How can I clear an error "The expression to the left of the equals sign is not a valid target for an assignment?"

I tried to solve a problem (see the attachments). But I gor an error "The expression to the left of the equals sign is not a valid target for an assignment". How to solve this issue.

function plot_velocity_vs_eta()
    % Parameters
    A = 0.5;
    Gr = 0.7;
    Gc = 0.5;
    Kp = 3.0;
    beta = 0.5;
    Pr = 0.3;
    Df = 0.2;
    Sc = 0.1;
    L0 = 0.5;
    Sr = 0.3;
    M = [1,2,3,4,5];
    

    % Initialize plot
    figure;
    hold on;

    % Iterate over each Prandtl number
    for i = 1:length(M_values)
        M = M_values(i);

        % Define the system of equations
        equations = @(eta, y) [
            dy(1)=y(2);                                                       
            dy(2)=y(3);                                                       
            dy(3)=(0.5*eta*A*y(3)-Gr*y(4)-Gc*y(6)+(M+Kp+A)*y(2))/(1+(1/beta));
            dy(4)=y(5);
            dy(5)=(Pr*(0.5*A*eta*y(5))-Pr*Df*Sc*(0.5*eta*A*y(7)+2*A*y(6)-L0*y(6)))/(1-(Pr*Df*Sc*Sr));
            dy(6)=y(7);
            dy(7)=(Sc*(0.5*eta*A*y(7)+2*A*dy(6))-Sr*Pr(0.5*A*eta*y(5)+2*A*y(4)))/(1-(Sr*Df*Pr));
        ];

        % Define the boundary conditions function
        bc = @(ya, yb) boundary_conditions(ya, yb);

        % Solve the equations
        eta_span = [0, 10];
        initial_conditions = [0; 1; 0; 0; 1; 0; 1]; % Initial guesses for f, f', theta, phi, f'', phi'
        options = odeset('RelTol', 1e-6, 'AbsTol', 1e-9);
        [eta, solution] = ode45(equations, eta_span, initial_conditions, options);

        % Calculate velocity (f'(eta))
        velocity = solution(:, 2);

        % Plot velocity against eta
        plot(eta, velocity, '-', 'LineWidth', 2, 'DisplayName', sprintf('M = %.2f', M));
    end

    % Add labels and legend
    xlabel('\eta');
    ylabel('Velocity (f''(\eta))');
    title('Velocity Profile for Different M Numbers');
    legend('Location', 'best');
    grid on;
    hold off;
end

function res = boundary_conditions(ya, yb)
    Bi1 = 0.5;
    Bi2 = 0.5;
    K0 = 0.3;
    % Boundary conditions
    res = [
        ya(2)= 1 + K0*ya(3); 
        ya(5) = - Bi1*(1-ya(4)); 
        ya(7) = - Bi2*(1-ya(6));  
        yb(2) = 0; 
        yb(4) = 0; 
        yb(6) = 0; 
        ];
end

Answer 1

[Torsten]

You have seven differential equations, but you only specify six boundary conditions. That's mathematically incorrect.

plot_velocity_vs_eta()

Error using bvparguments (line 103)
Error in calling BVP4C(ODEFUN,BCFUN,SOLINIT,OPTIONS):
The boundary condition function BCFUN should return a column vector of length 7.

Error in bvp4c (line 119)
bvparguments(solver_name,ode,bc,solinit,options,varargin);

Error in solution>plot_velocity_vs_eta (line 42)
sol = bvp4c(equations, bc, initial_conditions, options);

function plot_velocity_vs_eta()
    % Parameters
    A = 0.5;
    Gr = 0.7;
    Gc = 0.5;
    Kp = 3.0;
    beta = 0.5;
    Pr = 0.3;
    Df = 0.2;
    Sc = 0.1;
    L0 = 0.5;
    Sr = 0.3;
    M_values = [1,2,3,4,5];
    

    % Initialize plot
    figure;
    hold on;

    % Iterate over each Prandtl number
    for i = 1:length(M_values)
        M = M_values(i);

