Can pdepe solve a system of two second-order equations?

Updated 30.01.25, based on the given answers and the pde1dM_manual.pdf section 4.4, I updated the code below accordingly

I am trying to figure out pdepe and how to use it to solve a higher order equation written as multiple second order equations. Even though I am trying to solve a more complex equation in the future I want to start with a simple case

. (Updated based on the answers)

My idea is to rewrite it by including a second variable (in fact I now need three equations, since I got the second derivative in time)

, (Updated based on the answers)

I chose the spatial coordinate to range from x=linsapce(0,2*pi,N) to ensure that my initial conditions satisfy my boundary conditions (see below).

In pdepe terms I wrote

function [c,f,s] = pdex1pde(x,t,u,dudx)
    global D
    c = [1 ; 1; 0];
    f = [0; -D*dudx(3); -dudx(1)];
    s = [u(2); 0; u(3)]; 
    
    % Old code
    % c = [1 ; 0];
    % f = [D*dudx(2); dudx(1)];
    % s = [0; -u(2)];  %where q=u(2);
end

and providing some initial condition (from 0 to 2pi)

function u0 = pdex1ic(x) 
    % initial condition based on pde1dM_manual.pdf section 4.4
    init = 1-cos(x);
    d4init_dx4 = cos(x);
    u0 = [init; 0; d4init_dx4];
    
    % Old code
    % init = 1-cos(x);
    % d2init_dx2 = cos(x);
    % 
    % u0 = [init; d2init_dx2]
end

For the boundary conditions I just used a Neumann boundary for each boundary

function [pl,ql,pr,qr] = pdex1bc(xl,ul,xr,ur,t)
    % Updated boundary conditions equivalent to a simply supported beam
    % based on pde1dM_manual.pdf section 4.4.3
    pl = [ul(1); ul(2); ul(3)];
    ql = [0; 0; 0];
    pr = [ur(1); ur(2); ur(3)];
    qr = [0; 0; 0];
    
    % Old code
    % pl = [0;0];
    % ql = [1;1];
    % pr = [0;0];               
    % qr = [1;1];   
end

Now pdepe gives me a warning that it could not converge at time t=0 and I don't get a result.

Is there something that I am missing?

I had a look at other questions about this topic and already stumbled upon this github page (https://github.com/wgreene310/pdepe-examples), but I don't really understand it without the written equations.

Thank you very much for answering.

Update: Pdepe converged and was able to solve the updated equation as a system of two spatialy second order equations

[James Blanchard]

What boundary conditions are you trying to use?

[Torsten]

@James Blanchard

As written, first and third derivative of u are set to 0 at both endpoints.

[Schmieje]

I just thought to set Neumann boundaries on the ends of both equations, so .

[James Blanchard]

I don't think that system has a unique solution.

[Schmieje]

You are right, the equation should read following the pde1dm_manual.

Answer 1

[Torsten]

Goto

https://uk.mathworks.com/matlabcentral/fileexchange/97437-pde1dm

download the code, goto folder "documents" and study example 4.4 in "pde1dm_manual".

[Schmieje]

Thank you @Torsten, that is exactly what I was looking for

[Torsten]

Note that the equation in the document uses the second derivative with respect to t, not the first.

[Torsten]

@Schmieje

Since you modified your question I'm not sure whether you could solve your problem with the code given or whether you still encounter difficulties.

[Schmieje]

Yes, I got a solution for the updated code.

These are the results for every

i = 1:10:numel(t)
sol = pdepe(0,@pdex1pde,@pdex1ic,@pdex1bc,x,t);
result = sol(i,:,1);

with the given model parameters

x = linspace(0,2*pi,100);
t = linspace(0,10,151);
D = 0.01;