The renormalizing series of some integral equations

We consider integral equations for which the perturbation expansion gives a power series in a parameter h whose coefficients are divergent integrals. We eliminate the divergent integrals by introducing a renormalizing Z(t, h) series in the minimal subtraction scheme. We investigate the convergence of the formal Z series in relation to the kernels of the integral equations. We find a relation of the renormalizing series to the Lagrange inversion series and also some other relations.     

On the determination op forces op constraint reaction : PMM vol. 39, n 6, 1975, pp. 1129–1134

The equations of motion of mechanical systems with multipliers are reduced to the form enabling the separation of these equations into two groups, the first group describing the motions of the system, and the second group defining the multipliers. Each multiplier is determined independently of the remaining multipliers, and this makes it easy to assess the dynamic effect of each constraint on the system. On the basis of this approach, we study the following problems: determination of the constraint reactions [1], study of the motion of controlled systems with prescribed constraints [2, 3] and utilization of the method of nonholonomic mechanical systems in the case when the first integrals exist [4].     

A divergent system of non-stationary equations of motion of viscoelastic media in euler coordinates

A system of divergent equations of non-stationary motions of viscoelastic media is presented. It is shown that for continuous flows it is equivalent to the well-known system of equations in /1/. Divergent equations are preferable, for instance, from the viewpoint of their utilization in calculational algorithms. On the basis of the divergent forms obtained, relationships at discontinuities are analysed.     

Extensions of the Appelrot classes for the generalized gyrostat in a double force field

For the integrable system on e(3, 2) found by Sokolov and Tsiganov we obtain explicit equations of some invariant 4-dimensional manifolds on which the induced systems are almost everywhere Hamiltonian with two degrees of freedom. These subsystems generalize the famous Appelrot classes of critical motions of the Kowalevski top. For each subsystem we point out a commutative pair of independent integrals, describe the sets of degeneration of the induced symplectic structure. With the help of the obtained invariant relations, for each subsystem we calculate the outer type of its points considered as critical points of the initial system with three degrees of freedom.     

Higher-Dimensional Integrable Newton Systems with Quadratic Integrals of Motion

Newton systems     , with integrals of motion quadratic in velocities, are considered. We show that if such a system admits two quadratic integrals of motion of the so-called cofactor type , then it has in fact n quadratic integrals of motion and can be embedded into a  (2 n + 1)  -dimensional bi-Hamiltonian system, which under some non-degeneracy assumptions is completely integrable. The majority of these cofactor pair Newton systems are new, but they also include conservative systems with elliptic and parabolic separable potentials, as well as many integrable Newton systems previously derived from soliton equations. We explain the connection between cofactor pair systems and solutions of a certain system of second-order linear PDEs (the fundamental equations ), and use this to recursively construct infinite families of cofactor pair systems.     

The solution of infinitely ill-conditioned weakly-singular problems

It is demonstrated on examples that a weak singularity (i.e., with converging improper integral) may produce in computations (depending on the algorithm employed) an infinitely ill-conditioned situation when arbitrarily small imprécisions introduced by the algorithm or by a software create divergent approximations for mathematically convergent integrals. The possibility of hidden singularities is shown, and the double error phenomenon is identified and demonstrated in a simple example. Construction of test problems is proposed to check the applicability of existing software prior to its use for the solution of real life problems with weakly-singular equations. It is shown that the application of the integration by parts formula to weakly-singular integrals may create strong singularities (i.e., unbounded terms or divergent improper integrals). Methods of removal of singularities with and without compensation are studied for the numerical solution of infinitely ill-conditioned weakly-singular problems.     

