Класифікація диференціальних рівнянь за симетрійними властивостями (за матеріалами наукового повідомлення на засіданні Президії НАН України 5 липня 2017 р.)

У доповіді розглянуто задачу класифікації ліївських симетрій у класах нелінійних диференціальних рівнянь з частинними похідними. Такі симетрії, зокрема, дозволяють відібрати фізично важливі рівняння з певного класу, а також побудувати їх точні розв'язки. Для багатьох класів рівнянь, що є важливими для застосувань, класичні методи групового аналізу не дозволяють отримати вичерпну класифікацію симетрій. Такі задачі потребують нових підходів, більшість з яких ґрунтуються на використанні невироджених точкових перетворень. На прикладах групової класифікації узагальнених рівнянь Кавахари та квазілінійних рівнянь реакції—дифузії показано ефективність нещодавно розроблених методів, зокрема відшукання найбільш широких груп еквівалентності та відображень між класами.The report is devoted to the problem of Lie symmetry classification for classes of nonlinear partial differential equations. Such symmetries allow one, in particular, to select equations of potential physical interest and to construct their exact solutions. For many classes of partial differential equations which are important for applications classical methods of group analysis do not result in exhaustive group classification. Such complicated group classification problems require new tools to be solved completely. Majority of the modern approaches are based on the usage of nondegenerate point transformations. Using the group classifications of variable coefficient generalized Kawahara equations and quasilinear reaction—diffusion equations as illustrative examples, we show the effectiveness of the recently developed approaches. These approaches include, in particular, the construction of the widest possible equivalence groups and the method of mapping between classes

Lie symmetry group, exact solutions and conservation laws for multi-term time fractional differential equations

In this paper, the time fractional Benjamin-Bona-Mahony-Peregrine (BBMP) equation and time-fractional Novikov equation with the Riemann-Liouville derivative are investigated through the use of Lie symmetry analysis and the new Noether's theorem. Then, we construct their group-invariant solutions by means of Lie symmetry reduction. In addition, the power-series solutions are also obtained with the help of the Erdélyi-Kober (E-K) fractional differential operator. Furthermore, the conservation laws for the time-fractional BBMP equation are established by utilizing the new Noether's theorem

Exact and Approximate Symmetries and Approximate Conservation Laws of Differential Equations with a Small parameter

The frameworks of Baikov-Gazizov-Ibragimov (BGI) and Fushchich-Shtelen (FS) approximate symmetries have proven useful for many examples where a small perturbation of an ordinary or partial differential equation (ODE, PDE) destroys its local exact symmetry group. For the perturbed model, some of the local symmetries of the unperturbed equation may (or may not) re-appear as approximate symmetries. Approximate symmetries are useful as a tool for systematic construction of approximate solutions. While for algebraic and first-order differential equations, to every point symmetry of the unperturbed equation, there corresponds an approximate point symmetry of the perturbed equation, for second and higher-order ODEs, this is not the case: a point symmetry of the original ODE may be unstable, that is, not have an analogue in the approximate point symmetry classification of the perturbed ODE. We show that such unstable point symmetries correspond to higher-order BGI approximate symmetries of the perturbed ODE, and can be systematically computed. We present a relation between BGI and FS approximate point symmetries for perturbed ODEs. Multiple examples of computations of exact and approximate point and local symmetries are presented, with two detailed examples that include a fourth-order nonlinear Boussinesq ODE reduction. Examples of the use of higher-order approximate symmetries and approximate integrating factors to obtain approximate solutions of higher-order ODEs, including Benjamin-Bona-Mahony ODE reduction are provided. The frameworks of BGI and FS approximate symmetries are used to study symmetry properties of partial differential equations with a small parameter. In general, we show that unlike in the ODE case, unstable point symmetries of an unperturbed PDE do not necessarily yield local approximate symmetries for the perturbed equation. We classify stable point symmetries of a one-dimensional wave model in terms of BGI and FS frameworks. We find a connection between BGI and FS approximate local symmetries for a PDE family. We classify approximate point symmetries for a family of one-dimensional wave equations with a small nonlinear term, and construct a physical approximate solution for a family that includes a one dimensional wave equation describing the wave motion in a hyperelastic material with a single family of fibers. For this model, we find wave breaking times numerically and using the approximate solution. A complete classification of exact and approximate point symmetries of the two-dimensional wave equation with a general small nonlinearity is presented. We investigate approximate conservation laws of systems of perturbed PDEs. We apply the direct mul tiplier method to obtain new approximate conservation laws for perturbed PDEs including nonlinear heat and wave equations. We show that the direct method generalizes the Noether’s theorem for construction of approximate conservation laws by proving that an approximate multiplier corresponds to an approximate local symmetry of an approximately variational problem. We present two formulas relating to construct ad ditional approximate conservation laws for a system of perturbed PDEs. We illustrate these formulas using perturbed wave equation and nonlinear telegraph system. An application for using approximate conservation laws to construct potential systems and approximate potential symmetries is provide

