how to solve nonlinear equations using boundary conditions?

Hi all,

I have two sets of equation.

 m_c_theta0=(m_c0-1/M+1)*cos(theta0)+j_L0*sin(theta0)+1/M-1;
j_L_theta0=(-m_c0+1/M-1)*sin(theta0)+j_L0*cos(theta0);
j_M0=j_L_theta0-l*theta0;
m_m_theta0=1;
m_c_gama=(m_c_theta0-1/M-1)*cos(gama-theta0)... 
          +j_L_theta0*sin(gama-theta0)+1/M+1;
j_L_gama=(-m_c_theta0+1/M+1)*sin(gama-theta0) ...
          +j_L_theta0*cos(gama-theta0);
j_M_gama=j_L_theta0-l*(gama-theta0);
m_m_gama=-1;

boundary conditions are

    [ j_L_gama+x(1); %curernt
             m_c_gama+x(2); %voltage
             j_M0+j_M_gama; 
          m_m_theta0+1;      
            ];

where,

     l=0.2, F=0.5:0.1:1.5; gama=pi/F;

I need to find the initial,

j_L0=x(1);
m_c0=x(2);
theta0=x(3);
M=x(4);
for above F range and plot(F,M)

I want to do it by using fsolve.

can anyone please help?

[safisay]

Can anyone please help?

[Walter Roberson]

How are those boundary conditions to be interpreted? Is there an implicit "= 0" on each one? So j_L_gama+x(1) = 0, m_c_gama+x(2)=0, and so on? If so then the equation

m_m_theta0=1;

contradicts m_m_theta0+1 = 0

[safisay]

yes, they are like j_L_gama+x(1) = 0, m_c_gama+x(2)=0.

Sorry, I did not realize that. it should be like this

    m_m_theta0=-1;

[Walter Roberson]

If I ignore m_m_theta0+1=0 as a boundary condition in favour of the explicitly stated m_m_theta0=1 then I appear to derive two formulas for M from the equations:

M = 10*F*(sin((1/2)*(10*F*j_M_gama-Pi)/F)+sin((1/2)*(10*F*j_M_gama+Pi)/F))/(10*F*(m_c_theta0-1)*sin((1/2)*(10*F*j_M_gama-Pi)/F)+Pi*cos((1/2)*(10*F*j_M_gama-Pi)/F)+10*F*(m_c_theta0+1)*sin((1/2)*(10*F*j_M_gama+Pi)/F)+cos((1/2)*(10*F*j_M_gama+Pi)/F)*Pi)

M = 10*F*(cos((1/2)*(10*F*j_M_gama-Pi)/F)+cos((1/2)*(10*F*j_M_gama+Pi)/F)-2)/(10*F*(m_c_theta0-1)*cos((1/2)*(10*F*j_M_gama-Pi)/F)-Pi*sin((1/2)*(10*F*j_M_gama-Pi)/F)+10*F*(m_c_theta0+1)*cos((1/2)*(10*F*j_M_gama+Pi)/F)-sin((1/2)*(10*F*j_M_gama+Pi)/F)*Pi)

provided that I assume all quantities are real-valued.

At one point an equation seems to fall out that

j_M_gama = 2*j_L_theta0-Pi/(5*F)+j_M_gama

which implies that 2*j_L_theta0 = Pi/(5*F)

If the task is to rewrite all of the equations in terms of j_L0, m_c0, theta0, M and F then I am not sure that is possible, but I have not gone through to rigorously check.

[safisay]

note that: m_m_theta0=-1.

all quantities are real valued. Is it possible to find the solutions numerically using fsolve rather than finding a symbolic equation?

and plot(M vs F)

[Walter Roberson]

The value of m_m_theta0 does not matter to the code, as you never use the value.

"Is it possible to find the solutions numerically using fsolve rather than finding a symbolic equation?"

I would need to go back through my steps, but I do not think this can be done without there either being more equations or at least one variable (two, more likely) having been assigned a numeric value. The number of equations effectively usable for solving appears to be less than the number of variables to be solved for.

[safisay]

I have solved the equations for specific F. I have attached the code.

I am just not quite sure how to do that for a range of F.

[safisay]

Can anyone please help?

Answer 1

[Walter Roberson]

options=optimset('Display','on',...
    'Algorithm','trust-region-reflective','LargeScale','off','MaxFunEvals',20000);   

l=0.4; 

Fvals = 0.5:0.1:1.5;
numF = length(Fvals);
j_L0 = zeros(1, numF);
m_c0 = zeros(1, numF);
theta0 = zeros(1, numF);
M = zeros(1,numF);

for F_idx = 1 : numF
  F = Fvals(F_idx);
  func_CC=@(x)myfun_CC(x,l,F);
  x0=[ 1, -1.5, 2 1]; % Make a starting guess at the solution
  x = fsolve(func_CC, x0, options);  % Call solver
  j_L0(F_idx) = x(1);
  m_c0(F_idx) = x(2);
  theta0(F_idx) = x(3);
  M(F_idx) = x(4);
end