Python - 求解瞬态矩阵微分方程

Question

I am facing some problems solving a time-dependent matrix differential equation. The problem is that the time-dependent coefficient is not just following some time-dependent shape, rather it is the solution of another differential equation.

Up until now, I have considered the trivial case where my coefficient G(t) is just G(t)=pulse(t) where this pulse(t) is a function I define. Here is the code:

# Matrix differential equation
def Leq(t,v,pulse): 
    v=v.reshape(4,4) #covariance matrix 
    M=np.array([[-kappa,0,E_0*pulse(t),0],\.  #coefficient matrix 
                [0,-kappa,0,-E_0*pulse(t)],\
                [E_0*pulse(t),0,-kappa,0],\
                [0,-E_0*pulse(t),0,-kappa]])
    
    Driff=kappa*np.ones((4,4),float) #constant term
    
    dv=M.dot(v)+v.dot(M)+Driff #solve dot(v)=Mv+vM^(T)+D
    return dv.reshape(-1)  #return vectorized matrix

#Pulse shape
def Gaussian(t):
    return np.exp(-(t - t0)**2.0/(tau ** 2.0))

#scipy solver
cov0=np.zeros((4,4),float) ##initial vector
cov0 = cov0.reshape(-1);   ## vectorize initial vector
Tmax=20 ##max value for time
Nmax=10000 ##number of steps
dt=Tmax/Nmax  ##increment of time
t=np.linspace(0.0,Tmax,Nmax+1)

Gaussian_sol=solve_ivp(Leq, [min(t),max(t)] , cov0, t_eval= t, args=(Gaussian,))

And I get a nice result. The problem is that is it not exactly what I need. Want I need is that dot(G(t))=-kappa*G(t)+pulse(t), ie the coefficient is the solution of a differential equation.

I have tried to implement this differential equation in a sort of vectorized way in Leq by returning another parameter G(t) that would be fed to M, but I was getting problems with the dimensions of the arrays.

Any idea of how should I proceed?

Thanks,

Answer 1

In principle you have the right idea, you just have to split and join the state and derivative vectors.

def Leq(t,u,pulse): 
    v=u[:16].reshape(4,4) #covariance matrix 
    G=u[16:].reshape(4,4) 
    ... # compute dG and dv
    return np.concatenate([dv.flatten(), dG.flatten()])

The initial vector has likewise to be such a composite.