Article

    Ufa Mathematical Journal
    Volume 3, Number 2, pp. 79-84

    Explicit solution of the Cauchy problem for motion equation of the subterranean waters with free surface


    Umarov Kh.G.

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    For linear partial differential equation, modeling evolution of filtering liquids' free surface $$\lambda u_t-\Delta_2 u_t=\alpha\Delta_2 u-\beta \Delta_2^2u +f ,$$ where $u=u(x,y,t)$ -- searched function, characterizing liquids' pressure, $f=f(x,y,t)$ --— given function, calculating external influence on filtration flow, $\Delta_2=\delta^2/\delta x^2+\delta^2/\delta y^2$ —-- Laplace's differential operator, $\lambda,\ \alpha,\ \beta$ —-- positive constants dependent on characteristics of watery soil, the evident type of the Cauchy problem solution in space $L_p(R^2),\ 1