and Computational Methods

doi: 10.18698/2309-3684-2014-1-131144

The paper deals with the citation index and h-index, which are the main scientometric indices, currently used for evaluating the performance of scientists and university professors. The author indicates their main disadvantages and considers a number of illustra-tive examples. The study shows that the normalized citation index (taking into account the presence of co-authors) has a number of important advantages in comparison with other scientometric indices. The author proposes new indices — the maximum citation indices, which can be easily calculated, have a simple and clear interpretation and have a number of distinct advantages in comparison with the h-index.

Polyanin A. Disadvantages of citation index and Hirsch and using other scientometrics. Маthematical Modeling and Coтputational Methods, 2014, №1 (1), pp. 131-144

doi: 10.18698/2309-3684-2019-1-6597

A brief review of existing modifications of the method of functional separation of varia-bles is given. A new more general approach is proposed for construction of ex-act solu-tions of nonlinear equations of mathematical physics and mechanics, which is based on implicit transformations of integral type in combination using the split-ting principle. The effectiveness of this approach is illustrated on nonlinear diffusion equations that contain reaction and convective terms with variable coefficients. The focus is on equations of a fairly general form that depend on two or three arbitrary functions (such nonlinear equations are the most difficult to analyze). Many new exact solutions with functional separation of variables and generalized traveling wave type solutions are described. The obtained solutions can be used to test various numerical and approximate analytical methods of mathematical physics

Полянин А.Д. Методы функционального разделения переменных и их применение в математической физике. Математическое моделирование и численные методы, 2019, № 1, с. 67–97.

doi: 10.18698/2309-3684-2014-4-5373

In the article we explored nonlinear hyperbolic delay reaction-diffusion equations with varying transfer coefficients. A number of generalized separable solutions were obtained. Most of the equations considered contain arbitrary functions. Global nonlinear instability conditions of solutions of hyperbolic delay reaction-diffusion systems were determined. The generalized Stokes problem for a linear delay diffusion equation with periodic boundary conditions was solved.

Polyanin A., Sorokin V., Vyazmin A. Nonlinear delay reaction-diffusion equations of hyperbolic type: Exact solutions and global instability. Маthematical Modeling and Coтputational Methods, 2014, №4 (4), pp. 53-73

doi: 10.18698/2309-3684-2015-4-337

We present a number of new simple separable, generalized separable, and functional separable solutions to one-dimensional nonlinear delay reaction-diffusion equations with varying transfer coefficients of the formut = [G(u)ux ]x F(u,w),where w = u(x,t) and w = u(x,t ), with denoting the delay time. All of the equations considered contain one, two, or three arbitrary functions of a single argument. The generalized separable solutions are sought in the form =1 = () () N

n n n u x t , withn (x) and n (t) to be determined in the analysis using a new modification of the functional constraints method. Some of the results are extended to nonlinear delay reaction-diffusion equations with time-varying delay = (t). We also present exact solutions to more complex, three-dimensional delay reactiondiffusion equations of the formut = div[G(u)u] F(u,w).Most of the solutions obtained involve free parameters, so they may be suitable for solving certain problems as well as testing approximate analytical and numerical methods for non-linear delay PDEs.

Polyanin A., Zhurov A. Nonlinear delay reaction-diffusion equations with varying transfer coefficients: generalized and functional separable solutions. Маthematical Modeling and Coтputational Methods, 2015, №4 (8), pp. 3-37

doi: 10.18698/2309-3684-2021-2-96116

The problems of estimating the similarity index of inhomogeneous scientific publications containing equations and formulas are discussed for the first time. It is shown that the presence of equations and formulas (as well as figures, drawings, and tables) is a complicating factor that significantly complicates the study of such texts. It has been proved that the method for determining the similarity index of publications, based on taking into account individual mathematical symbols and parts of equations and formulas, is ineffective and can lead to erroneous and even completely absurd conclusions. Possibilities of the most popular analytical systems Antiplagiat and iThenticate, currently used in scientific journals, are investigated for detecting plagiarism and self–plagiarism. The results of processing by the iThenticate system of specific examples and specific test problems containing equations and formulas are presented. It has been established that this analytical system, when analyzing heterogeneous texts, is often unable to distinguish self– plagiarism from pseudo-self-plagiarism, seeming real (but false and imaginary) self– plagiarism. A model complex situation is considered, in which the identification of self–plagiarism requires the involvement of highly qualified specialists of a narrow profile. Various ways to improve the work of analytical systems for comparing inhomogeneous texts are proposed. This article will be useful to researchers and university teachers in physics, mathematics, and engineering, programmers dealing with problems in image recognition and research topics of digital image processing, as well as a wide range of readers who are interested in issues of plagiarism and self–plagiarism.

Полянин А.Д., Шингарева И.К. Индекс подобия математических и других научных публикаций с уравнениями и формулами и проблема идентификации самоплагиата. Математическое моделирование и численные методы, 2021, № 2, с. 96–116.

doi: 10.18698/2309-3684-2024-1-124142

Rather general nonstationary strongly nonlinear partial differential equations with three independent variables are investigated, which contain the first time derivative and a quadratic combination of the second derivatives with respect to spatial variables of the Monge–Ampere type (such equations are often called parabolic Monge–Ampere equations). Some equations of this type are found in differential geometry and electron magnetohydrodynamics. This paper describes multiparameter transformations that preserve the form of the considered class of nonlinear equations, which is given by an arbitrary function. Two-dimensional and one-dimensional reductions leading to simpler partial differential equations with two independent variables or ordinary differential equations are also considered. Using methods of generalized separation of variables, a number of exact solutions have been constructed, many of which can be represented in elementary functions. The obtained results and exact solutions can be used to assess the accuracy and analyze the adequacy of numerical methods for solving problems described by strongly nonlinear partial differential equations.

Полянин А.Д. Преобразования, редукции и точные решения широкого класса нестационарных уравнений с нелинейностью типа Монжа – Ампера. Математическое моделирование и численные методы, 2024, № 1, с. 124–142.