Nonlinear Waves in a Rod: Results for Incompressible Elastic Materials

A nonlinear intrinsic theory is used to describe the motions of a straight round elastic rod including the influence of radial shear and inertia. Consideration of steady wave motions reduces the two coupled partial differential equations to ordinary differential equations for which two integrals of the motion may be found. For incompressible elastic materials with the restriction of small strain gradients, but arbitrary finite strains, a large variety of exact solutions may be found by quadrature. These include large amplitude periodic waves (which may contain shocks), solitary waves, and in some cases waves that are transitional from one stress level to another. Such solutions may be found for uniform stress strain curves that are concave up or down or that contain inflections, and even for nonmontonic curves, which have been used to represent phase transitions.     

The motion of a body supported at three points on a rough plane

The problem of the motion of a heavy rigid body, supported on a rough horizontal plane at three of its points, is considered. The contacts at the support points are assumed to be unilateral and subject to the law of dry (Coulomb) friction. The dynamics of possible motions of such a body under the action of gravity forces and dry friction is investigated. In the case of a plane body, it is possible to obtain particular integrals of the equations of motion.     

Hereditary effects of exponentially damped oscillators with past histories

This paper presents hereditary effects of exponentially damped oscillators with past histories. Unlike the classical viscously damped oscillators, the nonviscously damped ones involve damping forces which depend on time-histories of vibrating motions via convolution integrals. As a result, equations of motion of such systems are a set of coupled second-order Volterra integro-differential equations. In this work, initial value problems for the integro-differential equations are revisited. The initial conditions should contain time-histories of vibrating motions. Then, initialization response of exponentially damped oscillators is obtained. It is used to characterize the hereditary effects on the dynamic response. At last, stability of initialization response is proved from the theoretical viewpoint and verified by numerical simulations. This reveals that the hereditary effects gradually recede with increasing of time.     

A special case of the motion of a point mass

When investigating the motion of a point mass in a plane. Zhukovskii [1] pointed out a case when, without finding the general integrals of the equations of motion, one can specify particular integrals of these motions. To obtain the particular integrals in explicit form, a certain constraint was imposed on the force function. The case of motion without this constraint is investigated.     

The steady motions of a disc on an absolutely rough plane

The branching of the steady motions of a heavy circular disc on an absolutely rough horizontal plane is investigated. The motions corresponding to critical points of the energy integral at fixed levels of two other integrals having the form of hypergeometric series are considered.     

A strong approximation for double subfractional integrals

A Riemann–Stieltjes integral strong approximation to double Stratonovich integrals with respect to odd and even fractional Brownian motions is considered. We prove the convergence in quadratic mean, uniformly on compact time intervals, of the ordinary double integral process obtained by linear interpolation of the odd and even fractional Brownian motions, to the double Stratonovich integral. The deterministic integrands are continuous or are given by bimeasures.     

Stabilization of the motions of non-autonomous mechanical systems

The problem of stabilizing the motions of mechanical systems that can be described by non-autonomous systems of ordinary differential equations is considered. The sufficient conditions for stabilizing of the motions of mechanical systems with assigned forces due to forces of another structure are obtained by constructing a vector Lyapunov function and a reference system. Examples of the solution of the problems of stabilizing the rotational motion of an axisymmetric satellite in an elliptic orbit, a non-tumbling gyro horizon, etc. are considered ©2009     

First Integrals and Periodic Solutions of a System with Power Nonlinearities

Under consideration is some system of ordinary differential equations with power nonlinearities. These systems are widely used in mathematical biology and chemical kinetics, and can also occur by reduction of more sophisticatedmodels. We formulate conditions on the system parameters which guarantee the existence of first integrals defined by the combinations of power and logarithmic functions of the phase variables. Using the first integrals, we construct periodic solutions for the three-variable systems. A few examples are given illustrating the results.     

Logarithmic matrix norms in motion stability problems

The problem of the stability of the motions of mechanical systems, described by non-linear non-autonomous systems of ordinary differential equations, is considered. Using the logarithmic matrix norm method, and constructing a reference system, the sufficient conditions for the asymptotic and exponential stability of unperturbed motion and for the stabilization of progammed motions of such systems are obtained. The problem of the asymptotic stability of a non-conservative system with two degrees of freedom is solved, taking for parametric disturbances into account. Examples of the solution of the problem of stabilizing programmed motions – for an inverted double pendulum and for a two-link manipulator on a stationary base – are considered.     

