how to convert second order ODE to first in the below FDTD problem

for index=1:iterationsNum

I am currently workin on this particlar FDTD problem where it is required to input the above second order ODE into the below code in the rohNew line.Thougt about using OdeToVectorfield but could not understand how to integrate it in this context.

You can just ignore the constants.

    ELz=(EL(2:end)-EL(1:end-1))/dz;
    ELnew(2:end)=EL(2:end)+dt*c/n*(1i*kappa*roh(2:end).*ES(2:end)-ELz);
    ESz=(ES(2:end)-ES(1:end-1))/dz;
    croh=conj(roh);
    ESnew(1:end-1)=ES(1:end-1)+dt*c/n*(1i*kappa*croh(1:end-1).*EL(1:end-1)+ESz);
    f = wgn(1,N,Q/dt/dz,'complex','linear');%f=sqrt(Q/dz)*(randn(1,N))/sqrt(dt);%sqrt(Q/dz)*(randn(1,N)+1i*randn(1,N))/sqrt(2)/sqrt(dt);
    rohNew=roh+dt*(1i*LAMBDA*EL.*conj(ES)+f-GAMMAb*roh/2);
        
    EL=ELnew;
    ES=ESnew;
    roh=rohNew;
end

Unrecognized function or variable 'iterationsNum'.

Answer 1

[UDAYA PEDDIRAJU]

Hi Yogesh,

1) To convert a second-order ODE to a first-order system, introduce a new variable for the derivative of the original variable. For the given ODE:

• Introduce “z(t) = y'(t)” to represent the first derivative of y(t).
• Rewrite the second-order ODE as two first-order ODEs:

y'(t) = z(t)
z'(t) = f(t, y(t), z(t))

In your case, you can try introducing a new variable “roh'(t) = rohNew(t) - roh(t)” and rewrite the second-order ODE as a system of two first-order ODEs:

ELz=(EL(2:end)-EL(1:end-1))/dz;
ELnew(2:end)=EL(2:end)+dt*c/n*(1i*kappa*roh(2:end).*ES(2:end)-ELz);
ESz=(ES(2:end)-ES(1:end-1))/dz;
croh=conj(roh);
ESnew(1:end-1)=ES(1:end-1)+dt*c/n*(1i*kappa*croh(1:end-1).*EL(1:end-1)+ESz);
f = wgn(1,N,Q/dt/dz,'complex','linear');%f=sqrt(Q/dz)*(randn(1,N))/sqrt(dt);%sqrt(Q/dz)*(randn(1,N)+1i*randn(1,N))/sqrt(2)/sqrt(dt);
rohNew=roh+dt*(1i*LAMBDA*EL.*conj(ES)+f-GAMMAb*roh/2);
rohPrime = rohNew - roh;
EL=ELnew;
ES=ESnew;
roh=rohNew;   

2) Regarding the usage of “ odeToVectorField” yes, it is useful to convert the higher order ODEs to a system of first-order ODEs, here is an example.

syms y(t) z(t)
Dy = diff(y, t);
Dz = diff(z, t);
ode = Dz == f(t, y, z); % Replace f(t, y, z) with your specific function
V = odeToVectorField(ode);
M = matlabFunction(V, 'vars', {'t', 'Y'}); 

You can refer to https://www.mathworks.com/help/symbolic/odetovectorfield.html for more details.

I hope this gives you an idea on how to convert 2nd order ODE to 1st order system!