Non-linear differential and integro-differential equations and systems describe various processes in physical, economic, chemical, technological, and other sciences. Undoubtedly, it is very important to study the qualitative and structural characteristics of the solutions of initial-boundary value problems of these equations and systems; construct and study discrete analogs; and investigate numerical algorithms. Systems of integro-differential equations arise, for example, in mathematical modeling of the process of propagation of the magnetic field into a substance.
in the paper by Gordeziani D.G., Dzhangveladze T.A., Korshia T.K. Existence and Uniqueness of a Solution of Certain Nonlinear Parabolic Problems. Dfferential'nye Uravnenyia, 1983, V.19, N7, p.1197-1207, the reduction of the well-known system of Maxwell equations to the integro-differential form is shown.
In the work by Laptev G.I. Quasilinear Evolution Partial Differential Equations with Operator Coefficients. Doctoral Dissertation, Moscow, 1990 (Russian), a generalization of the above-mentioned model is given. In particular, if the temperature along the body is considered to be a constant, i.e. dependent on time and independent of spatial variables, then the so-called averaged integro-differential model is obtained to model the process of magnetic field propagation.
For such a one-dimensional averaged integro-differential model, the large time behavior of the solution of the initial-boundary value problem is investigated. The convergence of the corresponding semidiscrete scheme for the case of nonlinearity, which extends the previously studied cases, is also studied.

Project members:

• Temur Jangveladze (Principal Investigator)