Papers by Andrei Polyanin
Mathematics
The study considers a nonlinear multi-parameter reaction–diffusion system of two Lotka–Volterra-t... more The study considers a nonlinear multi-parameter reaction–diffusion system of two Lotka–Volterra-type equations with several delays. It treats both cases of different diffusion coefficients and identical diffusion coefficients. The study describes a few different techniques to solve the system of interest, including (i) reduction to a single second-order linear ODE without delay, (ii) reduction to a system of three second-order ODEs without delay, (iii) reduction to a system of three first-order ODEs with delay, (iv) reduction to a system of two second-order ODEs without delay and a linear Schrödinger-type PDE, and (v) reduction to a system of two first-order ODEs with delay and a linear heat-type PDE. The study presents many new exact solutions to a Lotka–Volterra-type reaction–diffusion system with several arbitrary delay times, including over 50 solutions in terms of elementary functions. All of these are generalized or incomplete separable solutions that involve several free para...
Mathematics, 2020
The study gives a brief overview of existing modifications of the method of functional separation... more The study gives a brief overview of existing modifications of the method of functional separation of variables for nonlinear PDEs. It proposes a more general approach to the construction of exact solutions to nonlinear equations of applied mathematics and mathematical physics, based on a special transformation with an integral term and the generalized splitting principle. The effectiveness of this approach is illustrated by nonlinear diffusion-type equations that contain reaction and convective terms with variable coefficients. The focus is on equations of a fairly general form that depend on one, two or three arbitrary functions (such nonlinear PDEs are most difficult to analyze and find exact solutions). A lot of new functional separable solutions and generalized traveling wave solutions are described (more than 30 exact solutions have been presented in total). It is shown that the method of functional separation of variables can, in certain cases, be more effective than (i) the n...
Applied Mathematics Letters, 2019
The paper describes a new approach for constructing exact solutions to nonlinear partial differen... more The paper describes a new approach for constructing exact solutions to nonlinear partial differential equations that employs separation of variables using special (nonlinear integral) transformations and the splitting principle. To illustrate its effectiveness, the method is applied to nonlinear reaction-diffusion type equations that involve variable coefficients and arbitrary functions. New exact functional separable solutions as well as generalized traveling wave solutions are obtained.
Theoretical Foundations of Chemical Engineering, 2015
New classes of exact solutions to nonlinear hyperbolic reaction—diffusion equations with delay ar... more New classes of exact solutions to nonlinear hyperbolic reaction—diffusion equations with delay are described. All of the equations under consideration depend on one or two arbitrary functions of one argument, and the derived solutions contain free parameters (in certain cases, there can be any number of these parameters). The following solutions are found: periodic solutions with respect to time and space variable, solutions that describe the nonlinear interaction between a standing wave and a traveling wave, and certain other solutions. Exact solutions are also presented for more complex nonlinear equations in which delay arbitrarily depends on time. Conditions for the global instability of solutions to a number of reaction—diffusion systems with delay are derived. The generalized Stokes problem subject to the periodic boundary condition, which is described by a linear diffusion equation with delay, is solved.
Theoretical Foundations of Chemical Engineering, 2013
ABSTRACT Differential-difference heat conduction and diffusion models and equations with a finite... more ABSTRACT Differential-difference heat conduction and diffusion models and equations with a finite relaxation time are described that give a finite disturbance propagation rate. The modified Biot-Fourier law with delay is used for the heat flux, which leads to the differential-difference heat-conduction equation $\left. {\frac{{\partial T}} {{\partial t}}} \right|_{t + \tau } = a\Delta T, $ where the left-hand side is calculated at t + τ (τ is the relaxation time) and the right-hand side is calculated, as usual, at t (without a time shift). At τ = 0, the differential-difference heat-conduction equation turns into the classical parabolic heat-conduction equation; if the left-hand side is expanded into a series in τ and two main terms of expansion are retained, we obtain the Cattaneo-Vernotte hyperbolic heat-conduction equation. An exact solution to the differential-difference heat-conduction equation is derived for a one-dimensional problem without the initial conditions with an arbitrary periodic boundary condition. The approximate solution is constructed to the general three-dimensional initial-boundary value problem of heat propagation with a finite relaxation time in a bounded domain with arbitrary initial heat distribution and the boundary condition of the third kind. It is shown that the differential-difference model makes it possible to derive the Oldroyd-type differential heat-conduction model.
