System Of Fourth-order Differential Equations Research Articles

An Approach to the Solution of the Initial Boundary-Value Problem for Systems of Fourth-Order Hyperbolic Equations

The initial boundary-value problem for systems of fourth-order partial differential equations with two independent variables is considered. By using a new unknown eigenfunction, the problem under consideration is reduced to an equivalent nonlocal problem for a system of second-order hyperbolic-type integro-differential equations with integral conditions. An algorithm for finding an approximate solution of the resulting equivalent problem is proposed, and its convergence is proved. Conditions for the existence of a unique classical solution of the initial boundary-value problem for systems of fourth-order differential equations are established in terms of the coefficients of the system and the boundary matrices.

Positive solutions for a system of fourth-order differential equations with integral boundary conditions and two parameters

In this work, we investigate a class of nonlinear fourth-order systems with coupled integral boundary conditions and two parameters. We give the Green's functions for the system with boundary conditions, and then obtain some useful properties of the Green's functions. By using the Guo–Krasnosel'skii fixed-point theorem and the Green's functions, some sufficient conditions for the existence of positive solutions are presented. As applications, two examples are presented to illustrate the application of our main results.

On periodic motions of a gyrostat satellite with a large inner (gyrostatic) angular momentum

Abstract Rotational motion of an axially symmetric gyrostat satellite under the action of its gravitational torque in a circular orbit is considered. Periodic motions of the symmetry axis of the satellite relative to the orbital coordinate system are investigated. In absolute space, these motions appear as a slow precession about the normal to the orbital plane. Such motions are described by an autonomous system of fourth-order differential equations. The gyrostatic angular momentum is assumed to be large, which allows us to introduce a large parameter into the equations of motion. Solutions that are a rest in absolute space, with the symmetry axis forming a nonzero angle with the orbital plane, serve as the generating solutions. The period of the found solutions depends on this angle. Earlier, the limit case of such periodic motions when the symmetry axis of the satellite lies in the orbital plane in the generating solutions was investigated. The limit solutions describe small oscillations of the symmetry axis of the satellite in absolute space, and their period is equal to half the orbital period. To prove the existence of the new motions, we reduce the boundary value problem configuring the periodic solutions to a system of integral equations, which is solved by the method of successive approximations. This reduction is carried out according to the same scheme as in the degenerate case, but the necessary solutions of the integral equations are constructed differently. The result obtained explains the appearance of the limit solutions although the latter cannot be constructed within the framework of the considered general case.

Chaotic Convection in a Viscoelastic Fluid Saturated Porous Medium with a Heat Source

Chaotic convection in a viscoelastic fluid saturated porous layer, heated from below, is studied by using Oldroyd’s type constituting relation and in the presence of an internal heat source. A modified Darcy law is used in the momentum equation, and a heat source term has been considered in energy equation. An autonomous system of fourth-order differential equations has been deduced by using a truncated Fourier series. Effect of internal heat generation on chaotic convection has been investigated. The asymptotic behavior can be stationary, periodic, or chaotic, depending upon the flow parameters. Construction of four-scroll, or “two-butterfly,” and chaotic attractor has been examined.

Periodic motions of a gyrostat satellite with a large gyrostatic moment about the centre of mass

An axisymmetric gyrostat satellite, whose centre of mass moves in a fixed circular orbit about an attractive centre is considered. The satellite moves about the centre of mass under the action of the gravitational moment. Under the assumption that the gyrostatic moment of the satellite is large and is directed along its axis of symmetry, the periodic motions of this axis are investigated in a small vicinity of a fixed direction, lying in the orbital plane in absolute space. Such motions are described by a system of fourth-order differential equations with periodic coefficients that contains a large parameter. A theorem for the existence and uniqueness of a symmetric periodic solution of this system that has the required properties is proved using methods for constructing periodic solutions of differential equations with a large parameter.

TRANSITIONAL DYNAMICS OF AN ENDOGENOUS GROWTH MODEL WITH AN EROSION EFFECT

The convergence features of an endogenous growth model with physical capital, human capital and R&D have been studied. We add an erosion effect (supported by empirical evidence) to this model, and fully characterize its convergence properties. The dynamics is described by a fourth-order system of differential equations. We show that the model converges along a one-dimensional stable manifold and that its equilibrium is saddle-path stable, for most plausible values for the parameters. We also argue that one of the implications of considering this ‘erosion effect’ is the increase in the adherence of the model to data.

