We have a vector space of a graph, so we can relate graph theory to algebra. Why can we not relate a graph with a differential equation?
Of course! For instance, that is exactly what is being done in discrete resonance area of nonlinear wave theory. Nodes represent wavevectors, each storing some energy, edges - the paths, along which energy flows from node to node, and time evolution of energy flow is governed by a system of differential equations, usually one equation per node.
You can consult the book of E.Kartashova for details. A short overview of the present state-of-the-art you find in the latest articles of A.Kartashov and E.Kartashova (or in reverse order).
Sorry for a bit of PR ;)
Of course! For instance, that is exactly what is being done in discrete resonance area of nonlinear wave theory. Nodes represent wavevectors, each storing some energy, edges - the paths, along which energy flows from node to node, and time evolution of energy flow is governed by a system of differential equations, usually one equation per node.
You can consult the book of E.Kartashova for details. A short overview of the present state-of-the-art you find in the latest articles of A.Kartashov and E.Kartashova (or in reverse order).
Sorry for a bit of PR ;)
There's a lot written on graphs and partial differential equations. Let's exploit the Poisson equation d*d phi = rho, or the same in classical language div grad phi = rho as an example. When this is solved in finite dimensional spaces with finite element kind of approaches, one creates a mesh. Let G stand for the incidence data between the nodes and edges. That for every edge e from node i to j, the entry G_ei is -1, G_ej is 1 and otherwise 0. Now, the "discrete counterpart" of the d*d operator (that is, what is written div grad in the classical language) is G^T M G, where G is the mass matrix of so called Whitney 1-forms called also by the name edge elements.
If you like to know more on these matters, check what Alain Bossavit has written about Whitney forms.
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