On January 10, 2019, at 10:30 the “Optimal Control” department of IMM will held its regular seminar. The head of the laboratory “Mathematical problems of control” of the Institute of Control Systems of ANAS, prof. of ANAS, doctor of mathematics Ilgar Gurbat oglu Mamedov will give a talk on “Some usual and fractional model differential and integral equations of mathematical biology and their applications”.

The talk deals with main model differential and integral equations of mathematical biology and their applications. Information about non-local character (with memory) of the considered biological processes in mathematical biology will be given.

One of the model differential equations in mathematical biology describing the filtration process in humidification of the roots of plants the Aller equation, the relation between this equation and canonical form of the second order linear hyperbolic equations, the telegraph equation used in communication, the string vibration equation describing the oscillation processes will be touched. The talk also deals with the Bianchi equation used in modeling vibration processes used in study of radiowaves, one of the main model differential equations of mathematical biology used in hydrodynamics, the generalized Manjeron equation and the relation of this equation with Boussinesq-Love equation describing the longitudinal waves.

Then the mathematical relation of the Boussinesq-Love equation with the Laplace equation describing stationary filtration processes for liquids will be substantiated. The mathematical relation of the Laplace equation with the first order elliptic differential equation and relations of the Cauchy-Riemann equation with fractional order differential equations will be shown.

Then one of the fractional model integral equations describing the motion of the material point under the influence of the gravity, the Abel integral equation (Tautochrone problem) will be considered. The talk also deals with importance of the gravity force for the existence of living world, on water demand of living world in biology, of human, plants and animals on non-local character of water, its information carrier character. The fact that water circulation in nature is in fact information exchange, will be revealed.

The importance of Volterra integral equations in mathematical biology and medicine, the fact that biological processes have memory, in other words, are of non-local character, the dependence of biological processes on its backward also will be discussed.
It will be talked about the fact that as a negative derivative function the Volterra integral equation in fact is a special case of fractional equation in Riemann-Liouville sense.

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