ODE45 shooting method

Hello

I want to solve a system of 1st order ODE's using ODE45. To apply the shooting method I want to solve for the inital values z0 = [7 z]. The last y-value of the interval y(2) should then be a function of z. I want to plot this y-end-value function with z = linspace(-60,0,60).

In the following I am trying to find this function, but without luck.

clc; clear
syms z
tspan = [0 2];  
z0 = [7 z]; 
[x,y] = ode45(@fun,tspan,z0)

, where @fun is:

function dy = fun(x,y)
dy = zeros(2,1);
    dy(1) = y(2);
    dy(2) = y(1)^2/7 - x/7 - y(2)/7;
 end

[darova]

ode45 is numerical solver. You can't find solution if you use symbolic variables

You have second order ODE (second derivative). Where is the second initial/boundar condition?

Answer 1

[Ameer Hamza]

It appears that a symbolic solution to you equation does not exist and ode45 cannot solve in term of a symbolic variable. To solve it for different initial conditions, you will need to use a for-loop.

z = linspace(-60,0,60);

y2_last = zeros(size(z));
for i=1:numel(z)
    tspan = [0 2];  
    z0 = [7 z(i)]; 
    [x,y] = ode45(@fun,tspan,z0);
    y2_last(i) = y(end,2);
end
plot(z, y2_last);

function dy = fun(x,y)
dy = zeros(2,1);
    dy(1) = y(2);
    dy(2) = y(1)^2/7 - x/7 - y(2)/7;
 end

[Frederik Rolighed Christensen]

Exactly, I figured it out at last. Thanks for the reply though!

/Frederik

[Ameer Hamza]

Glad to be of help.

Answer 2

[Bùi Danh]

function shooting_method()
    % Set the initial guess for initial condition 
    y0_guess = 0;
    
    % Set the shooting parameter guess
    lambda_guess = 1;
    
    % Set the desired boundary condition
    yf_desired = 2;
    
    % Set the tolerance for convergence
    tol = 1e-6;
    
    % Set the maximum number of iterations
    max_iter = 100;
    
    % Initialize the error and iteration counter
    error = 1;
    iter = 1;
    
    % Main loop
    while error > tol && iter <= max_iter
        % Solve the initial value problem using the guessed initial condition
        [t, y] = ode45(@ode_func, [0, 1], [y0_guess, lambda_guess]);
        
        % Get the final value of y from the solution
        yf = y(end, 1);
        
        % Compute the error
        error = abs(yf - yf_desired);
        
        % Compute the derivative of y at t=1
        y_deriv = y(end, 2);
        
        % Compute the update for the initial condition guess
        y0_guess = y0_guess - (yf - yf_desired) / y_deriv;
        
        % Update the iteration counter
        iter = iter + 1;
    end
    
    % Print the results
    disp(['Final value of y: ', num2str(yf)]);
    disp(['Number of iterations: ', num2str(iter)]);
    
    % Plot the solution
    plot(t, y(:, 1));
    xlabel('t');
    ylabel('y');
    title('Solution of the boundary value problem');
end

function dydt = ode_func(t, y)
    % Define the ODE system
    dydt = zeros(2, 1);
    dydt(1) = y(2);
    dydt(2) = y(1) + t;
end