My Math Forum Complex system of ODE

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 July 21st, 2011, 01:14 AM #1 Newbie   Joined: Jul 2011 Posts: 2 Thanks: 0 Complex system of ODE Hello! I have to solve a system of differential equations like this $\dot {\mathbf{c}}(t)= \mathbf{G}(\mathbf{R}(t)) \mathbf{c}(t)$ where $\mathbf{c}= \{c_i(t)\} \quad and \quad \mathbf{G} = \{\gamma_{ij}\} \in \mathcal{C}$ --- complex vector and matrix, respectively; assume that $\mathbf{c}(t=0)$ are known. Let's find a solution $\mathbf{c}= \mathbf{v} exp(\lambda t)$. And so, we get $\mathbf{G(\mathbf{R}(t)) v}= \lambda \mathbf{v}$ the eigenvalue and eigenvector problem with general complex non-symmetric matrix $\mathbf{G}$. I code it using FORTRAN and I use libraries to find the eigenvalues and eigenvectors. And so, the general solution looks like this $\mathbf c(t)= \sum_i \alpha_i \mathbf v_i exp(\lambda_i t)$ I have some doubt about matrix $\mathbf G$ and about my system of ODE. First of all. Does it the SODE with constant coefficients $\gamma_{ij}(\mathbf R(t))$? The matrix depends parametrically from $R(t)$. Secondly. Are the general solution coefficients $\alpha_i$ different at each $t \in [t_1; t_2]$? And Must I find it from several conditions, for example, $\mathbf{c}^{*} \mathbf{c} \equiv 1$? Finally. Should I solve this SODE with conventional numerical methods, like Runge-Kutta 4th order (I have the initial conditions for vector $\mathbf{c}(t=0)$)? Thank you very much.

 Tags complex, ode, system

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