Doklady Physics, 2010
Solid isotropic waveguides are frequently used for generation, transmission, and mechanicalvibrat... more Solid isotropic waveguides are frequently used for generation, transmission, and mechanicalvibration amplification, for example, in low frequency acoustic transducers. The theoretical investigation of low drequency acoustic transducers, and electromagnetic waveguides is conventionally based on using second� order wave equations (1-3). This approach is justified for describing the wavepropagation effects in rela� tively thin and long solid rods. As Rayleigh showed (4), the error caused by neglecting the effects of the trans� verse motion of the rod is proportional to the square of the ratio between the characteristic crosssection radius and the length of the rod. In (5) for more exact analysis of the longitudinal vibrations of a relatively thick and short rod, we used the more accurate Rayleigh model, which took into account the effects of inertia of the transverse cross section. It was shown that the eigenvalues for har� monic vibrations of the rod with constant and variable cross sections are limited from above. Therefore, the Rayleigh model has a restricted field of application. In this work, for analyzing the longitudinal vibra� tions of a conical rod, we used the Rayleigh-Bishop model (6, 7), which generalizes the Rayleigh model and takes into account both lateral displacements and the shear stress in the transverse cross section. The rod vibrations are described by a linear partial differential equation with the variable coefficients containing a mixed fourthorder derivative. Free and forced vibra� tions of cylindrical and conical rods are considered. It has been shown that the classical model of longitudi� nal vibrations of the rod described by the secondorder wave equation can substantially overestimate the fre� quencies of the rod free of vibrations in comparison with the Rayleigh-Bishop model. It should be noted that transverse vibrations of a rod described by a linear partial differential equation of the fourth order were considered in many works (see, for example, (8-11)).
Communications in Nonlinear Science and Numerical Simulation, 2014
We present a number of exact multiplicative and additive separable solutions to nonlinear delay r... more We present a number of exact multiplicative and additive separable solutions to nonlinear delay reaction-diffusion equations of the form u t = ku xx + F (u, w), where u = u(x, t), w = u(x, t − τ), and τ is the delay time. All of the equations involve one arbitrary function of one argument. We also give a number of exact solutions to more complex nonlinear partial functional-differential equations with a time-varying delay τ = τ (t) as well as to equations with several delay times. We extend the results to nonlinear partial differential-difference equations containing arbitrary linear differential operators of any order in the independent variables x and t; in particular, these equations include the nonlinear delay Klein-Gordon equation.
Applied Mathematical Modelling, 2008
In this paper we analyse the vibrations of an N-stepped Rayleigh bar with sections of complex geo... more In this paper we analyse the vibrations of an N-stepped Rayleigh bar with sections of complex geometry, supported by end lumped masses and springs. Equations of motion and boundary conditions are derived from the Hamilton's variational principal. The solutions for tapered and exponential sections are given. Two types of orthogonality for the eigenfunctions are obtained. The analytic solution to the N-stepped Rayleigh model is constructed in terms of Green function.
Doklady Physics, 2007
We consider one-dimensional longitudinal vibrations of a rigid rod with a nonuniform cross-sectio... more We consider one-dimensional longitudinal vibrations of a rigid rod with a nonuniform cross-section, fixed at its ends with lumped masses and springs. The cross-section inertia effects are taken into account on the basis of the Rayleigh theory. The equation of motion and the boundary conditions are derived from Hamilton’s variational principle. The characteristic equation is constructed and the eigenvalues for
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We propose a new method for constructing exact solutions to n onlinear delay reaction–diffusion e... more We propose a new method for constructing exact solutions to n onlinear delay reaction–diffusion equations of the form ut = kuxx + F (u,w), whereu = u(x, t),w = u(x, t−τ), andτ is the delay time. The method is based on searching for solutions in the formu = ∑N n=1 ξn(x)ηn(t), where the functions ξn(x) andηn(t) are determined from additional functional constraints (which are difference o r functional equations) and the original delay partial differential equation. All of the equations consid ered contain one or two arbitrary functions of a single argument. We describe a considerable number of new e xact generalized separable solutions and a few more complex solutions representing a nonlinear su perposition of generalized separable and traveling wave solutions. All solutions involve free pa rameters (in some cases, infinitely many parameters) and so can be suitable for solving certain probl ems and testing approximate analytical and numerical methods for nonlinear delay PDEs. The resul...
New exact solutions of equations of longitudinal vibration of conical and exponential rod are obt... more New exact solutions of equations of longitudinal vibration of conical and exponential rod are obtained for the Rayleigh-Love model. These solutions are used as reference results for checking accuracy of the method of lines. It is shown that the method of lines generates solu-tions, which are very close to those that are predicted by the exact theory. It is also shown that the accuracy of the method of lines is improved with increasing the number of intervals on the rod. Reliability of numerical methods is very important for obtaining approximate so-lutions of physical and technical problems. In the present paper we consider the Rayleigh-Love model of longitudinal vibrations of rods with conical and exponential cross-sections. It is shown that exact solution of the problem of longitudinal vibration of the conical rod is ob-tained in Legendre spherical functions and the corresponding solution for the rod of exponen-tial cross-section is expressed in the Gauss hypergeometric functions....
This note considers fairly general quasi-homogeneous systems of first-order nonlinear ODEs and ho... more This note considers fairly general quasi-homogeneous systems of first-order nonlinear ODEs and homogeneous systems of second-order nonlinear ODEs that contain arbitrary functions of several arguments. It presents several exact solutions to these systems in terms of elementary functions.
