Marco Garcia, Dr. Ricardo Suarez, & Dr. Cynthia Flores
Partial differential equations are often used to model the behavior of systems in many different fields of study. For instance is physics they can used to model particle behavior, corrosion models for biological quantities as well as electricity and magnetism.
Here for this topic, the significance of this topic/study is based upon nonlocal gradient operators where non local differential operators are very important in the studies of nonlocal models especially nonlocal diffusion models which apply to many different areas in research.
For the background it is composed of partial differential equations in 3-D dimensions. Also it is where the variational setting is the nonlocal Dirichlet energies within the energy densities which are quadratic in nonlocal gradients. For the gradient dependent second order equations where the influence on the solution can arise in different ways on one hand there are semi-linear equations and the idea of drift or transport.
We will present the difference between local and nonlocal heat kernels with respect to linear and nonlinear models. Also we want to distinguish the difference between the homogenous versus nonhomogeneous equations in particular their applications .