = 0.5;

Gbar = 12; P1 = 309; P2 = 1338;

gam3 = 0.0492288; gam4 = 0.0243931; d1 = 1.2521405; delta3 = 0.0250461;

Gbar1 = (P1/(P1+P2))*Gbar; Gbar2 = (P2/(P1+P2))*Gbar; M1 = (g1*exp(-rhom1*t)-chi1*(Kg1/(K1+Kg1))^xi1)*E1; W1 = (Pw-E1-Pg*(P1./(P1+P2))*Gbar)^2+4*beta31*Pw*Q1*M1*Pg; J1 = (Pg*(P1./(P1+P2))*Gbar-Pw*E1)^2+4*beta31*Pw*Q1*M1*Pg; Pe1 = ((Pw*E1-Pg*Gbar1) + sqrt(W1))/(2*E1);

Me1 = ((Pg*(P1./(P1+P2))*Gbar-Pw*E1)+sqrt(J1))/(2*Pg ); Me2 = (g2*exp(-rhom2*t)-chi2*(Kg2./(K2+Kg2))^xi2)*beta32*Q2; Y1 = E1^2*Pe1^2+beta31*Pw*Q1*M1*Pg; QQe1 = alpha1*exp(rhoq1*t)*(K1+Kg1)^beta11*L1^beta21*E1^beta31; Op = Me1 - (P1./(P1+P2))*Gbar; PP = P1*P2;

syms lamda1 qg1 qp1 qn1 qc1 qk1

eq1 = lamda1*( -phi/(2*tlast) )... +qp1*( psi*(P1+P2)*Pg/(sqrt(PP)*Gbar) )*( (chi1*xi1*beta31*Pw*Q1*E1^2*Pe1*Kg1^xi1)/(Y1*(K1+Kg1)^(xi1+1)) ) +qn1*chi1*xi1*E1*(Kg1^xi1)/(K1+Kg1)^(xi1+1) +qc1*a1*beta11*QQe1/(K1+Kg1) -qk1*b1*(beta31*Pw*Q1*Pg/(E1*Pe1^2))*( (chi1*xi1*beta31*Pw*Q1*E1^2*Pe1*Kg1^xi1)/(Y1*(K1+Kg1)^(xi1+1)) );

eq2 = qg1*( -phi/(2*tlast) )... +lamda1*( 1/(2*ag1) - Pg*( (chi1*xi1*beta31*Pw*Q1*E1^2*Pe1*Kg1^xi1)*(1/(K1+Kg1)-1/Kg1)/(Y1*(K1+Kg1)^xi1) ) ) +qp1*psi*(P1+P2)*Pg*(1/(sqrt(PP)*Gbar))* ( (chi1*xi1*beta31*Pw*Q1*E1^2*Pe1*Kg1^xi1)*(1/(K1+Kg1)-1/Kg1)/(Y1*(K1+Kg1)^xi1) ) +qn1*chi1*xi1*E1*((Kg1/(K1+Kg1))^xi1)*(1/(K1+Kg1)-1/Kg1) +qc1*a1*beta11*QQe1/(K1+Kg1) -qk1*b1*beta31*Pw*Q1*Pg*(1/(E1*Pe1^2))*( (chi1*xi1*beta31*Pw*Q1*E1^2*Pe1*Kg1^xi1)*(1/(K1+Kg1)-1/Kg1)/(Y1*(K1+Kg1)^xi1) );

eq3 = qp1*( -phi/(2*tlast) )... -lamda1*( (Me1-Gbar1) - Pg*beta31*Pw*Q1*M1*(Me1-Gbar1)/Y1 ) -qk1*b1*beta31*Pw*Q1*M1*(Me1-Gbar1)/Y1;

eq4 = qn1*( phi/(2*tlast) )... +lamda1^((eta11*gam1+gam1)/(eta11*gam1+gam1-1)) * eta21*(ep/eta11)^(eta11*gam1/(gam1+gam1*eta11-1)) * GHGs^(-eta21*gam1/(eta11*gam1+gam1-1)) * GHG^(((eta21-eta11)*gam1-gam1+1)/(eta11*gam1+gam1-1));

eq5 = qc1*( -phi/(2*tlast) )... +lamda1*(1-Pg*beta31*Pw+M1*E1*Pe1/Y1) +qp1*psi*(P1+P2)*Pg*(1/(sqrt(PP)*Gbar))*beta31*Pw+M1*E1*Pe1/Y1 +qk1*b1*beta31*Pw+E1^2*Pe1/Y1;

eq6 = qk1*( phi/(2*tlast) )... +lamda1*Pg*M1*(1/E1)*beta31*Pw*Q1*Pg*(Me1-Gbar1)/Y1 -qp1*psi*(P1+P2)*Pg*(1/(sqrt(PP)*Gbar))*M1*(1/E1)*beta31*Pw*Q1*Pg*(Me1-Gbar1)/Y1 -qn1*M1/E1 -qc1*a1*beta31*QQe1/E1;

sol = solve( eq1,eq2, eq3, eq4, eq5, eq6 ); sol. lamda1; sol. qg1; sol. qp1; sol. qn1; sol. qc1; sol. qk1;

disp('lamda1, qg1, qp1, qn1, qc1, qk1')

Answer 5

[Alex Sha]

try numerical solution:

lamda1: 1.48312225679967E-24

qp1: 54185.6537859035

qn1: -133.888674663691

qc1: 0.188502621518643

qk1: -6630.74381075639

qg1: 282.666798246712

Me1=76.2725002858213