Solve matrix differential equations using Matlab

[quansnow]

For d V/d t=AV + VA^{T}+ D, where A and D are the given 6*6 matrices,

A=[-0.5*K 0 0 0 0 G1;0 -0.5*K 0 0 -G1 0;0 0 -0.5*K 0 0 -G2;0 0 0 -0.5*K -G2 0;0 G1 0 -G2 -0.5*RM 0;-G1 0 -G2 0 0 -0.5*RM]
D=[0.5*K 0 0 0 0 0;0 0.5*K 0 0 0 0;0 0 0.5*K 0 0 0;0 0 0 0.5*K 0 0;0 0 0 0 0 0;0 0 0 0 0 0.5*RM*(2*N+1)];
in which the values of K,G1,G2,RM,N are known.


My way:

let V= [V(1) V(2) V(3) V(4) V(5) V(6);
V(7) V(8) V(9) V(10) V(11) V(12);
V(13) V(14) V(15) V(16) V(17) V(18);
V(19) V(20) V(21) V(22) V(23) V(24);
V(25) V(26) V(27) V(28) V(29) V(30);
V(31) V(32) V(33) V(34) V(35) V(36)]

then plug V into d V/d t=AV + VA^{T}+D to get 36 equations as follows
dV(1)/dt =K/2 - (K*V(1))/2 + V(6)*conj(G1) - (V(1)*conj(K))/2 + G1*V(31);
dV(2)/dt =G1*V(32) - V(5)*conj(G1) - (V(2)*conj(K))/2 - (K*V(2))/2;
dV(3)/dt =G1*V(33) - V(6)*conj(G2) - (V(3)*conj(K))/2 - (K*V(3))/2;
dV(4)/dt =G1*V(34) - V(5)*conj(G2) - (V(4)*conj(K))/2 - (K*V(4))/2;
dV(5)/dt =V(2)*conj(G1) - (K*V(5))/2 - V(4)*conj(G2) - (V(5)*conj(RM))/2 + G1*V(35);
dV(6)/dt =G1*V(36) - V(1)*conj(G1) - V(3)*conj(G2) - (V(6)*conj(RM))/2 - (K*V(6))/2;
dV(7)/dt = V(12)*conj(G1) - (K*V(7))/2 - (V(7)*conj(K))/2 - G1*V(25);
dV(8)/dt = K/2 - (K*V(8))/2 - V(11)*conj(G1) - (V(8)*conj(K))/2 - G1*V(26);
dV(9)/dt =- (K*V(9))/2 - V(12)*conj(G2) - (V(9)*conj(K))/2 - G1*V(27);
dV(10)/dt = - (K*V(10))/2 - V(11)*conj(G2) - (V(10)*conj(K))/2 - G1*V(28);
dV(11)/dt =V(8)*conj(G1) - (K*V(11))/2 - V(10)*conj(G2) - (V(11)*conj(RM))/2 - G1*V(29);
dV(12)/dt =- (K*V(12))/2 - V(7)*conj(G1) - V(9)*conj(G2) - (V(12)*conj(RM))/2 - G1*V(30);
dV(13)/dt =V(18)*conj(G1) - (K*V(13))/2 - (V(13)*conj(K))/2 - G2*V(31);
dV(14)/dt =- (K*V(14))/2 - V(17)*conj(G1) - (V(14)*conj(K))/2 - G2*V(32);
dV(15)/dt =K/2 - (K*V(15))/2 - V(18)*conj(G2) - (V(15)*conj(K))/2 - G2*V(33);
dV(16)/dt =- (K*V(16))/2 - V(17)*conj(G2) - (V(16)*conj(K))/2 - G2*V(34);
dV(17)/dt = V(14)*conj(G1) - (K*V(17))/2 - V(16)*conj(G2) - (V(17)*conj(RM))/2 - G2*V(35);
dV(18)/dt=- (K*V(18))/2 - V(13)*conj(G1) - V(15)*conj(G2) - (V(18)*conj(RM))/2 - G2*V(36);
dV(19)/dt=V(24)*conj(G1) - (K*V(19))/2 - (V(19)*conj(K))/2 - G2*V(25);
dV(20)/dt= - (K*V(20))/2 - V(23)*conj(G1) - (V(20)*conj(K))/2 - G2*V(26);
dV(21)/dt=- (K*V(21))/2 - V(24)*conj(G2) - (V(21)*conj(K))/2 - G2*V(27);
dV(22)/dt=K/2 - (K*V(22))/2 - V(23)*conj(G2) - (V(22)*conj(K))/2 - G2*V(28);
dV(23)/dt=V(20)*conj(G1) - (K*V(23))/2 - V(22)*conj(G2) - (V(23)*conj(RM))/2 - G2*V(29);
