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 them.

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 mechanics.

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 in biological and physical media can be simulated.

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 illustrated by the presented graphic material.

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

to the class of singular generalized functions in spaces with odd dimensions, which makes it impossible to apply classical methods of potential theory for such BVPs.

asymptotic interior and exterior Padé approximations for these problems are constructed. In the last problem, the influence of the Mach number on asymptotic

Chapter 6 is expository for the importance of using programming in algebraic calculations. Although complete binomial and multinomial construction can be a hard task, there exist some mathematical formulas that can be deployed to calculate binomial and multinomial coefficients, in order to make it quicker. A main aim here is the development of an alternative method to carry out the calculation of binomial

The analytic formulas are presented, that yield binomial coefficients, by means of summation series, and the equation targeted at binomial calculations, is deduced which is convenient for calculations. Finally an algorithm set up on Computing Algebra System (CAS) Maxima is raised. The Appendix explains all calculus and logic deductions in this chapter. This algorithm checks if it is faster than usual one

Chapter 7 is devoted to problems of interpolation of nondifferential functions, which are typical for real and experimental signals by study of different physical processes and others. The authors provide an overview of several types of fractal interpolation functions. The connections between fractal interpolation functions resulting from Banach contractions as well as those resulting from Rakotch contractions are considered. The theoretical and practical significance for the generation of fractal functions for interpolation purposes in 2D and 3D spaces is

given. The new methods presented can be extended to piecewise fractal

The book would be interesting for specialists in the field of mathematical and theoretical physics, mechanics and biophysics, students of mechanics, mathematics, physics and biology departments of higher educational institutions. The thoughtful reader will find in it a lot that is necessary and useful for his scientific research

**Lyudmila Alexeyeva**

Kazakhstan

Institute of Mathematics and Mathematical Modeling,

of the solutions is also investigated.

and multinomial coefficients.

by calculations.

interpolation functions.

work.

The method of generalized functions, used in Chapter 3 for solving the boundary value problems, allows construction of regular integral representations of solutions, which determine the solution inside the domain through the boundary values of the solution and their derivatives. Some of them are known from boundary conditions, and to determine the unknown boundary functions, the resolving singular boundary integral equations are constructed in 2D and 3D spaces. The correctness of the posed problems is proved, taking into account the appearance of shock waves.

Chapter 4 is devoted to the development of the method of generalized functions (GFM) for construction of solutions of BVPs for hyperbolic systems of mathematical physics, which describe wave processes and dynamics of continue media, in particular, dynamics of elastic solids and media.

This very constructive method is based on the idea of transition from the classical formulation of initial boundary value problems to its formulation in the space of generalized functions. This allows reducing a process of BVP solving to solving the differential equations system with a singular right-hand side in the space of generalized functions. This singular part contains simple and double layers, the densities of which are determined by the value of the solution and its derivatives on the boundary. By using the Green matrix (tensor) of these equations, a generalized solution of the BVP can be obtained in the form of a convolution of the right-hand side with this matrix. The regularization of solutions and transition to their regular representations makes it possible to construct a classical solution for the BVP. The asymptotic properties of Green tensor and the tensors derived from it, which are the kernels of these integral representations, make it possible to construct resolving singular boundary integral equations.

The method of generalized functions is universal and can be applied to differential equations of any types. It allows us to study processes accompanied by shock waves, which is often impossible using classical methods. All these issues are considered in sufficient detail in this chapter.

The next part of this book covers the construction of various approximations for solutions of differential equations and functions. Note that the use of various approximations in the form of series and sequences of elementary and special functions to construct solutions to equations and boundary value problems is one of the most common ways to solve them, and the choice of such approximations is closely related to the specifics of the problems being solved.

The Padé approximation method is one of the most promising nonlinear methods for summing power series and localizing its singular points. It is convenient to use it when constructing solutions of equations based on asymptotic expansions of solutions in a small parameter in the vicinity of singular points.

Chapter 5 is devoted to applications of asymptotic methods in solving the nonlinear BVPs of mechanics using the Padé approximation. Here three boundary value problems for nonlinear ordinary differential equations are considered: the Airy boundary value problem, the BVP for Blasius equation, which describes laminar flow of boundary layers, and BVP for equations of laminar boundary layer near a semi-infinite plate in super-sonic flow of viscous perfect gas. In this chapter,

asymptotic interior and exterior Padé approximations for these problems are constructed. In the last problem, the influence of the Mach number on asymptotic of the solutions is also investigated.

Chapter 6 is expository for the importance of using programming in algebraic calculations. Although complete binomial and multinomial construction can be a hard task, there exist some mathematical formulas that can be deployed to calculate binomial and multinomial coefficients, in order to make it quicker. A main aim here is the development of an alternative method to carry out the calculation of binomial and multinomial coefficients.

The analytic formulas are presented, that yield binomial coefficients, by means of summation series, and the equation targeted at binomial calculations, is deduced which is convenient for calculations. Finally an algorithm set up on Computing Algebra System (CAS) Maxima is raised. The Appendix explains all calculus and logic deductions in this chapter. This algorithm checks if it is faster than usual one by calculations.

