Linear and nonlinear differential and functional-differential equations and their applications

Linear and nonlinear differential and functional-differential equations and their applications

Year 2018

**Area of science:**

Mathematics, mechanics and informatics.

**Scientific direction:**

Mathematics and theoretical physics: modeling of dynamic systems.

The project is devoted to the development of new qualitative and geometric methods for the study of boundary value problems for differential and functional differential equations, their application to the Vlasov equations (kinetics of high-temperature plasma), the Kato problem of the square root of the operator, mathematical biology and mathematical medicine.

Project goals

- The study of boundary value problems for elliptic functional-differential equations containing in the upper part of compression with incommensurable logarithms.
- To obtain the terms for the unique solvability of the elliptic functional-differential equations with orthotropic contractions in the scale of Kondratiev weight spaces.
- To find classes of elliptic differential-difference equations for which the Dirichlet problem is correct in the half-space, to obtain integral representations and qualitative properties of its solutions.
- The study of Fredholm and index problems of G-operators associated with groups of integral Fourier operators.
- Dynamic properties of spatially discrete and continuous reaction-diffusion systems with hysteresis in the multidimensional case.
- Existence and stability of traveling waves in systems with fast oscillating coefficients.
- The study of the interaction between the nonpotential operators, bivariation and algebraic structures.
- The study of the capability information, calculate the volume of hyperbolic 5-simplex in terms of the lengths of edges by calculating one-dimensional integrals of some elementary functions.

Project leader
All participants

Andrey Shishkov

Doctor of Physics and Mathematics, professor

Project results

Algebraic terms for the unique solvability of the Dirichlet problem, as well as the continuous dependence of the solution on the compression parameters, will be obtained for the functional-differential equation with incommensurable compressions. The terms that guarantee the existence of an infinite-dimensional kernel will be obtained. In addition, the following effect will be set: the spectral properties of functional operators with compressions are unstable in regard to small perturbations of the compression parameters.

Sufficient terms for unambiguous solvability in the Kondratiev weight spaces of elliptic functional-differential equations with orthotropic compressions (stretching in one variable and compression in another) considered on the plane will be obtained, and the terms depend on the parameters of the spaces.

Poisson kernel development and obtaining an integral representation of the solution of the Dirichlet problem in the half-space of elliptic differential-difference equations of general form. The proof of the theorems on asymptotic proximity of the obtained solutions.

It is assumed to set Fredholm character for G-operators associated with groups of integral Fourier operators, both in the case of discrete groups and in the case of Lie groups. As an application, we plan to review boundary value problems for hyperbolic equations with data on the entire boundary.

Using non-Eulerian functionals, variational formulations of new classes of evolutionary problems with non-potential operators will be obtained, the interaction of their symmetry generators will be analyzed, and the corresponding non-classical integral invariants will be constructed.

Are received asymptotic formulas for the moments of the switching hysteresis in the nodes of the two-dimensional spatial lattice for discrete equations of reaction-diffusion with hysteresis will be obtained. In the spatially continuous case, the existence of self-similar solutions will be proved. For traveling waves in porous media, their proximity to traveling waves in a two-scale averaged limit system will be proved. In addition, the existence of waves in the limit system will be proved.

It is planned to obtain the volume formulas of the hyperbolic 5-simplex through the coordinates of the vertices and the lengths of the edges.

Sufficient terms on the functional-differential operator with partial derivatives will be obtained, containing the logarithms are incommensurable contraction arguments, providing strong ellipticity of these operators and justice for them, the Kato hypothesis about the square root of the operator.

It will also be proved that elliptic differential-difference operators with degeneration satisfy the Kato hypothesis about the square root of the operator.

New classes of stationary solutions of Vlasov-Poisson equations will be obtained.

It is assumed to prove the existence of solutions of mixed problems for the Vlasov-Poisson equations with an external magnetic field having a carrier lying strictly inside the domain. At the same time, sufficient terms for the physical parameters included in the equations will be constructive.

An extension of the applications of the H-theorem for the coagulation-crushing equations and the Becker-Doering equation, for the Liouville type equations in Hamiltonian mechanics and for the Hamilton-Jacobi method will be obtained.

Multiscale models of the processes of interaction between immune systems and pathogens, taking into account various variants of the heterogeneity of the component processes will be developed.

Nonlocal reaction-diffusion equations and delay equations will be studied. The spectral properties of the corresponding linear operators will be analyzed. Properties of nonlinear problems will be studied and a topological degree will be constructed. The existence of reaction-diffusion waves and pulses will be considered for blood clotting and infection propagation models.

Application area

- To find new classes of operators satisfying Kato hypothesis square root from the operator: strongly elliptic functional-differential operators with incommensurable logarithms of argument contractions and elliptic differential-difference equations with degeneration.
- To study stationary solutions of the system of Vlasov-Poisson equations describing the equilibrium state of high-temperature plasma in a thermonuclear reactor, to study the kinetics of high-temperature plasma, as well as the growth of entropy and its consequences in mechanics and kinetics.
- To study the dynamics of distributed systems (in physical space or feature space) describing the interaction of immune cells and pathogens in humans and animals.
- To study of reaction-diffusion waves and pulses to predict thrombosis and spread of infection in tissue.