Stochastic parabolic equations on graphs

A biodegradable stent is a mesh that is used to treat the narrow or closed part of the artery to open and restore normal blood flow, which is made of a biodegradable material.{However, since the struts in the stent are thin, a simple structural model (called the one-dimensional curved rod model) can be used to mathematically model the stent.So differential diffusion equations can describe the degradation of the stent. The movement of the blood causes microscopic trembling of the edges of the wall. Therefore, the perturbation of the corresponding differential equation by a white noise type term should correspond to the real situation. We consider some linear parabolic equations on graphs with white noise terms. To investigate the initial-boundary value problem for these equations, we reduce it to the corresponding deterministic problem.First, we prove the existence and uniqueness of the we ak solution to deterministic problem for parabolic equations on graphs.Finally, same results are obtained for linear stochastic parabolic equations on graphs.