solving a system of equations

clc
clear all
M1=3
P1=101325 %pa or 1 atm
theta2 = .349 %rad or 20 degrees
theta3 =.262 %rad or 15 degrees
 %first get P2 and P3
 %area 2
gamma=1.4
lambda2=sqrt((((M1.^2)-1).^2)-3.*(1+((gamma-1)/2).*(M1.^2)).*(1+((gamma+1)/2).*(M1.^2)).*(tan(theta2)).^2);
chi2=(1./(lambda2.^3)).*((((M1.^2)-1).^3)-9.*(1+((gamma-1)./2).*(M1.^2)).*(1+((gamma-1)/2).*(M1.^2)+((gamma+1)./4).*(M1.^4)).*((tan(theta2)).^2));

beta2= atan(((M1.^2)-1+2.*lambda2.*cos((4.*pi+acos(chi2))./3))./(3.*(1+((gamma-1)/2).*(M1.^2)).*tan(theta2)))

M1n2=M1*sin(beta2)
M2n=(((M1n2^2) +(2/(gamma-1)))/((((2*gamma)/(gamma-1))*(M1n2^2)-1)))^(1/2)
M2=M2n/sin(beta2-theta2)
P2=P1*(1+((2*gamma)/(gamma+1))*(M1n2^2 -1))

% repeat for P3
lambda3=sqrt((((M1.^2)-1).^2)-3.*(1+((gamma-1)/2).*(M1.^2)).*(1+((gamma+1)/2).*(M1.^2)).*(tan(theta3)).^2);
chi3=(1./(lambda3.^3)).*((((M1.^2)-1).^3)-9.*(1+((gamma-1)./2).*(M1.^2)).*(1+((gamma-1)/2).*(M1.^2)+((gamma+1)./4).*(M1.^4)).*((tan(theta3)).^2));

beta3= atan(((M1.^2)-1+2.*lambda3.*cos((4.*pi+acos(chi3))./3))./(3.*(1+((gamma-1)/2).*(M1.^2)).*tan(theta3)))

M1n3=M1*sin(beta3)
M3n=(((M1n3^2) +(2/(gamma-1)))/((((2*gamma)/(gamma-1))*(M1n3^2)-1)))^(1/2)
M3=M3n/sin(beta3-theta3)
P3=P1*(1+((2*gamma)/(gamma+1))*(M1n3^2 -1))

%solve for 5 variables 5 unknowns


syms P4 P4p theta4 theta4p beta4 beta4p
P4=P4p

P4=P3*(1+((2*gamma)/(gamma+1))*((M3^2)*((sin(beta4))^2)-1))
P4p=P2*(1+((2*gamma)/(gamma+1))*((M2^2)*((sin(beta4p))^2)-1))
theta4=atan(2*cot(beta4)*((((M3^2)*((sin(beta4))^2)-1))/((M3^2)*(gamma+cos(2*beta4))+2)))
theta4p=atan(2*cot(beta4p)*((((M2^2)*((sin(beta4p))^2)-1))/((M2^2)*(gamma+cos(2*beta4p))+2)))

theta2+theta4p=theta3+theta4

above is my code, i need to solve for theta4, theta4p, beta4, beta4p, and either P4 or P4p, since these two are equal.

not sure which solver would work for this

Answer 1

[Torsten]

M1=3

M1 = 3

P1=101325 %pa or 1 atm

P1 = 101325

theta2 = .349 %rad or 20 degrees

theta2 = 0.3490

theta3 =.262 %rad or 15 degrees

theta3 = 0.2620

%first get P2 and P3
%area 2
gamma=1.4

gamma = 1.4000

lambda2=sqrt((((M1.^2)-1).^2)-3.*(1+((gamma-1)/2).*(M1.^2)).*(1+((gamma+1)/2).*(M1.^2)).*(tan(theta2)).^2);
chi2=(1./(lambda2.^3)).*((((M1.^2)-1).^3)-9.*(1+((gamma-1)./2).*(M1.^2)).*(1+((gamma-1)/2).*(M1.^2)+((gamma+1)./4).*(M1.^4)).*((tan(theta2)).^2));
beta2= atan(((M1.^2)-1+2.*lambda2.*cos((4.*pi+acos(chi2))./3))./(3.*(1+((gamma-1)/2).*(M1.^2)).*tan(theta2)))

beta2 = 0.6590

M1n2=M1*sin(beta2)

M1n2 = 1.8370

M2n=(((M1n2^2) +(2/(gamma-1)))/((((2*gamma)/(gamma-1))*(M1n2^2)-1)))^(1/2)

M2n = 0.6084

M2=M2n/sin(beta2-theta2)

M2 = 1.9943

P2=P1*(1+((2*gamma)/(gamma+1))*(M1n2^2 -1))

P2 = 3.8204e+05

% repeat for P3
lambda3=sqrt((((M1.^2)-1).^2)-3.*(1+((gamma-1)/2).*(M1.^2)).*(1+((gamma+1)/2).*(M1.^2)).*(tan(theta3)).^2);
chi3=(1./(lambda3.^3)).*((((M1.^2)-1).^3)-9.*(1+((gamma-1)./2).*(M1.^2)).*(1+((gamma-1)/2).*(M1.^2)+((gamma+1)./4).*(M1.^4)).*((tan(theta3)).^2));
beta3= atan(((M1.^2)-1+2.*lambda3.*cos((4.*pi+acos(chi3))./3))./(3.*(1+((gamma-1)/2).*(M1.^2)).*tan(theta3)))

beta3 = 0.5629

M1n3=M1*sin(beta3)

M1n3 = 1.6009

M3n=(((M1n3^2) +(2/(gamma-1)))/((((2*gamma)/(gamma-1))*(M1n3^2)-1)))^(1/2)

M3n = 0.6682

M3=M3n/sin(beta3-theta3)

M3 = 2.2543

P3=P1*(1+((2*gamma)/(gamma+1))*(M1n3^2 -1))

P3 = 2.8609e+05

%solve for 5 variables 5 unknowns
syms P4 P4p theta4 theta4p beta4 beta4p
eqn1 = P4==P4p

eqn1 = 

eqn2 = P4==P3*(1+((2*gamma)/(gamma+1))*((M3^2)*((sin(beta4))^2)-1))

eqn2 = 

eqn3 = P4p==P2*(1+((2*gamma)/(gamma+1))*((M2^2)*((sin(beta4p))^2)-1))

eqn3 = 

eqn4 = theta4==atan(2*cot(beta4)*((((M3^2)*((sin(beta4))^2)-1))/((M3^2)*(gamma+cos(2*beta4))+2)))

eqn4 = 

eqn5 = theta4p==atan(2*cot(beta4p)*((((M2^2)*((sin(beta4p))^2)-1))/((M2^2)*(gamma+cos(2*beta4p))+2)))

eqn5 = 

eqn6 = theta2+theta4p==theta3+theta4

eqn6 = 

S = vpasolve([eqn1,eqn2,eqn3,eqn4,eqn5,eqn6],[P4 P4p theta4 theta4p beta4 beta4p])%,[0 inf;0 inf;0 2*pi;0 2*pi;0 2*pi;0 2*pi])

S = struct with fields:
         P4: 286008.97317717601854893168488697
        P4p: 286008.97317717601854893168488697
     theta4: 0.00008460289275915508457608111229808
    theta4p: -0.086915397107240844915423918887702
      beta4: -163.82235899238345755662732994989
     beta4p: 9.8849808344785758940601584211874