Determination of contact stresses in problems on bending of a package of transversely isotropic bars

A mechanomathematical model for bending of packages of transversely isotropic bars of rectangular cross section is proposed. Adhesion, slippage, and separation zones between the bars are considered. The resolving equations for deflections and tangential displacements are supplemented with a system of linear differential equations for determining the normal and tangential contact stresses, and boundary conditions are formulated. A scheme for analytical solution of two contact problems—a package under the action of a distributed load and a round stamp—is considered. For these packages, a transition is performed from the initial system of differential equations for determining the contact stresses, where the unknown functions are interrelated by recurrent relationships, to one linear differential equation of fourth order and then to a system of linear algebraic equations. This transition allows us to integrate the initial system and get expressions for the contact stresses.Translated from Mekhanika Kompozitnykh Materialov, Vol. 40, No. 6, pp. 761–778, November–December, 2004.     

Some exact solutions of the non-linear Dirac-Hamilton system

We obtain a family of substitutions which reduce a system of multi-dimensional Dirac—Hamilton differential equations to ordinary differential equations. We construct large classes of exact solutions to this system.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 610–616, May, 1990.     

Fourier–Bessel-type series: The fourth-order case

The structured higher-order Bessel-type linear ordinary differential equations were first discovered in 1994. There is a denumerable infinity of these higher-order equations, all of then of even-order.These differential equations possess many of the properties of the classical second-order Bessel differential equation, but these higher-order cases bring remarkable new analytic structures. In many ways it is sufficient to study the properties of the fourth-order Bessel-type differential equation to be able to assess the corresponding properties of the sixth-and higher-order cases.This paper follows a number of earlier papers devoted to the study of the fourth-order case. These publications show the connections between the special function properties of solutions of the differential equation, and the properties of linear differential operators generated by the associated linear differential expression in certain weighted Lebesgue, and Lebesgue–Stieltjes function spaces.To follow the earlier papers on the study of the fourth-order Bessel-type differential equation, this present paper determines the form of the Fourier–Bessel-type series which best extends the classical theory of the second-order Fourier–Bessel series.In fact the Fourier–Bessel-type series are based on a new orthogonal system in terms of the regular eigensolutions of the fourth-order Bessel-type equation. The corresponding eigenvalues are obtained by restricting the spectral parameter to the zeros of an analytic function arising already in the Dini boundary conditions.     

Numerical solutions for some coupled nonlinear evolution equations by using spectral collocation method

In this paper, we introduce a spectral collocation method based on Lagrange polynomials for spatial derivatives to obtain numerical solutions for some coupled nonlinear evolution equations. The problem is reduced to a system of ordinary differential equations that are solved by the fourth order Runge–Kutta method. Numerical results of coupled Korteweg–de Vries (KdV) equations, coupled modified KdV equations, coupled KdV system and Boussinesq system are obtained. The present results are in good agreement with the exact solutions. Moreover, the method can be applied to a wide class of coupled nonlinear evolution equations.     

Runge–Kutta–collocation methods for systems of functional–differential and functional equations

Systems of functional–differential and functional equations occur in many biological, control and physics problems. They also include functional–differential equations of neutral type as special cases. Based on the continuous extension of the Runge–Kutta method for delay differential equations and the collocation method for functional equations, numerical methods for solving the initial value problems of systems of functional–differential and functional equations are formulated. Comprehensive analysis of the order of approximation and the numerical stability are presented.     

Explicit Formulas for Generalized Action–Angle Variables in a Neighborhood of an Isotropic Torus and Their Application

Different versions of the Darboux–Weinstein theorem guarantee the existence of action–angle-type variables and the harmonic-oscillator variables in a neighborhood of isotropic tori in the phase space. The procedure for constructing these variables is reduced to solving a rather complicated system of partial differential equations. We show that this system can be integrated in quadratures, which permits reducing the problem of constructing these variables to solving a system of quadratic equations. We discuss several applications of this purely geometric fact in problems of classical and quantum mechanics.     

Max-Plus Stochastic Processes

This paper is concerned with processes which are max-plus counterparts of Markov diffusion processes governed by Ito sense stochastic differential equations. Concepts of max-plus martingale and max-plus stochastic differential equation are introduced. The max-plus counterparts of backward and forward PDEs for Markov diffusions turn out to be first-order PDEs of Hamilton–Jacobi–Bellman type. Max-plus additive integrals and a max-plus additive dynamic programming principle are considered. This leads to variational inequalities of Hamilton–Jacobi–Bellman type.     

Complex Lie symmetries for scalar second-order ordinary differential equations

A scalar complex ordinary differential equation can be considered as two coupled real partial differential equations, along with the constraint of the Cauchy–Riemann equations, which constitute a system of four equations for two unknown real functions of two real variables. It is shown that the resulting system possesses those real Lie symmetries that are obtained by splitting each complex Lie symmetry of the given complex ordinary differential equation. Further, if we restrict the complex function to be of a single real variable, then the complex ordinary differential equation yields a coupled system of two ordinary differential equations and their invariance can be obtained in a non-trivial way from the invariance of the restricted complex differential equation. Also, the use of a complex Lie symmetry reduces the order of the complex ordinary differential equation (restricted complex ordinary differential equation) by one, which in turn yields a reduction in the order by one of the system of partial differential equations (system of ordinary differential equations). In this paper, for simplicity, we investigate the case of scalar second-order ordinary differential equations. As a consequence, we obtain an extension of the Lie table for second-order equations with two symmetries.     

