On a functional-differential equation (in a historical context) (Q2785050)

The paper is devoted to historical aspects of the development and some applications of the theory of functional-differential equations. The author presents some historical remarks concerning the beginning of modern mathematics in Poland at the end of the XIX-th century and a further survey of it during the XX-th century. Some special aspects of the history of research and teaching in mathematics are mentioned. NEWLINENEWLINENEWLINEFurthermore, the paper is devoted to the theory of functional-differential equations generalizing an integro-differential equation which was investigated by J. Bodziony, S. Gołab and J. Szarski in connection with some technical problems arising when screening certain granular bodies. This was described for the first time by J. Bodziony who introduced the following equation: NEWLINE\[NEWLINE \Biggl[ \int_{a}^{b}u(t,y) dy\Biggr]\cdot u_t(t,x)=-\lambda (t,x)\cdot u(t,x), NEWLINE\]NEWLINE where \(a,b\in \mathbb{R}\), \(\lambda :[0,t^0)\times (a,b)\to [0,\infty)\) are given and \(u(.,.)\) is an unknown function. The quantity \(\int_{a}^{b}u(t,y) dy\) has the physical meaning of the total volume of the material contained in the screen at the moment \(t\).