Is a general method for solving a system of linear partial differential equation with trigonometric function as coefficients exist ?
For example something like that: $q$ is the unknown function, $2 \pi $ periodic, from $\mathfrak{R}^2\times [0,T]$ to $\mathfrak{C}$ and $A$, $B$, $C$ function of $\mathfrak{R}^2$ to $\mathfrak{C}$ are the coefficient of the system. $i$ is an integer varying from $1$ to $4$. The equation look like, using Einstein summation convention: \begin{equation} \partial_t q_i(y,z)= A_j(y,z)\partial_{j} q_{i}(y,z) + q_j(y,z) B_{ij}(y,z) + C_{i}(y,z) \end{equation} $A$ is of the from: \begin{equation} A_j(y,z)=a^{(1)}_j \sin(2\pi y) + a^{(2)}_j \sin(2\pi z) + a^{(3)}_j \cos(2\pi y) + a^{(4)}_j \cos(2\pi z)+a_j^{(5)} \end{equation} with $a_{j}^{(k)}$ in $\mathfrak{C}$. Functions $B_{ij}$ and $C_{j}$ are define in the same way that $A_{j}$. The initial condition $q(y,z,t=0)$ is given.