The Bi-Hamiltonian Theory of the Harry Dym Equation

We describe how the Harry Dym equation fits into the the bi-Hamiltonian formalism for the Korteweg–de Vries equation and other soliton equations. This is achieved using a certain Poisson pencil constructed from two compatible Poisson structures. We obtain an analogue of the Kadomtsev–Petviashivili hierarchy whose reduction leads to the Harry Dym hierarchy. We call such a system the HD–KP hierarchy. We then construct an infinite system of ordinary differential equations (in infinitely many variables) that is equivalent to the HD–KP hierarchy. Its role is analogous to the role of the Central System in the Kadomtsev–Petviashivili hierarchy.     

Analytical properties of solutions to a system of nonlinear partial differential equations

Certain analytical properties of self-similar solutions to a system of N partial differential equations, which is the generalized Liouville equation, are investigated.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 2, pp. 208–213, November, 1995.     

Semilinear Kolmogorov Equations and Applications to Stochastic Optimal Control

Semilinear parabolic differential equations are solved in a mild sense in an infinite-dimensional Hilbert space. Applications to stochastic optimal control problems are studied by solving the associated Hamilton–Jacobi–Bellman equation. These results are applied to some controlled stochastic partial differential equations.     

Numerical solution of the propagation problem of longitudinal elastic waves in an orthotropic hollow cylinder

Wave processes in an orthotropic cylinder are analyzed. Harmonic oscillations are considered for which the original problem is reduced to a system of ordinary differential equations. This system is solved by the numerically stable discrete-orthogonalization method. Discrete curves for specific problems are given.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 56, pp. 75–78, 1985.     

Some mathematical models of ion-exchange processes in countercurrent columns

The article investigates some one-dimensional mathematical models of nonequilibrium ion-exchange processes in countercurrent columns in accordance with standard assumptions of mathematical physics. Existence and uniqueness are proved in the boundary-value problem for a system of partial differential equations corresponding to these models. Stability and convergence of the difference schemes used to solve these boundary-value problems are analyzed. Results of model calculations are presented.Translated from Matematicheskoe Modelirovanie i Reshenie Obratnykh Zadach Matematicheskoi Fiziki, pp. 65–75, 1993.     

Nonlocal ansatze and solutions of a nonlinear system of heat-conduction equations

By a nonlocal substitution, a nonlinear system of heat-conduction equations is reduced to a scalar nonlinear heat-conduction equation. The Lie and conditional invariance of the scalar equation is used to find nonlocal ansatze which reduce the original system to systems of ordinary differential equations.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 2, pp. 293–302, February, 1993.     

Exact solutions of the nonlinear Dirac equation in terms of Bessel,Gauss and Legendre functions and Chebyshev-Hermite polynomials

Substitutions are proposed, reducing a system of nonlinear Dirac equations to ordinary differential equations, integrable in special functions. It is established that the class of special functions in which the solution of the Dirac equation is written essentially depends on the form of nonlinearity.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 564–568, April, 1990.     

On the attainability and stability of an invariant set of a system of stochastic differential equations

One investigates questions of attainability and stability of an invariant set of a linear system of stochastic differential equations for the corresponding nonlinear system. Conditions, under which stability in probability is possible, are singled out.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 4, pp. 568–572, April, 1992.     

Classical Symmetry Reductions of the Schwarz–Korteweg–de Vries Equation in 2+1 Dimensions

Classical reductions of a (2+1)-dimensional integrable Schwarz–Korteweg–de Vries equation are classified. These reductions to systems of partial differential equations in 1+1 dimensions admit symmetries that lead to further reductions, i.e., to systems of ordinary differential equations. All these systems have been reduced to second-order ordinary differential equations. We present some particular solutions involving two arbitrary functions.     

Runge–Kutta methods for first-order periodic boundary value differential equations with piecewise constant arguments

In this paper we deal with the numerical solutions of Runge–Kutta methods for first-order periodic boundary value differential equations with piecewise constant arguments. The numerical solution is given by the numerical Green’s function. It is shown that Runge–Kutta methods preserve their original order for first-order periodic boundary value differential equations with piecewise constant arguments. We give the conditions under which the numerical solutions preserve some properties of the analytic solutions, e.g., uniqueness and comparison theorems. Finally, some experiments are given to illustrate our results.     

Solving systems of ODEs by homotopy analysis method

This paper applies the homotopy analysis method (HAM) to systems of ordinary differential equations (ODEs). The systems investigated include stiff systems, the chaotic Genesio system and the matrix Riccati-type differential equation. The HAM gives approximate analytical solutions which are of comparable accuracy to the seven- and eight-order Runge–Kutta method (RK78).     

The determining equations for the nonclassical method of the nonlinear differential equation(s) with arbitrary order can be obtained through the compatibility

In this paper, firstly we show that the determining equations of the (1+1) dimension nonlinear differential equation with arbitrary order for the nonclassical method can be derived by the compatibility between the original equation and the invariant surface condition. Then we generalize this result to the system of the (m+1) dimension differential equations. The nonlinear Klein–Gordon equation, the (2+1)-dimensional Boussinesq equation and the generalized Nizhnik–Novikov–Veselov equation serve as examples illustrating this method.