How to convert a fourth-order ordinary differential equation into a system of first-order ordinary differential equations in order to solve it.

Hello everyone,

I am currently trying to define and convert a fourth-order ordinary differential equation into a system of first-order ordinary differential equations in order to solve it using the Runge-Kutta method.

The fourth-order ordinary differential equation I am trying to solve is as follows.

y is a function of x, and a, b, c, and d are constants.

To convert it into a system of first-order ordinary differential equations, I differentiated the left-hand side and rearranged the equation in terms of y′′′′.

However, I am not obtaining the desired solution at this point.

Could you please guide me on the correct way to define the function?

Thank you for your assistance.

[Torsten]

What are your initial or boundary conditions ?

[James Tursa]

What is it about your current solution that you don't like? Can you show us your code?

[啓輝]

Thank you all for your comments. I apologize for the delayed response.

The initial conditions are as follows. We assume y'(0) here and envision a problem where we search for the optimal value.

y(0) = 2e-9

y'(0) = 1e-11

y''(0) = 0

y'''(0) = 3.5e5

The code is as follows.

function dydx = f(x, y, A_Ham, sigma, a, b, c, d)
    
    y1 = y(1);
    y2 = y(2);
    y3 = y(3);
    y4 = y(4);
    
    % term1
    term1 = a * (y3 * y1 - y2 * y2) /  y1^2;
    
    % term2
    term2 = -b * ( ...
        (3 * y1^2 * y2 * y4 / (1 + y2^2)^1.5) ...
        - (3 * y1^3 * y2 * y3 * y4 / (1 + y2^2)^2.5)...
        );
    
    % term3
    term3 = c * ( ...
        ((3 * y1^2 * y2^2 * y3^2 + y1^3 * y3^3 + 2 * y1^3 * y2 * y3 * y4) / (1 + y2^2)^2.5) ...
        - (5 * y1^3 * y2^2 * y3^3 / (1 + y2^2)^3.5)...
        );
    
    % term4
    term4 = d * sqrt(1 + y2^2);
    
    % term5
    term5 = -b * (sigma * delta^3) / (1 + delta_prime^2)^1.5;
    
    y4 = (term4 - (term1 + term2 + term3)) / term5;
    
    dydx = [y1;
        y2;
        y3;
        y4];
end

Answer 1

[Torsten]

If all conditions are given at x = 0, it's an easy problem:

Define

z(x) = integral_{x'=0}^{x'=x} d*sqrt(1+y'(x')^2) dx'

The system of differential equations to be solved is then given by

z' = d*sqrt(1+y'^2), z(0) = 0        (1)
(a*y'/y - b*y^3*y'''/(1+y'^2) + c*y^3*y'*y''^2/(1+y'^2)^2.5) - (a*y'(0)/y(0) - b*y(0)^3*y(0)'''/(1+y'(0)^2) + c*y(0)^3*y'(0)*y''(0)^2/(1+y'(0)^2)^2.5) = z  (2)

Now solve (2) for y''' and convert it to a system of three first-order equations for y,y' and y''.

Then use a MATLAB ode-integrator to solve (1) and (2) as a system of four first-order equations for z, y, y' and y''.