I'm trying to transform the first order differential equation
$\dot{x} = k x(t)$
into the corresponding set of stochastic differential equations. I have two independent uniformly distributed variables, that is, the initial condition $x_0 \in [x_{0,l}, x_{0,u}]$ and $k \in [k_{l}, k_{u}]$.
we can take expansions for both variables as
$x = \sum_{i=0}^p x_i \psi_i(\zeta)$ $k = \sum_{i=0}^p k_i \psi_i(\zeta)$
and if we plug these expressions in the original differential equation we get
$\sum_{k=0}^p \dot{x}_k \psi_k(\zeta) = \sum_{i=0}^p\sum_{j=0}^p k_i x_j \psi_i(\zeta) \psi_j(\zeta)$
Then, we perform a Galerkin projection, which turns into
$\dot{x}_k = \frac{1}{\gamma_k}<\left(\sum_{i=0}^p\sum_{j=0}^p k_i x_j \psi_i(\zeta) \psi_j(\zeta)\right),\psi_k(\zeta) >$
where $\gamma_k$ is $1/2\int_{-1}^1 P_k(\zeta)^2 d\zeta$
first question: is the procedure correct? in 1D (only initial condition or only parameter $k$) results are perfect. What is not clear to me is if it is still ok to follow this procedure for two independent variables, which 2d Pdf I should plug in (the product of the two constant pdf? something like
$pdf_{2d} = 1/(x_{0,u}-x_{0,l})\cdot 1/(k_{u}-k_{k,l})$
or
$pdf_{2d} = 1/2\cdot 1/2$
by considering the pdf over the normalized domain $\zeta \in [-1,1]$?
Moreover, do we always need quadrature even for $n$ independent variables? or $n$-dimensional integrations techniques are required for the multivariable Galerkin projection?
Thanks in advance!
