Transcendental equation in two variables

I have a transcendental equation in two variables x and y as (x^2-y^2)*(-y^2+1+0.015*x^4)*xi*besselj(1,xi)+y^2*0.005*besselj(0,xi))-0.0091x^2*y^2*besselj(1,xi)*xi=0, where xi=sqrt(0.041*y^2-x^2), y varies from 0 to 3. I want to solve this equation numerically such that I can get the purely real x, purely imaginary x and complex x. And each type of roots have four or five branches. I tried to solve it using fsolve but I am not getting the solutions separately.

[Torsten]

And each type of roots have four or five branches. I tried to solve it using fsolve but I am not getting the solutions separately.

Given a value for y, you expect four or five different solutions for x ? And you don't get them all in one call to fsolve ? Or what exactly do you mean here ?

Answer 1

[Hassaan]

To run this code in MATLAB:

1. Save the function in a .m file, for example, complex_root_finding.m.
2. Execute the function in MATLAB command window by typing complex_root_finding.

This script defines a function complex_trans_eq that computes the real and imaginary parts of the transcendental equation and uses fsolve to find the roots. The results will be printed to the MATLAB command window. The initial guess z0 may need to be adjusted based on the specifics of the equation to ensure proper convergence.

complex_root_finding()

Solutions (x_real, x_imag, y):
         0         0         0
         0         0    0.0303
         0         0    0.0606
         0         0    0.0909
         0         0    0.1212
         0         0    0.1515
         0         0    0.1818
         0         0    0.2121
         0         0    0.2424
         0         0    0.2727
         0         0    0.3030
         0         0    0.3333
         0         0    0.3636
         0         0    0.3939
         0         0    0.4242
         0         0    0.4545
         0         0    0.4848
         0         0    0.5152
         0         0    0.5455
         0         0    0.5758
         0         0    0.6061
         0         0    0.6364
         0         0    0.6667
         0         0    0.6970
         0         0    0.7273
         0         0    0.7576
         0         0    0.7879
         0         0    0.8182
         0         0    0.8485
         0         0    0.8788
         0         0    0.9091
         0         0    0.9394
         0         0    0.9697
         0         0    1.0000
         0         0    1.0303
         0         0    1.0606
         0         0    1.0909
         0         0    1.1212
         0         0    1.1515
         0         0    1.1818
         0         0    1.2121
         0         0    1.2424
         0         0    1.2727
         0         0    1.3030
         0         0    1.3333
         0         0    1.3636
         0         0    1.3939
         0         0    1.4242
         0         0    1.4545
         0         0    1.4848
         0         0    1.5152
         0         0    1.5455
         0         0    1.5758
         0         0    1.6061
         0         0    1.6364
         0         0    1.6667
         0         0    1.6970
         0         0    1.7273
         0         0    1.7576
         0         0    1.7879
         0         0    1.8182
         0         0    1.8485
         0         0    1.8788
         0         0    1.9091
         0         0    1.9394
         0         0    1.9697
         0         0    2.0000
         0         0    2.0303
         0         0    2.0606
         0         0    2.0909
         0         0    2.1212
         0         0    2.1515
         0         0    2.1818
         0         0    2.2121
         0         0    2.2424
         0         0    2.2727
         0         0    2.3030
         0         0    2.3333
         0         0    2.3636
         0         0    2.3939
         0         0    2.4242
         0         0    2.4545
         0         0    2.4848

function complex_root_finding
    % Define the range for y
    y_values = linspace(0, 3, 100);

    % Store the solutions
    solutions = [];

    % Options for fsolve
    options = optimoptions('fsolve', 'Display', 'none');

    % Loop over the range of y values
    for y = y_values
        % Define the system of equations for real and imaginary parts
        fun = @(z) complex_trans_eq(z, y);

        % Initial guess: Change this if necessary
        z0 = [0, 0];

        % Solve the system of equations using fsolve
        [z_sol, fval, exitflag] = fsolve(fun, z0, options);

        % If a solution is found, store it
        if exitflag > 0  % Solution converged
            solutions = [solutions; [z_sol, y]];
        end
    end

    % Display the solutions
    disp('Solutions (x_real, x_imag, y):');
    disp(solutions);
end

function F = complex_trans_eq(z, y)
    x_real = z(1);
    x_imag = z(2);
    x = x_real + 1i * x_imag;
    xi = sqrt(0.041*y^4*x^2 - x^2);

    % Real part of the function
    F_real = real((x^2 - y^2)*(y^2 + 1 + 0.015*x^4)*x*besselj(1, xi) + ...
              y^2*0.005*besselj(0, xi) - 0.0091*x^2*y^2*besselj(1, xi));

    % Imaginary part of the function
    F_imag = imag((x^2 - y^2)*(y^2 + 1 + 0.015*x^4)*x*besselj(1, xi) + ...
              y^2*0.005*besselj(0, xi) - 0.0091*x^2*y^2*besselj(1, xi));

    % Return a vector of the real and imaginary parts
    F = [F_real; F_imag];
end

Code may need to be adjusted as per your needs. Thank you.

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