Nonhomogeneous Linear Differential Equations Research Articles

Splitting of a system of nonhomogeneous linear differential equations that depend on two small parameters

Splitting of a system of nonhomogeneous linear differential equations that depend on two small parameters

The load-carrying capacity of plain finite partial journal bearings in the turbulent regime

Serious errors occur in hydrodynamic bearings when the lubricant film becomes turbulent if they are analysed/designed according to laminar theory. Thus there is a need for an accurate description of the load capacity of plain journal bearings in the turbulent regime. It has been shown that the governing turbulent-laminar lubrication equation for plain partial journal bearings of finite length under dynamic operation is a second-order linear non-homogeneous partial differential equation with analytically known variable coefficients which can be split by the classical method of separation of variables. One of the resulting ordinary equations is integrated directly while the other is solved by a direct method. The solution is subject to zero-pressure boundary conditions. All the results of interest such as pressure distribution, load-carrying capacity and attitude angle are dimensionless and are obtained in the closed form which is simple and accurate and permits straightforward calculation of numerical data over a wide range of parameters.

Growth, Oscillation and Comparison Theorems for Second Order Linear Difference Equations

This paper studies homogeneous and nonhomogeneous second order linear difference equations. Comparison theorems based on the coefficients are proven for the homogeneous equation. Existence of so called r-type solutions for the forced equation is established. Oscillation and nonoscillation properties of the forced equation based on the forcing term and the associated homogeneous equation are discussed.

On the almost periodic solutions of certain second-order abstract differential equations

In a Hilbert space, certain solutions of second-order nonhomogeneous linear differential equations with a Stepanov almost periodic forcing function are shown to be almost periodic.

Oscillation of Second Order Nonhomogeneous Linear Differential Equations

This paper is a study of the oscillatory and asymptotic behavior of solutions of the second order, nonhomogeneous linear differential equation $( {r( x )y'} )^\prime + q( x )y = f( x )$, and the associated homogeneous equation $( {r( x )u'} )^\prime + q( x )u = 0$. Conditions are determined, under which the nonhomogeneous equation is oscillatory if and only if the homogeneous equation is oscillatory. Separation theorems for these two equations are also established.

Oscillation theorems for second-order nonhomogeneous linear differential equations

Oscillation theorems for second-order nonhomogeneous linear differential equations

SOME PROBLEMS OF THE EXISTENCE AND BEHAVIOUR OF SOLUTIONS OF NON-HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS IN HILBERT SPACE

SOME PROBLEMS OF THE EXISTENCE AND BEHAVIOUR OF SOLUTIONS OF NON-HOMOGENEOUS LINEAR DIFFERENTIAL EQUATIONS IN HILBERT SPACE

Classification of Bounded Solutions of a Linear Nonhomogeneous Differential Equation

Classification of Bounded Solutions of a Linear Nonhomogeneous Differential Equation

Classification of bounded solutions of a linear nonhomogeneous differential equation

An elementary criterion, depending only upon the initial data of a solution, is formulated to determine the boundedness of solutions of a nonhomogeneous linear system of ordinary differential equations. The associated homogeneous linear differential equation is required to be either conditionally stable or conditionally asymptotically stable.

On the estimation of numerus clausus

In this paper a method for estimating the necessary number of enrollments is derived, when the future need of graduates and the probabilities of ever graduating and of graduating in a certain time are given. The method consists of constructing a nonhomogeneous linear difference equation and of determining the optimal values of the unknown coefficients in its solution.

On the solutions of certain linear nonhomogeneous second-order differential equations

(1971). On the solutions of certain linear nonhomogeneous second-order differential equations. Applicable Analysis: Vol. 1, No. 1, pp. 57-63.

Noniterative Solutions of Integral Equations for Scattering. V. Auxiliary T(kj) Matrix Formalism

