MATLAB: How to find theta_k in terms of theta_i and theta_j in the following formula

numerical solution

 Theta_k *sqrt(1+theta_k^2) + arcsinh(theta_k) = {theta_j* sqrt(1+theta_j^2)+arcsinh(theta_j)}+{theta_i * sqrt(1+theta_i^2)+arcsinh(theta_i)}

Best Answer

 F = @(theta_k) (theta_k .* sqrt(1+theta_k.^2) + asinh(theta_k)) - ( (theta_j .* sqrt(1+theta_j.^2) + asinh(theta_j)) + (theta_i .* sqrt(1+theta_i.^2) + asinh(theta_i)));
 
 guess = rand();
 
 fsolve(F, guess)

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MATLAB: Solving non linear equation using fsolve

You are not passing theta_n and other parameters to root2d(). One fix is to make root2d a Nested Function. Also, you must choose a different initial point. phi0 = [10,10] worked for me.

function solve_it()
    beta_a = 10;
    alpha_n = 10 ;
    eta= 18;
    i = 18;
    theta_i = 10;
    theta_n = 10 ;
    
    fun = @root2d;
    phi0 = [10,10];
    phi = fsolve(fun,phi0);
    phi_n = phi(1),
    phi_i = phi(2),
    function F = root2d(phi)
    F = zeros(2,1);
    F(1) = (pi*sin((pi*(phi(1) + theta_n))/180)*cos((pi*phi(2))/180)*(cos((pi*theta_n)/180) + tan((pi*i)/180)*tan((pi*theta_i)/180)))/(180*sin((pi*phi(1))/180)*(cos((pi*(phi(1) + theta_n))/180)*cos((phi(2)*pi)/180) + sin((phi(2)*pi)/180)*tan((theta_i*pi)/180))^2) - (pi*cos((pi*phi(1))/180)*(cos((pi*theta_n)/180) + tan((pi*i)/180)*tan((pi*theta_i)/180)))/(180*sin((phi(1)*pi)/180)^2*(cos((pi*(phi(1) + theta_n))/180)*cos((pi*phi(2))/180) + sin((pi*phi(2))/180)*tan((pi*theta_i)/180)));
    F(2) = ((cos((pi*theta_n)/180) + tan((pi*i)/180)*tan((pi*theta_i)/180))*((pi*cos((pi*(phi(1) + theta_n))/180)*sin((pi*phi(2))/180))/180 - (pi*cos((pi*phi(2))/180)*tan((pi*theta_i)/180))/180))/(sin((pi*phi(1))/180)*(cos((pi*(phi(1) + theta_n))/180)*cos((phi(2)*pi)/180) + sin((phi(2)*pi)/180)*tan((theta_i*pi)/180))^2);
    end
end

MATLAB: Plot of theta_new and cos theta

After the first

    plot(theta_new,cos(theta_new),'*')

add

   hold on

and remove the second plot() call.

Best Answer

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MATLAB: Solving non linear equation using fsolve

MATLAB: Plot of theta_new and cos theta

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