Partial differential equations on graphs
This project with Annie Rak started in the summer 2016 as a HCRP project. The topic is ``differential equations on graphs". We explored in the summer 2016 first various dynamical systems on networks. The topic of discrete PDEs allows for expository, experimental as well as theoretical investigations in an area of mathematics which links discrete mathematics with dynamical system theory and discrete stochastic processes (as PDEs related to the Laplacian are always closely related to diffusion processes). Here are some topics we played with:- Heat and wave equations u'=-Lu, u''=-Lu
- Linear and nonlinear ave equations like u''=-Lu+sin(u)
- Geometric evolution in the geometry of graphs like PDEs on forms
- Transport and advection consensus dynamics like u'=-L_{+}u on a directed graph.
- Burger and fluid like equations u'= T(u,u) with quadratic difference operator
- Integrability, like Sine-Gordon which generalizes Volterra-Toda like systems.
- PDE on graphs: Some discussion notes from Summer 2016 [PDF].
- >Senior thesis scan
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