Solution of the third mixed problem on filtration of a fluid in a crack-porous medium (Q1584158)

In this paper a solution is constructed for the third mixed problem for the equation \[ \frac{\partial p}{\partial t} = k\,\frac{\partial^2p}{\partial x^2} + \eta\, \frac{\partial^3p}{\partial x^2\partial t},\tag{1} \] which satisfies the conditions \[ \begin{gathered} \frac{\partial p(t,x)}{\partial x}\bigg| _{x=0} - \lambda[ p(t,0)-h(t)] = 0,\tag{2}\\ p(0,x) = A\,\exp (-x/\sqrt{\eta}).\tag{3} \end{gathered} \] It is indicated that the problem (1)\,--\,(3) can be reduced to the first and second boundar y problems on filtration in cracks [see \textit{A.~M.~Ametov}, Prikl. Mat. Tekhn. Fiz. 1994, No.~2, 85--88 (1994)].