        % Define the system of equations
        equations = @(eta,y) ...
            [y(2);...
             y(3);...
             (0.5*eta*A*y(3)-Gr*y(4)-Gc*y(6)+(M+Kp+A)*y(2))/(1+1/beta);...
             y(5);...
             (Pr*0.5*A*eta*y(5)-Pr*Df*Sc*(0.5*eta*A*y(7)+2*A*y(6)-L0*y(6)))/(1-Pr*Df*Sc*Sr);...
             y(7);...
             (Sc*(0.5*eta*A*y(7)+2*A*y(7))-Sr*Pr*(0.5*A*eta*y(5)+2*A*y(4)))/(1-Sr*Df*Pr)];

        % Define the boundary conditions function
        bc = @(ya, yb) boundary_conditions(ya, yb);

        % Solve the equations
        eta_span = linspace(0,10,100);
        initial_conditions = bvpinit(eta_span,[0; 1; 0; 0; 1; 0; 1]); % Initial guesses for f, f', theta, phi, f'', phi'
        options = odeset('RelTol', 1e-6, 'AbsTol', 1e-9);
        sol = bvp4c(equations, bc, initial_conditions, options);

        % Calculate velocity (f'(eta))
        velocity = sol.y(2,:);
        eta = sol.x;

        % Plot velocity against eta
        plot(eta, velocity, '-', 'LineWidth', 2, 'DisplayName', sprintf('M = %.2f', M));
    end

    % Add labels and legend
    xlabel('\eta');
    ylabel('Velocity (f''(\eta))');
    title('Velocity Profile for Different M Numbers');
    legend('Location', 'best');
    grid on;
    hold off;
end

function res = boundary_conditions(ya, yb)
    Bi1 = 0.5;
    Bi2 = 0.5;
    K0 = 0.3;
    % Boundary conditions
    res = [ya(2)-(1+K0*ya(3));...
           ya(5)-(-Bi1*(1-ya(4)));...
           ya(7)-(-Bi2*(1-ya(6)));...
           yb(2);...
           yb(4);...
           yb(6)];
end

Answer 2

[Steven Lord]

You cannot use assignment inside the body of an anonymous function. Change this:

% Define the system of equations
equations = @(eta, y) [
    dy(1)=y(2);                                                       
    dy(2)=y(3);                                                       
    dy(3)=(0.5*eta*A*y(3)-Gr*y(4)-Gc*y(6)+(M+Kp+A)*y(2))/(1+(1/beta));
    dy(4)=y(5);
    dy(5)=(Pr*(0.5*A*eta*y(5))-Pr*Df*Sc*(0.5*eta*A*y(7)+2*A*y(6)-L0*y(6)))/(1-(Pr*Df*Sc*Sr));
    dy(6)=y(7);
    dy(7)=(Sc*(0.5*eta*A*y(7)+2*A*dy(6))-Sr*Pr(0.5*A*eta*y(5)+2*A*y(4)))/(1-(Sr*Df*Pr));
];

to:

% Define the system of equations
equations = @(eta, y) [
    y(2);                                                       
    y(3);                                                       
    (0.5*eta*A*y(3)-Gr*y(4)-Gc*y(6)+(M+Kp+A)*y(2))/(1+(1/beta));
    y(5);
    (Pr*(0.5*A*eta*y(5))-Pr*Df*Sc*(0.5*eta*A*y(7)+2*A*y(6)-L0*y(6)))/(1-(Pr*Df*Sc*Sr));
    y(7);
    (Sc*(0.5*eta*A*y(7)+2*A*dy(6))-Sr*Pr(0.5*A*eta*y(5)+2*A*y(4)))/(1-(Sr*Df*Pr));
];

You will also need to change your boundary condition function. Or actually, since you never use it except in the definition of the anonymous function bc and never use that anonymous function, just eliminate bc and your boundary_conditions function entirely.

Or if your use of ode45 is just for practice in learning how to call the differential equation solvers and you really want to solve this boundary value problem rather than an ordinary differential equation, see the bvp4c or bvp5c functions. The examples on those documentation pages or on the "Solving Boundary Value Problems" topic linked from the end of those function pages should provide you with examples you can adapt for your equations.