Chapter 7 is devoted to problems of interpolation of nondifferential functions, which are typical for real and experimental signals by study of different physical processes and others. The authors provide an overview of several types of fractal interpolation functions. The connections between fractal interpolation functions resulting from Banach contractions as well as those resulting from Rakotch contractions are considered. The theoretical and practical significance for the generation of fractal functions for interpolation purposes in 2D and 3D spaces is given. The new methods presented can be extended to piecewise fractal interpolation functions.

The book would be interesting for specialists in the field of mathematical and theoretical physics, mechanics and biophysics, students of mechanics, mathematics, physics and biology departments of higher educational institutions. The thoughtful reader will find in it a lot that is necessary and useful for his scientific research work.

> **Lyudmila Alexeyeva** Institute of Mathematics and Mathematical Modeling, Kazakhstan

to the class of singular generalized functions in spaces with odd dimensions, which makes it impossible to apply classical methods of potential theory for such BVPs.

The method of generalized functions, used in Chapter 3 for solving the boundary value problems, allows construction of regular integral representations of solutions, which determine the solution inside the domain through the boundary values of the solution and their derivatives. Some of them are known from boundary conditions, and to determine the unknown boundary functions, the resolving singular boundary integral equations are constructed in 2D and 3D spaces. The correctness of the posed problems is proved, taking into account the appearance of shock waves.

Chapter 4 is devoted to the development of the method of generalized functions (GFM) for construction of solutions of BVPs for hyperbolic systems of mathematical physics, which describe wave processes and dynamics of continue media, in

This very constructive method is based on the idea of transition from the classical formulation of initial boundary value problems to its formulation in the space of generalized functions. This allows reducing a process of BVP solving to solving the differential equations system with a singular right-hand side in the space of generalized functions. This singular part contains simple and double layers, the densities of which are determined by the value of the solution and its derivatives on the boundary. By using the Green matrix (tensor) of these equations, a generalized solution of the BVP can be obtained in the form of a convolution of the right-hand side with this matrix. The regularization of solutions and transition to their regular representations makes it possible to construct a classical solution for the BVP. The asymptotic properties of Green tensor and the tensors derived from it, which are the kernels of these integral representations, make it possible to construct resolving

The method of generalized functions is universal and can be applied to differential equations of any types. It allows us to study processes accompanied by shock waves, which is often impossible using classical methods. All these issues are considered in

The next part of this book covers the construction of various approximations for solutions of differential equations and functions. Note that the use of various approximations in the form of series and sequences of elementary and special functions to construct solutions to equations and boundary value problems is one of the most common ways to solve them, and the choice of such approximations is

The Padé approximation method is one of the most promising nonlinear methods for summing power series and localizing its singular points. It is convenient to use it when constructing solutions of equations based on asymptotic expansions of solu-

Chapter 5 is devoted to applications of asymptotic methods in solving the nonlinear BVPs of mechanics using the Padé approximation. Here three boundary value problems for nonlinear ordinary differential equations are considered: the Airy boundary value problem, the BVP for Blasius equation, which describes laminar flow of boundary layers, and BVP for equations of laminar boundary layer near a semi-infinite plate in super-sonic flow of viscous perfect gas. In this chapter,

closely related to the specifics of the problems being solved.

tions in a small parameter in the vicinity of singular points.

particular, dynamics of elastic solids and media.

singular boundary integral equations.

sufficient detail in this chapter.

**IV**

**Chapter 1**

**Abstract**

in a future study.

**1. Introduction**

2016.

**1**

Keller-Segel model, chemotaxis

*Maria Vladimirovna Shubina*

Exact Traveling Wave Solutions

Parabolic Models of Chemotaxis

In this chapter we consider several different parabolic-parabolic systems of chemotaxis which depend on time and one space coordinate. For these systems we obtain the exact analytical solutions in terms of traveling wave variables. Not all of these solutions are acceptable for biological interpretation, but there are solutions that require detailed analysis. We find this interesting, since chemotaxis is present in the continuous mathematical models of cancer growth and invasion (Anderson, Chaplain, Lolas, et al.) which are described by the systems of reaction–diffusiontaxis partial differential equations, and the obtaining of exact solutions to these systems seems to be a very interesting task, and a more detailed analysis is possible

**Keywords:** parabolic-parabolic system, exact solution, soliton solution, Patlak-

1.Exact Traveling Wave Solutions of One-Dimensional Parabolic-Parabolic Models of Chemotaxis, Russian J Math Phys., Maik Nauka/Interperiodica

2.The 1D parabolic-parabolic Patlak-Keller-Segel model of chemotaxis: The particular integrable case and soliton solution, J Math Phys., 57(9), 091501,

Chemotaxis, or the directed cell (bacteria or other organisms) movement up or down a chemical concentration gradient, plays an important role in many biological and medical fields such as embryogenesis, immunology, cancer growth, and invasion. The macroscopic classical model of chemotaxis was proposed by Patlak in 1953 [1] and by Keller and Segel in the 1970s [2–4]. Since then, the mathematical modeling of chemotaxis has been widely developed. This model is described by the system of coupled nonlinear partial differential equations. Proceeding from the study of the properties of these equations, it is concluded that the model demonstrates a deep

mathematical structure. The survey of Horstmann [5] provides a detailed

This chapter uses the publications of Shubina M.V.:

Publishing (Russian Federation), 25(3), 383–395, 2018.

of One-Dimensional Parabolic-
