Environmental contaminant dispersion models based on the Vladimirov-Taibleson p-adic pseudo-differential operator

Abstract This research develops a new framework for modeling the dispersion of contaminants in non-Archimedean media using the Taibleson-Vladimirov $$D^{\alpha }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>D</mml:mi> <mml:mi>α</mml:mi> </mml:msup> </mml:math> p -adic pseudo-differential operator. A p -adic differential equation is proposed and solved, modeling the dispersion of contaminants in a p -adic medium initially confined to a specific region. Additionally, it is analyzed how the p -adic heat kernel associated with the Taibleson-Vladimirov $$D^{\alpha }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>D</mml:mi> <mml:mi>α</mml:mi> </mml:msup> </mml:math> operator serves as a powerful tool for understanding how contaminants disperse over time, making the heat kernel a cornerstone for mathematical models aimed at environmental applications in p -adic settings. Furthermore, through the heat equation associated with the Taibleson-Vladimirov $$D^{\alpha }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>D</mml:mi> <mml:mi>α</mml:mi> </mml:msup> </mml:math> operator, the evolution of contaminant concentration over time is described. Finally, a stochastic fractional p -adic diffusion equation is formulated and solved, incorporating initial conditions, external sources, and random noise.