Two new methods of numerical integration of Cauchy problems for ODEs with blow-up solutions are d... more Two new methods of numerical integration of Cauchy problems for ODEs with blow-up solutions are described. The first method is based on applying a differential transformation, where the first derivative (given in the original equation) is chosen as a new independent variable. The second method is based on introducing a new non-local variable that reduces ODE to a system of coupled ODEs. Both methods lead to problems whose solutions do not have blowing-up singular points; therefore the standard numerical methods can be applied. The efficiency of the proposed methods is illustrated with several test problems.
Mathematics
We study nonlinear pantograph-type reaction–diffusion PDEs, which, in addition to the unknown u=u... more We study nonlinear pantograph-type reaction–diffusion PDEs, which, in addition to the unknown u=u(x,t), also contain the same functions with dilated or contracted arguments of the form w=u(px,t), w=u(x,qt), and w=u(px,qt), where p and q are the free scaling parameters (for equations with proportional delay we have 0
Mathematics
This paper describes a number of simple but quite effective methods for constructing exact soluti... more This paper describes a number of simple but quite effective methods for constructing exact solutions of nonlinear partial differential equations that involve a relatively small amount of intermediate calculations. The methods employ two main ideas: (i) simple exact solutions can serve to construct more complex solutions of the equations under consideration and (ii) exact solutions of some equations can serve to construct solutions of other, more complex equations. In particular, we propose a method for constructing complex solutions from simple solutions using translation and scaling. We show that in some cases, rather complex solutions can be obtained by adding one or more terms to simpler solutions. There are situations where nonlinear superposition allows us to construct a complex composite solution using similar simple solutions. We also propose a few methods for constructing complex exact solutions to linear and nonlinear PDEs by introducing complex-valued parameters into simpl...
Journal of Nonlinear Mathematical Physics
The paper is concerned with different classes of nonlinear Klein-Gordon and telegraph type equati... more The paper is concerned with different classes of nonlinear Klein-Gordon and telegraph type equations with variable coefficients c(x)u tt + d(x)u t = [a(x)u x ] x + b(x)u x + p(x) f (u), where f (u) is an arbitrary function. We seek exact solutions to these equations by the direct method of symmetry reductions using the composition of functions u = U(z) with z = ϕ(x,t). We show that f (u) and any four of the five functional coefficients a(x), b(x), c(x), d(x), and p(x) in such equations can be set arbitrarily, while the remaining coefficient can be expressed in terms of the others. The study investigates the properties and finds some solutions of the overdetermined system of PDEs for ϕ(x,t). Examples of specific equations with new exact functional separable solutions are given. In addition, the study presents some generalized traveling wave solutions to more complex, nonlinear Klein-Gordon and telegraph type equations with delay.
Доклады Академии наук
A new method for constructing exact solutions of nonlinear equations of mathematical physics, whi... more A new method for constructing exact solutions of nonlinear equations of mathematical physics, which is based on nonlinear integral type transformations in combination with the splitting principle, is proposed. The effectiveness of the method is illustrated on nonlinear equations of the reaction-diffusion type, which depend on two or three arbitrary functions. New exact functional separable solutions and generalized traveling wave solutions are described.
Journal of Physics: Conference Series
We present a method of numerical integration of singularly perturbed boundaryvalue problems with ... more We present a method of numerical integration of singularly perturbed boundaryvalue problems with a small parameter, based on introducing a new nonlocal independent variable. As a result, we obtain more suitable problems that allow the application of standard fixed-step numerical methods. Two test boundary-value problems for second-order ODEs that have monotonic and non-monotonic exact or asymptotic solutions, expressed in elementary functions, are considered. Comparison of numerical, exact, and asymptotic solutions showed the high efficiency of the method based on generalized nonlocal transformations for solving singular boundary-value problems with a small parameter.
Applied Mathematics and Computation
Several new methods of numerical integration of Cauchy problems with blow-up solutions for nonlin... more Several new methods of numerical integration of Cauchy problems with blow-up solutions for nonlinear ordinary differential equations of the first-and secondorder are described. Solutions of such problems have singularities whose positions are unknown a priori (for this reason, the standard numerical methods for solving problems with blow-up solutions can lead to significant errors). The first proposed method is based on the transition to an equivalent system of equations by introducing a new independent variable chosen as the first derivative, t = y ′ x , where x and y are independent and dependent variables in the original equation. The second method is based on introducing a new auxiliary nonlocal variable of the form ξ = ∫ x x 0 g(x, y, y ′ x) dx with the subsequent transformation to the Cauchy problem for the corresponding system of ODEs. The third method is based on adding to the original equation of a differential constraint, which is an auxiliary ODE connecting the given variables and a new variable. The proposed methods lead to problems whose solutions are represented in parametric form and do
Доклады Академии наук
The study describes a new modification of the method of functional separation of variables for no... more The study describes a new modification of the method of functional separation of variables for nonlinear equations of mathematical physics. Solutions are sought in an implicit form that involves several free functions; the specific expressions of these functions are determined in the subsequent analysis of the arising functional differential equations. The effectiveness of the method is illustrated by examples of nonlinear reaction-diffusion equations and Klein-Gordon type equations with variable coefficients that depend on one or more arbitrary functions. A number of new exact functional separable solutions and generalized traveling-wave solutions are obtained.
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Papers by Andrei Polyanin