2 second order differential equations with boundary conditions using bvp4c

problem statement 
(d^2 u(x))/〖dx〗^2 =(γu(x))/(1+αu(x))
(d^2 v(x))/〖dx〗^2 =a〖∆v〗^*  (dv(x))/dx-2/ε  (γu(x))/((1+αu(x)))
at x=0,u=1,v=0
at x=1,du/dx=0,dv/dx=0
α=0.001,γ=100 ,epsilon=1,〖∆v〗^*=0.1 a=[0,10,100,500,1000]  plot v verses x


Matlab Code:

alpha = 0.001;
    gamma = 100;
    epsilon = 1;
    delta_v = 0.1;
    
a_values = [0, 10, 100, 500, 1000];
for i = 1:length(a_values)
    %% options = bvpset('RelTol', 1e-6, 'AbsTol', 1e-6); % Set options
    sol = bvp4c(@equations, @boundary_conditions, initial_guess);

    x = linspace(0, 1, 100); 
    y = deval(sol, x); 
    plot(x, y(2, :)); % Plot x versus v
    hold on;
    xspan = [0, 1]; % Define the interval
initial_guess = [1, 0, 0, 0]; % Initial guess for u, v, du/dx, dv/dx
legend('a = 0', 'a = 10', 'a = 100', 'a = 500', 'a = 1000');
xlabel('x');
ylabel('v(x)');
title('Plot of x versus v(x)');
end

function res = boundary_conditions(ya, yb)
    res = [ya(1) - 1; ya(2); yb(3); yb(4)]; % Boundary conditions
end

function dydx = equations(x, y)
    u = y(1);
    v = y(2);
  
  dydx = zeros(4, 1);
        dydx(1) = y(3); % du/dx
        dydx(2) = y(4); % dv/dx
        dydx(3) = (gamma * y(1)) / (1 + alpha * y(1));
        dydx(4) = a * delta_v_star * y(4) - (2 / epsilon) * (gamma * y(1)) / (1 + alpha * y(1));
end
getting error Unrecognized function or variable 'initial_guess'.

Answer 1

[Torsten]

Almost the same code as in the preceeding problem:

% Parameters
alpha = 0.001;
gamma = 100;
epsilon = 1;
delta_v = 0.1;
a=[0,10,100,500,1000];
hold on
for i = 1:numel(a)
    fcn = @(x,y)[y(3);y(4);(gamma*y(1))/(1+alpha*y(1));a(i)*delta_v*y(4)-(2/epsilon)*(gamma*y(1))/(1+alpha*y(1));];
    bc = @(ya,yb)[ya(1)-1;ya(2);yb(3);yb(4)];
    guess = @(x)[1;0;0;0];
    xmesh = linspace(0,1,20);
    solinit = bvpinit(xmesh, guess);
    sol = bvp4c(fcn, bc, solinit);
    plot(sol.x,sol.y(2,:))
end
hold off
grid on