Solving direct and inverse problems for Fredholm-type integro-differential equations with application to pollution diffusion modeling

We analyze a particular Fredholm-type partial integro-differential equation. We study the direct problem and prove existence and uniqueness of the solution via a fixed-point argument for generalized contractive maps. This approach also allows us to formulate a collage-type result that can be used to solve inverse problems. We provide numerical examples and we also show how these equations can be used to model pollution diffusion of heavy pollutants and non-volatile substances such as heavy metals, chemical spills, radioactive isotopes, and others.