Mathematical model of pollution and self-purification of the environment (Q2703269)

The authors investigate the following mathematical model of pollution of the environment with turbulent diffusion, adsorption and disintegration of active admixtures NEWLINE\[NEWLINE {\partial^{2}u\over\partial x^{2}}- {\partial u\over\partial t}=x^{1-\beta}u^{\beta},\quad 0<x<s(t),\quad t>0, \quad u(x,0)=0,\quad 0<x<s(0), NEWLINE\]NEWLINE NEWLINE\[NEWLINE u(s(t),t)=u_{x}(s(t),t)=0,\quad t>0,\quad \int\limits_{0}^{s(t)}\left(u_{t}+x^{1-\beta}u^{\beta}\right)x dx= q(t),\quad t>0. NEWLINE\]NEWLINE In the cases \(\beta=0,1\) exact solutions of the corresponding stationary problems are obtained. For the non-stationary problems a constructive method for the approximate solution is proposed.