dV(24)/dt =- (K*V(24))/2 - V(19)*conj(G1) - V(21)*conj(G2) - (V(24)*conj(RM))/2 - G2*V(30);
dV(25)/dt=V(30)*conj(G1) - (RM*V(25))/2 - (V(25)*conj(K))/2 + G1*V(7) - G2*V(19);
dV(26)/dt= G1*V(8) - V(29)*conj(G1) - (V(26)*conj(K))/2 - (RM*V(26))/2 - G2*V(20);
dV(27)/dt=G1*V(9) - V(30)*conj(G2) - (V(27)*conj(K))/2 - (RM*V(27))/2 - G2*V(21);
dV(28)/dt=G1*V(10) - V(29)*conj(G2) - (V(28)*conj(K))/2 - (RM*V(28))/2 - G2*V(22);
dV(29)/dt= V(26)*conj(G1) - (RM*V(29))/2 - V(28)*conj(G2) - (V(29)*conj(RM))/2 + G1*V(11) - G2*V(23);
dV(30)/dt=G1*V(12) - V(25)*conj(G1) - V(27)*conj(G2) - (V(30)*conj(RM))/2 - (RM*V(30))/2 - G2*V(24);
dV(31)/dt=V(36)*conj(G1) - (RM*V(31))/2 - (V(31)*conj(K))/2 - G1*V(1) - G2*V(13);
dV(32)/dt= - (RM*V(32))/2 - V(35)*conj(G1) - (V(32)*conj(K))/2 - G1*V(2) - G2*V(14);
dV(33)/dt=- (RM*V(33))/2 - V(36)*conj(G2) - (V(33)*conj(K))/2 - G1*V(3) - G2*V(15);
dV(34)/dt=- (RM*V(34))/2 - V(35)*conj(G2) - (V(34)*conj(K))/2 - G1*V(4) - G2*V(16);
dV(35)/dt =V(32)*conj(G1) - (RM*V(35))/2 - V(34)*conj(G2) - (V(35)*conj(RM))/2 - G1*V(5) - G2*V(17);
dV(36)/dt=(RM*(2*N + 1))/2 - V(31)*conj(G1) - V(33)*conj(G2) - (V(36)*conj(RM))/2 - (RM*V (36))/2 - G1*V(6) - G2*V(18);
then solve these 36 equations by using mathlab command ode45.

My method is cumbersome.

Is there any easy way to solve d V/d t=AV + VA^{T}+ D?

Thanks!

 

[mmwave]

Thank you for your question. The method you have described is one way to solve the given equation, however, it can be quite cumbersome and time-consuming. There are a few other methods that can be used to solve this type of equation.

1. Matrix Exponential Method: This method involves using the matrix exponential function to solve the equation. The general solution can be expressed as V(t) = exp(At) * V(0), where V(0) is the initial condition. This method is commonly used for linear systems of equations.

2. Laplace Transform Method: This method involves taking the Laplace transform of both sides of the equation and solving for the transformed variable. The inverse Laplace transform can then be used to obtain the solution in the time domain.

3. Numerical Methods: As you have mentioned, using numerical methods such as ode45 in MATLAB can also be used to solve the equation. There are also other numerical methods such as Runge-Kutta methods and Euler's method that can be used to solve the equation.

I would recommend trying out these methods and seeing which one works best for your specific problem. I hope this helps.