Vol. 7, No. 1 (2023): Regular Issue

This special topic section collects recent developments on methods in differential equations dedicated to Nikolai Yanenko's 100th birthday anniversary. The papers were presented at the International Conference `Analytical and Numerical Methods in Differential Equations (Yanenko 100)' which was held during 23-27 August 2021. This conference was a celebration of Yanenko's foundational contributions on analytical and numerical methods in differential equations. Its topics ranged from physical experiment and mathematical modelling of real problems all the way to abstract mathematical objects. The 6 papers published herein were selected from the 76 presentations to share these advanced developments in differential equations with the community of Journal of Vibration Testing and System Dynamics.

Vector field realizations of Lie algebras in connection with the representation theory and the branching rule problem associated to embeddings of semisimple Lie algebras are considered. This allows to determine if a subalgebra in a given realization corresponds to an irreducible embedding, as well as to determine multiplicities in the branching rules. The invariants of the realizations associated to such embeddings are used to construct (second-order) systems of ordinary differential equations possessing a fixed Lie algebra of point symmetries. It is shown that for any embedding $\mathfrak{g}^{\prime}\subset\mathfrak{g}$ and any faithful representation there exists an integer $k$ such that for any $n\geq k$, systems of order $n$ with exact Lie point symmetry $\mathfrak{g}^{\prime}$ can be constructed.

The model of a stationary flow in porous media stemming from hydraulic fracking and accounting for inertial phenomena is considered. The incompressible fluid is modeled by a nonlinear Darcy--Forchheimer (DF) equation under mixed boundary conditions, which are appropriate for a fluid-driven fracture. The classical DF equation is generalized with respect to a growth exponent $m$ and inhomogeneous coefficients. Using mixed variational formulation of the problem for unknown fluid velocity and fluid pressure, the well-posedness theorem is proved for arbitrary $m>1$. The developed Lagrange multiplier formalism is advantageous for optimal shape design of fractures.

In general, a system of differential equations is integrable if there exist `sufficiently many' first integrals (FIs) so that its solution can be found by means of quadratures. Therefore, the determination of the FIs is an important issue in order to establish the integrability of a dynamical system. In this work, we consider holonomic autonomous dynamical systems defined by equations $\ddot{q}^{a}= -\Gamma_{bc}^{a}(q) \dot{q}^{b}\dot{q}^{c} -Q^{a}(q)$ where $\Gamma^{a}_{bc}(q)$ are the coefficients of a symmetric (possibly non-metrical) connection and $-Q^{a}(q)$ are the generalized forces. We prove a theorem which produces the FIs of any order of such systems in terms of the `symmetries' of the geometry defined by the quantities $\Gamma_{bc}^{a}(q)$. We apply the theorem to compute quadratic and cubic FIs of various dynamical systems.

Of the Maxwell and Einstein equations in the framework of the Vlasov--Maxwell--Einstein equations from the classical, but more general principle of least action. The resulting derivation of the Vlasov--type equations gives the Vlasov--Einstein equations different from those proposed earlier. A method is proposed for the transition from kinetic equations to hydrodynamic--type equations. In the case of Hamiltonian mechanics, the transition to the Hamilton--Jacobi equation from the hydrodynamic consequences of the Liouville equation is possible, as was done already in quantum mechanics. Thus, in the nonrelativistic case, we obtain the Milne--McCrea{--type} solutions.

Of the Maxwell and Einstein equations in the framework of the Vlasov--Maxwell--Einstein equations from the classical, but more general principle of least action. The resulting derivation of the Vlasov--type equations gives the Vlasov--Einstein equations different from those proposed earlier. A method is proposed for the transition from kinetic equations to hydrodynamic--type equations. In the case of Hamiltonian mechanics, the transition to the Hamilton--Jacobi equation from the hydrodynamic consequences of the Liouville equation is possible, as was done already in quantum mechanics. Thus, in the nonrelativistic case, we obtain the Milne--McCrea{--type} solutions.

A new method for constructing reciprocal transformations is proposed. The method uses the same steps as for finding equivalence group of transformations. It provides a systematic tool for finding classes of reciprocal transformations. As an illustration the method is applied to the one-dimensional gas dynamics equations, and new reciprocal transformations are found.

This paper presents a theory for nonlinear dynamics of dynamical systems possessing variable-independent univariate quadratic vector fields. The dynamical systems with a constant vector field and a variable-independent quadratic vector field are presented first, and the 1-dimensional flows discussed. Dynamical systems with linear and quadratic variable-independent univariate vector fields are discussed, and the corresponding bifurcation and global dynamics are discussed. Dynamical systems with two variable-independent univariate quadratic vector fields are analyzed, and the corresponding bifurcations and global dynamics are discussed through the first integral manifolds.