Applications of symmetries and conservation laws to the study of nonlinear elasticity equations

Mooney-Rivlin hyperelasticity equations are nonlinear coupled partial differential equations (PDEs) that are used to model various elastic materials. These models have been extended to account for fiber reinforced solids with applications in modeling biological materials. As such, it is important to obtain solutions to these physical systems. One approach is to study the admitted Lie symmetries of the PDE system, which allows one to seek invariant solutions by the invariant form method. Furthermore, knowledge of conservation laws for a PDE provides insight into conserved physical quantities, and can be used in the development of stable numerical methods. The current Thesis is dedicated to presenting the methodology of Lie symmetry and conservation law analysis, as well as applying it to fiber reinforced Mooney-Rivlin models. In particular, an outline of Lie symmetry and conservation law analysis is provided, and the partial differential equations describing the dynamics of a hyperelastic solid are presented. A detailed example of Lie symmetry and conservation law analysis is done for the PDE system describing plane strain in a Mooney-Rivlin solid. Lastly, Lie symmetries and conservation laws are studied in one and two dimensional models of fiber reinforced Mooney-Rivlin materials

Applications of Mathematical Models in Engineering

The most influential research topic in the twenty-first century seems to be mathematics, as it generates innovation in a wide range of research fields. It supports all engineering fields, but also areas such as medicine, healthcare, business, etc. Therefore, the intention of this Special Issue is to deal with mathematical works related to engineering and multidisciplinary problems. Modern developments in theoretical and applied science have widely depended our knowledge of the derivatives and integrals of the fractional order appearing in engineering practices. Therefore, one goal of this Special Issue is to focus on recent achievements and future challenges in the theory and applications of fractional calculus in engineering sciences. The special issue included some original research articles that address significant issues and contribute towards the development of new concepts, methodologies, applications, trends and knowledge in mathematics. Potential topics include, but are not limited to, the following: Fractional mathematical models; Computational methods for the fractional PDEs in engineering; New mathematical approaches, innovations and challenges in biotechnologies and biomedicine; Applied mathematics; Engineering research based on advanced mathematical tools

Numerical and Analytical Methods in Electromagnetics

Like all branches of physics and engineering, electromagnetics relies on mathematical methods for modeling, simulation, and design procedures in all of its aspects (radiation, propagation, scattering, imaging, etc.). Originally, rigorous analytical techniques were the only machinery available to produce any useful results. In the 1960s and 1970s, emphasis was placed on asymptotic techniques, which produced approximations of the fields for very high frequencies when closed-form solutions were not feasible. Later, when computers demonstrated explosive progress, numerical techniques were utilized to develop approximate results of controllable accuracy for arbitrary geometries. In this Special Issue, the most recent advances in the aforementioned approaches are presented to illustrate the state-of-the-art mathematical techniques in electromagnetics