Graphs and reducibility(or something like that)

I'm trying to simplify a network of electrical components and I'm curious if
theres any way to do this.


The network is equivalent to a graph but each edge consists has a "weight"
to it that is a function. Every edge has a similar function that "looks" the
same. Some nodes are known values while others are not.


The function is actually a linear differential equation with constant
coefficients.

For example, I might have something like


+
/\
Z2 Z3
/ \
+--Z1--+


Where Z1, Z2, Z3 are differential equations.

In reality the graph is equivalent to a system of differential equations and
there are unknowns that exist which each branch and/or at some nodes.


I can convert the differential equations into algebraic equations by taking
the laplacian as one method but it still involves solving a large system.

I'm more interested in using the fact that each edge/branch is identical to
every other one except for some basic constants.

In essence I think I can treat each branch with a weight as a vector where
the elements of the vector are the coefficients of the DE.


so for example, I might have as a function L(t;....) = c*dV1/dt + a*V2 -
b*V3

and so I could treat that as a vector like (a,-b,c). Every branch will just
have different coefficients in the function L but all of the same "form". I
feel that I could somehow use this to my advantage at simplifying the
graph/system. (Although I think the size of the vector grows exponentially
based on the size of the graph)

Is there anything I can do to simplify my problem? Maybe there are some
results in graph theory(I think this is where the problem comes from) that
can help? I'm actually wondering if there is some recursive way to simplify
the elements. Most of the "vectors" will only have non-zero elements for
those that represent coefficients around its branch.

In any case just looking for some ideas,

Thanks,
Jon



.

Loading