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I'm thinking of solving a Partial differential algebraic equation using multidimensional polynomial (i.e. Taylor series). Consider the PDAE:

$$\mathbf F \left( \mathbf x, \mathbf y, \frac{\partial y_i}{\partial x_j}, \frac{\partial^2 y_i}{\partial x_j \partial x_n}, \ldots \right) = \mathbf 0 \, , \left\{1<i<m,1<j<n\right\} \, ,\tag{1}$$

with a boundary/initial condition of

$$\mathbf G \left( \mathbf y, \frac{\partial y_i}{\partial x_j}, \frac{\partial^2 y_i}{\partial x_j \partial x_n}, \ldots \right)_{\partial \Omega} = \mathbf 0 \, . \tag{2}$$

Now if we approximate $ \mathbf y$ with a multidimensional polynomial:

$$ y_i \approx \sum_{l_1=0}^{o_1} \ldots \sum_{l_n=0}^{o_n} a_{i\left(l_1, \ldots, l_n\right)}x_1^{l_1} \ldots x_n^{l_n} \, , \tag{3}$$

and substitute it in Eq. 1 , we get system of nonlinear equations of $a_{i\left(l_1, \ldots, l_n\right)}$s. Using the boundary conditions this system of nonlinear equations can be solved to find $a$s.

I want to write a Matematica macro to automate this process.

Where should I start?

Are there any example about that?

Has there ever been an attempt to do so? even with other symbolic CAS software

Thanks for your help in advance.