The homogeneous integral solution method is used to develop a scheme for direct computation of the T matrix. The method involves calculation of auxiliary matrices T(kj) in terms of which the true T matrix is obtained. The auxiliary T(kj) matrices can be obtained by solving either integral or first-order linear non-homogeneous differential equations. The relationship of calculations based on these equations to those based on the wavefunction integral equations is discussed. Numerical studies of the general behavior of the homogeneous integral solution procedure are presented. These studies deal with rotationally inelastic collisions of an atom and a rigid rotator interacting via a Morse-type potential. An investigation of some higher-order numerical schemes for solving the integral and differential equations for the auxiliary matrices and the integral equations for the wavefunction is reported. The results indicate that the use of higher-order procedures such as the Runge–Kutta method or Simpson's quadrature scheme permit one to use step sizes 5–10 times larger than those required with the trapezoidal quadrature scheme (or equivalently, the uncorrected Euler method for the differential equation) to achieve a given accuracy. The role of the potential function is studied by a comparison of the | T |2 matrix elements for Lennard-Jones (12–6) and Morse potentials having the same well depth, curvature, and equilibrium position. In addition, hybrid potentials constructed by splicing together Morse and Lennard-Jones potentials are studied. The results indicate that, at the energy considered, the long-range portion of the potential has a dominant effect on the elastic and inelastic scattering.

Mathematical analysis of isometric cardiac muscle contraction: The effects of stimulus interval, temperature, and calcium

The time course of oscillatory and nonoscillatory isometric rested-state contraction in rat ventricle has been described by solutions of a nonhomogeneous second-order linear differential equation. The experimental conditions constitute a definition of the initial conditions of the differential equation and thus determine the form of the transient solution. The response of the contractile system can be further characterized by a damping ratio defined in terms of the constants of the homogeneous part of the differential equation. Test contractions early in the recovery of contractility were more heavily damped than the rested-state contraction. During the first phase of recovery, their damping decreased to that of the rested-state contraction. Under conditions of oscillation both low temperature and high calcium decreased the damping, the latter by a specific effect on the angular frequency of oscillation.

The Nonhomogeneous Linear Difference Equation with Variable Difference Interval

The Nonhomogeneous Linear Difference Equation with Variable Difference Interval

On the algebraic independence of the values of E-functions satisfying nonhomogeneous linear differential equations

We prove a theorem which reduces the investigation of the algebraic independence of solutions of linear nonhomogeneous systems of differential equations to the investigation of homogeneous systems. We use this theorem to prove the algebraic independence of the values of certain E-functions.

On the asymptotic growth of the solutions of a system of nonhomogeneous linear differential equations

On the asymptotic growth of the solutions of a system of nonhomogeneous linear differential equations

The nonhomogeneous linear difference equation with variable difference interval

Certain ordinary linear difference equations whose coefficients can be expressed as factorial series have been treated by Norlund [1]. The corresponding nonhomogeneous equation can then be solved by the difference analogue of the Lagrange variation of parameter method or possibly in other ways. The whole process is very complicated in execution and is at most little more than an existence theorem. The present note treats a general linear difference equation where the constant difference interval is replaced by a variable difference interval. So far as the author knows this is the first time that this has been done. Results include the case where the difference interval is constant as a special case. Also the series used, called general factorial series, include ordinary factorial series as used by Norlund [1] as a special case. It is believed that the author was the first to study such series [2].

On singular problems of Goursat and Cauchy

Introduction. In this paper the problems of Goursat and Cauchy are investigated for equations of hyperbolic type with singular coefficients. Effective algorithms are presented for constructing Duhamel functions of such singular initial problems, reducing these problems to a recurrent sequence of ordinary linear nonhomogeneous differential equations of the second order. Using established differential properties of the operators of Kummer and Gauss, explicit formulas for inverting them are obtained~ as well as simple estimates characterizing the behavior, in the neighborhood of a singular curve of the equation, of the iterations obtained. Studied also are theorems of uniqueness and extremal properties of the solutions of the singular Goursat and Cauchy problems. w i. By the singular Goursat problem we shall understand the problem of finding, in the rectangle = OPMQO, with ver t ices at O(0, 0), P(x0, 0), M(x0, Y0), Q(0, y0), a solution z(x, y) E C2(~) of the equation: L [z, A, B, C] = xz~y + A (x) z~ + B (x) z~ + C (x) z = O, which assumes along the lines x= 0, y=0 , the values (1.1)

Transcendental Functions Satisfying Nonhomogeneous Linear Differential Equations

Transcendental Functions Satisfying Nonhomogeneous Linear Differential Equations

Vliv driftu a mřížkového proudu počítacích stejnosměrných zesilovačů na přesnost řešení nehomogenních lineárních diferenciálních rovnic s konstantními koeficienty

Vliv driftu a mřížkového proudu počítacích stejnosměrných zesilovačů na přesnost řešení nehomogenních lineárních diferenciálních rovnic s konstantními koeficienty