Contents


Preface

Mathematical modeling is the most effective method for studying a wide variety of processes: physical, biological, social, and many others. The mathematical model of any process includes a set of defining parameters and characteristics of the object under study, as well as the establishment of mathematical relationships between

Dynamic processes in various media and structures under the action of external and internal sources of perturbations are described, as a rule, by differential equations of various types, the solutions of which depend on the geometry of the object under study and conditions on its boundaries, which can be infinite. Mathematical models of such processes are boundary value problems of mathematical physics and

The main content of this book is related to construction of analytical solutions of differential equations and systems of mathematical physics, to the development of analytical methods for solving boundary value problems for such equations, and the study of properties of their solutions. A wide class of equations (elliptic, parabolic, and hyperbolic) is considered here, on the basis of which complex wave processes

Chapter 1 is devoted to construction and research of solutions to a complex multiparameter system of nonlinear partial differential equations of the parabolic type and their modifications with an application to the problems of hemotaxis process in living organisms. Transport solutions of these equations are constructed that describe traveling waves of the solitons type. For various partial values of the parameters, the exact solutions of these equations are constructed using the theory of Bessel and hypergeometric functions. The constructed solutions are well illus-

In Chapter 2, a two-component Biot medium is considered, which allows modeling the dynamics of liquid and gas saturated porous media and rods. By using Fourier transformation of generalized functions, Green tensors of these hyperbolic systems are constructed in spaces of different dimensions. The cases of nonstationary motion and periodic vibration are considered. The regular integral representations

of these solutions are given for acting regular and singular mass forces.

Chapter 3 is devoted to solving the boundary value problems for equations of hyperbolic type of theoretical physics, which describes the motion of elementary particles in potential fields. In particular, the Klein-Gordon equation is considered, whose solutions for various scalar fields have been studied by many authors.

Note that boundary value problems for hyperbolic equations and systems in domains with arbitrary boundary geometry are among the most complex problems of mathematical physics, since the classical potential theory, characteristic of solving boundary value problems for elliptic and parabolic systems, is not applicable in the initial space-time. This is due to singularity and hyper singularity of the fundamental solutions of hyperbolic equations on wave fronts, as well as their belonging

in biological and physical media can be simulated.

trated by the presented graphic material.

them.

mechanics.