Transitional Dynamics of an Endogenous Growth Model with an Erosion Effect

The convergence features of an endogenous growth model with physical capital, human capital and R&D have been studied. We add an erosion effect (supported by empirical evidence) to this model, and fully characterize its convergence properties. The dynamics is described by a fourth-order system of differential equations. We show that the model converges along a one-dimensional stable manifold and that its equilibrium is saddle-path stable, for most plausible values for the parameters. We also argue that one of the implications of considering this erosion effect is the increase in the adherence of the model to data.

The asymptotic representation of higher-order g$\mathsf{^+}$-modes in stars with a convective core

Aims. A first-order asymptotic representation of higher-order non-radial g + -modes in spherically symmetric stars with a convective core is constructed from the full fourth-order system of governing equations. Stars are considered that, besides their convective core, also contain a radiative envelope, or both an intermediate radiative zone and a convective envelope. At the same time, the earlier asymptotic theory of Willems et al. (1997, A&A, 318, 99) relative to stars consisting of a convective core and a radiative envelope is made more transparent. Methods. As in the asymptotic theory of Smeyers (2006, A&A, 451, 223) for low-degree, higher-order p -modes, two-variable expansion procedures and boundary-layer theory are applied to the fourth-order system of differential equations established by Pekeris (1938, ApJ, 88, 189). Results. Eigenfrequency equations are derived in terms of the radial order n of the g + -mode. The first $n - 1$ nodes of the radial component of the Lagrangian displacement coincide with the $n - 1$ nodes of the divergence of the Lagrangian displacement, and are situated in the radiative envelope or in the intermediate radiative zone according to the type of star considered. The radial displacement displays an n th node near the surface. In stars containing an intermediate radiative zone and a convective envelope, the n th node is situated in the envelope. Conclusions. As well as for higher-order p -modes of spherically symmetric stars, the divergence of the Lagrangian displacement plays a basic role in the development of the asymptotic theory.

Analytical and numerical solution for a elastic pipe bend at in-plane bending with consideration for the end effect

The authors proposed an analytical method for the analysis of the end effect in a pipe bend loaded by a bending moment with consideration for the action of internal pressure. The method is based on the use of simplifying hypotheses and is reduced to the solution of a system of fourth-order differential equations along the axial coordinate with respect to unknown coefficients in the expansion for tangential displacements. An approximate analytical solution, which has a trapezoidal structure and is written in terms of Krylov’s functions, has been obtained. Boundary conditions are formulated in terms of the tangential and longitudinal displacements and axial and shearing stress resultant. For the flexibility factor, analytical solutions are presented in the case where a bend is approximated by a rigid restraint on both ends. To verify the analytical solution and its applicability limits, two numerical procedures were developed, which are based on the finite difference method and the reduction to the Kochi problem by the expansion of the unknowns in the Fourier series over the circumferential coordinate. The authors compare the results obtained with data from the literature, discuss the advantages and disadvantages of the methods, and present recommendations for their practical application.

The exact discrete time representation of a system of fourth-order differential equations

The exact discrete time representations of both a closed and an open system of fourth-order differential equations are considered, in which all vector processes may contain both stock and flow variables. For extra generality, the closed system of equations contains seasonal dummies. The open system here contains levels and derivatives (up to fourth. order) of the vector of exogenous variables whose continuous time paths are approximated using a quartic interpolant.

Covariant perturbations of anisotropic cosmological models.

In this paper I use the covariant approach to study the evolution of linear perturbations in a Bianchi type-I cosmology. It is found that the evolution of density perturbations is described by a fourth-order system of differential equations as in the treatment of Perko, Matzner, and Shepley for the case of a pressureless perfect fluid and I verify that these equations are consistent using the set of linear constraint equations. I then show that when the shear is axially symmetric this system reduces to one second-order differential equation for the density gradient. Approximate solutions are found for the case of dust, which are in agreement with earlier studies. I also consider a recently developed dynamic systems approach to discuss the qualitative behavior of the solutions and the stability of Bianchi type-I models with respect to linear inhomogeneous perturbations.

Boundary-value problems for a fourth-order system of differential equations

Boundary-value problems for a fourth-order system of differential equations

Boundary-value problems for a certain class of systems of fourth-order differential equations

Boundary-value problems for a certain class of systems of fourth-order differential equations