First integrals and families of symmetric periodic motions of a reversible mechanical system

A reversible mechanical system which allows of first integrals is studied. It is established that, for symmetric motions, the constants of the asymmetric integrals are equal to zero. The form of the integrals of a reversible linear periodic system corresponding to zero characteristic exponents and the structure of the corresponding Jordan Boxes are investigated. A theorem on the non-existence of an additional first integral and a theorem on the structural stabilities of having a symmetric periodic motion (SPM) are proved for a system with m symmetric and k asymmetric integrals. The dependence of the period of a SPM on the constants of the integrals is investigated. Results of the oscillations of a quasilinear system in degenerate cases are presented. Degeneracy and the principal resonance: bifurcation with the disappearance of the SPM and the birth of two asymmetric cycles, are investigated. A heavy rigid body with a single fixed point is studied as the application of the results obtained. The Euler-Poisson equations are used. In the general case, the energy integral and the geometric integral are symmetric while the angular momentum integral turns out to be asymmetric. In the special case, when the centre of gravity of the body lies in the principal plane of the ellipsoid of inertia, all three classical integrals become symmetric. It is ascertained here that any SPM of a body contains four zero characteristic exponents, of which two are simple and two form a Jordan Box. In typical situation, the remaining two characteristic exponents are not equal to zero. All of the above enables one to speak of an SPM belonging to a two-parameter family and the absence of an additional first integral. It is established that a body also executes a pendulum motion in the case when the centre of gravity is close to the principal plane of the ellipsoid of inertia.     

A Series Approach to Stochastic Differential Equations with Infinite Dimensional Noise

This paper considers semilinear stochastic differential equations in Hilbert spaces with Lipschitz nonlinearities and with the noise terms driven by sequences of independent scalar Wiener processes (Brownian motions). The interpretation of such equations requires a stochastic integral. By means of a series of Itô integrals, an elementary and direct construction of a Hilbert space valued stochastic integral with respect to a sequence of independent scalar Wiener processes is given. As an application, existence and strong and weak uniqueness for the stochastic differential equation are shown by exploiting the series construction of the integral.     

Semi-analytic integration of hypersingular Galerkin BIEs for three-dimensional potential problems

An accurate and efficient semi-analytic integration technique is developed for three-dimensional hypersingular boundary integral equations of potential theory. Investigated in the context of a Galerkin approach, surface integrals are defined as limits to the boundary and linear surface elements are employed to approximate the geometry and field variables on the boundary. In the inner integration procedure, all singular and non-singular integrals over a triangular boundary element are expressed exactly as analytic formulae over the edges of the integration triangle. In the outer integration scheme, closed-form expressions are obtained for the coincident case, wherein the divergent terms are identified explicitly and are shown to cancel with corresponding terms from the edge-adjacent case. The remaining surface integrals, containing only weak singularities, are carried out successfully by use of standard numerical cubatures. Sample problems are included to illustrate the performance and validity of the proposed algorithm.     

Algorithms and numerical analysis of dc fields in a piecewise-homogeneous medium by the boundary integral equation method

Boundary integral equation methods are considered for computing dc fields in three-dimensional regions filled with a piecewise-homogeneous medium. The problem is formulated and a system of Fredholm boundary integral equations of first kind is constructed, following directly from Green’s formula. The numerical solution stages are considered in detail, including construction and triangulation of the numerical surfaces, evaluation of surface integrals, and solution of a system of block-matrix equations. Translated from Prikladnaya Matematika i Informatika, No. 30, pp. 35–45, 2008.     

On Bohl’s argument theorem

The classical Bohl argument theorem of a conditionally periodic function is generalized. Conditionally periodic motions on a torus are replaced by the solutions of a nonlinear system of differential equations with invariant measure. Cases in which this system is assumed ergodic or strictly ergodic are considered.