Loaded equations and their applications. (English. Russian original) Zbl 0536.35080

Differ. Equations 19, 74-81 (1983); translation from Differ. Uravn. 19, No. 1, 86-94 (1983).
The author studied various loaded equations and the connection between these equations and some problems which occur from the modelling of some phenomena in other sciences (agroecosystems, the particle-transport problem in planeparallel geometry, the stationary one - velocity transport equation and others). Among the main results; we remark here the reduction of the Bitsadze-Samarskij problem to a Dirichlet problem for a loaded equation and the reduction of the nonlocal problem \[ u_ t=u_{xx},\quad in\quad D=\{(x,t),\quad 0<x<1,\quad 0<t<T\},\quad u(0,t)=\tau(t),\quad 0\leq t\leq T, \]
\[ (\partial /\partial t)\int^{1}_{0}u(\xi,t)d\xi =\nu(t),\quad 0\leq t\leq T,\quad u(x,0)=\phi(x),\quad 0\leq x\leq 1 \] to a first boundary value problem for the loaded parabolic equation \[ v_ t=v_{xx}+v_ x/(1-x)+(x- 1)\quad(\partial /\partial t)\int^{x}_{0}(v(\xi,t)/(1-\xi)^ 2)d\xi, \]
\[ v(0,t)=\tau(t),\quad v(1,t)=\mu(t),\quad v(x,0)=\Phi(x) \] \(t\in [0,T],x\in [0,1]\), where \(\mu\) and \(\Phi\) are well defined functions by the functions \(\phi\) and \(\nu\). An interesting application of the loaded equations is the reducing of the Von-Verster equation from biomathematics to a local problem for the partial differential equation of the first order.
Reviewer: N.Luca


35S15 Boundary value problems for PDEs with pseudodifferential operators