Solutions Of Homogeneous Differential Equations Research Articles

ESTIMATION OF PARAMETERS OF ECONOMIC GROWTH AUTOREGRESSIVE MODELS

The construction of time series models based on statistical data is one of the central tasks of modern econometric studies. Numerous economic models based on differential equations are used to explain the patterns of economic growth. The article is devoted to the estimation of parameters of economic growth models based on solutions of homogeneous differential equations with constant coefficients. A comparative analysis of methods for estimating the parameters of autoregression of economic dynamics series with additive interference in the output signal is carried out. The simulation results showed that the full least squares method gives the most accurate estimates. The most commonly used least squares method gives the worst estimates.

Jordan — Gauss successive elimination method in solution of pneumohydraulic floatation water purification problems

The emphasis of the article is a potential usage of Jordan - Gauss successive elimination method for reduction of a differential equation system (DES) to one differential equation and determining its solution providing all other solutions of DES. This approach is different from standard method of determining of eigen vectors and eigen values of matrix of DES, which are necessary to determine fundamental system of solutions of homogeneous differential equations of n-th order with constant coefficients denoted in norm form. In the article analytical solution is searched for in general form and can be used for other similar mathematical problems.

An inverse problem for a heat equation with piecewise-constant thermal conductivity

The governing equation is ut=(a(x)ux)x, 0≤x≤1, t>0, u(x,0)=0, u(0,t)=0, a(1)u′(1,t)=f(t). The extra data are u(1,t)=g(t). It is assumed that a(x) is a piecewise-constant function and f≢0. It is proved that the function a(x) is uniquely defined by the above data. No restrictions on the number of discontinuity points of a(x) and on their locations are made. The number of discontinuity points is finite, but this number can be arbitrarily large. If a(x)∊C2[0,1], then a uniqueness theorem has been established earlier for multidimensional problem, x∊Rn,n>1 [see A. G. Ramm, Multidimensional inverse problems and completeness of the products of solutions to PDE, J. Math. Anal. Appl., 134, 211 (1988)] for the stationary problem with infinitely many boundary data. The novel point in this work is the treatment of the discontinuous piecewise-constant function a(x) and the proof of Property C for a pair of the operators {ℓ1,ℓ2}, where ℓj≔−(d2/dx2)+k2qj2(x), j=1,2, and qj2(x)>0 are piecewise-constant functions, and for the pair {L1,L2}, where Lju≔−[aj(x)u′(x)]′+λu, j=1,2, and aj(x)>0 are piecewise-constant functions. Property C stands for completeness of the set of products of solutions of homogeneous differential equations [see A. G. Ramm, Inverse Problems (Springer, New York, 2005)].

Nondegeneracy and uniqueness of positive solutions for Robin problem of second order ordinary differential equations and its applications

This article studies positive solutions of Robin problem for semi-linear second order ordinary differential equations. Nondegeneracy and uniqueness results are proven for homogeneous differential equations. Necessary and sufficient conditions for the existence of one or two positive solutions for inhomogeneous differential equations or differential equations with concave–convex nonlinearities are obtained by making use of the nondegeneracy and uniqueness results for positive solutions of homogeneous differential equations.

The Adomian decomposition method for the approximate solution of homogeneous differential equations with dual variable and dual coefficients

In this study, the decomposition method of Adomian has been applied for the approximate solution of homogeneous differential equations with dual variable and dual coefficients.

Some Ergodic Theorems for Solutions of Homogeneous Differential Equations

Consider $x' = f(x)$, where f is defined on a cone K with nonempty interior in a Banach space X. This paper studies the conditions on f that ensure that the directions of the solutions, ${{x(t)} / {\|x(t)\|}}$, converge to the same point $u \in K$ no matter what initial conditions are. Problems of this type often arise in some models from population biology. The results are obtained for maps that are homogeneous of degree 1. In the proofs Hilbert’s projective metric is used.

Natural Frequencies of Curved Girders

A general finite element displacement formulation is presented to determine the natural frequencies of spatial thin-walled curved girders including the warping contribution. Also included are the rotatory inertia effects with respect to flexure and torsion, and the effects of antisymmetry of cross section (found to be significant). The stiffness and inertia properties of the finite element are obtained using the solutions of homogeneous differential equations governing the static problem as deformation modes. The obtained element relationships were programmed for use in digital computers, and a few example problems were analyzed. Some charts are given to expedite the determination of natural frequencies of horizontally curved girders.

Solutions of homogeneous differential equations of infinite order

Solutions of homogeneous differential equations of infinite order

A consistent discrete elements technique for thinwalled assemblages

A consistent matrix formulation for the discrete element technique for linear and eigenvalue problems of structures assembled from thinwalled segments with open cross section is presented. Using the solutions of homogeneous differential equations governing the static problem as deformation modes the stiffness, load, stability and mass matrix are derived. The procedure is exemplified on few appropriate examples of bending coupled with torsion, stability and vibrations. Obtained results are exact for linear static problems and an extremely close upper bound for eigenvalue problems.Through a limiting process, it is shown that the presented technique is an extension of the procedure used for solid beam structures.

Approximation of solutions of homogeneous differential equation, I

Approximation of solutions of homogeneous differential equation, I