Study on the existence of tracking model for a d-dimensional random walk transportation radionuclide particle in a fractured medium

Abstract

This paper presents a novel tracking model to detect a [Formula: see text]-dimensional random walk radionuclides and toxic chemicals in a fractured medium with minimum time and maximum probability. This model improves the physical and chemical properties of a fractured medium for a long time. It depends on a coordinated search technique with multiple sensors. The radionuclide particle has a linear flows on one direction from [Formula: see text]-directions to spread. We consider this particle dispersed through a cylinder path in a very small time. The tracking or searching process in each cylinder path will be done by two sensors which start from any point in the fractured medium (space). The particle’s initial position is a vector of random variables with a known probability distribution. More than presenting the analytical solution which shows the existence of a finite tracking technique, we derive some auxiliary results related to optimal tracking technique. Besides that, we derive this optimal tracking technique which minimizes the expected value of the first meeting time between one of the sensors and the radionuclide or the toxic chemical particle. The effectiveness of this model appears in some numerical results.