Linear Inhomogeneous Differential Equation Research Articles

The effect of the Earth’s oblateness on the Moon’s physical libration in latitude

The Moon’s physical libration in latitude generated by gravitational forces caused by the Earth’s oblateness has been examined by a vector analytical method. Libration oscillations are described by a close set of five linear inhomogeneous differential equations, the dispersion equation has five roots, one of which is zero. A complete solution is obtained. It is revealed that the Earth’s oblateness: a) has little effect on the instantaneous axis of Moon’s rotation, but causes an oscillatory rotation of the body of the Moon with an amplitude of 0.072″ and pulsation period of 16.88 Julian years; b) causes small nutations of poles of the orbit and of the ecliptic along tight spirals, which occupy a disk with a cut in a center and with radius of 0.072″. Perturbations caused by the spherical Earth generate: a) physical librations in latitude with an amplitude of 34.275″; b) nutational motion for centers of small spiral nutations of orbit (ecliptic) pole over ellipses with semi-major axes of 113.850″ (85.158″) and the first pole rotates round the second one along a circle with radius of 28.691″; c) nutation of the Moon’s celestial pole over an ellipse with a semi-major axis of 45.04″ and with an axes ratio of about 0.004 with a period of T = 27.212 days. The principal ellipse’s axis is directed tangentially with respect to the precession circumference, along which the celestial pole moves nonuniformly nearly in one dimension. In contrast to the accepted concept, the latitude does not change while the Moon’s poles of rotation move. The dynamical reason for the inclination of the Moon’s mean equator with respect to the ecliptic is oblateness of the body of the Moon.

An approximation property of power functions

We will solve the inhomogeneous linear first order differential equation of the form, xy′(x) + λy(x) = ∑m=0 am(x− c)m , and prove an approximation property of power functions. More precisely, we prove the local Hyers-Ulam stability of linear first order differential equation, xy′(x)+λy(x) = 0 , in a special class of analytic functions. Mathematics subject classification (2010): Primary 39B82, 41A30, secondary 34A30, 34A25, 34A05.

A note on Ehrenpreis’ fundamental principle

Ehrenpreis’ Fundamental Principle is an existence theorem for inhomogeneous linear partial differential equations, and has great system-theoretic significance. Usually it is formulated for the equations defined on open convex sets. However, openness of domains of definition is somewhat restrictive in systems theory. In this article, we show that the principle is valid in the case of sets that are the unions of increasing sequences of convex compact sets. Moreover, we offer a simplified proof.

Higher-order nonlinear electron-acoustic solitary excitations in partially degenerate quantum electron-ion plasmas

Propagation of dressed solitary excitations are studied in a partially degenerate quantum plasma in the framework of quantum-hydrodynamics model using multiple scales technique. The evolution equation together with a linear inhomogeneous differential equation is solved using Kodama–Taniuti renormalizing technique. It is shown that the type of solitary excitations (bright or dark) is defined by two critical plasma parameter values.

Inhomogeneous degenerate Sobolev type equations with delay

Under consideration is the first order linear inhomogeneous differential equation in an abstract Banach space with a degenerate operator at the derivative, a relatively p-radial operator at the unknown function, and a continuous delay operator. We obtain conditions of unique solvability of the Cauchy problem and the Showalter problem by means of degenerate semigroup theory methods. These general results are applied to the initial boundary value problems for systems of integrodifferential equations of the type of phase field equations.

A New Differential Operator Method to Study the Mechanical Vibration

In this paper, we propose a unified differential operator method to study mechanical vibrations, solving inhomogeneous linear ordinary differential equations with constant coefficients. The main advantage of this new method is that the differential operator D in the numerator of the fraction has no effect on input functions (i.e., the derivative operation is removed) because we take the fraction as a whole part in the partial fraction expansion. The method in various variants is widely implemented in related fields in mechanics and engineering. We also point out that the same mistakes in the differential operator method are found in the related references [1-4].

Multisummability of formal solutions of inhomogeneous linear partial differential equations with constant coefficients

We consider the Cauchy problem for a general inhomogeneous linear partial differential equation with constant coefficients in two complex variables. We obtain necessary and sufficient conditions for the multisummability of formal solutions in terms of analytic continuation properties and growth estimates of some functions connected with the inhomogeneity. The results are presented in the general framework of 1/p-fractional equations.

Cauchy problem for a differential equation in the Banach space with generalized strongly positive operator coefficient

The concept of strongly positive operator is generalized and the properties of the introduced operators are analyzed. The solutions of the Cauchy problem for a linear inhomogeneous differential equation with generalized strongly positive operator coefficient are obtained.

Research and Analysis of Dynamic Behaviour in the Mathematical Models of the 6-Axle Locomotive

With the aiming of mathematically modeling dynamic behavior in latitudinal and vertical movements of the 6-axle locomotive, this paper introduces the Kalker’s Wheel-Rail Rolling Contact Theories and their implementation in multibody codes. This paper also highlights methodology for solving inhomogeneous linear second-order differential equations with MATLAB computer software aided. The calculation has reported that the dynamic behaviors of Diesel-Electric 6-axle locomotive are significantly demonstrated. The calculation has reported that the dynamic behaviors of Diesel-Electric 6-axle locomotive are significantly demonstrated the requirements on Rolling stock Dynamic behaviors of Vietnam Railways.

Generalized equation for describing the magnetization in spoiled gradient-echo imaging

The purpose of this study was to demonstrate a generalized equation for describing the magnetization in spoiled gradient-echo (SPGR) imaging in which the in-pulse relaxation and magnetization transfer (MT) effects are taken into account. First, the time-dependent Bloch equations for the two-pool exchange model with MT effect were reduced to an inhomogeneous linear differential equation, and then a simple equation was derived to solve it using a matrix operation. Second, the equations describing the magnetization before and after the radiofrequency (RF) pulse were derived based on the above solution for the RF-pulse excitation and evolution phases. Finally, a generalized equation describing the steady-state magnetization was derived. The validity of this equation was investigated by comparing with the transverse magnetization obtained by the regular Ernst equation and analytical solution in which the in-pulse transverse relaxation is considered. When the same assumption was made in our method, there were good agreements between them, indicating the validity of our method. The in-pulse transverse and longitudinal relaxations decreased the transverse magnetization compared to the case in which these effects were neglected, whereas MT increased it. In conclusion, we derived a generalized equation for describing the magnetization in SPGR imaging. This equation will provide a suitable basis for understanding the signal intensity in SPGR imaging and/or T 1 measurement using an SPGR sequence in cases in which the effect of in-pulse relaxation and/or MT cannot be neglected.

The method of approximate particular solutions for solving certain partial differential equations

AbstractA standard approach for solving linear partial differential equations is to split the solution into a homogeneous solution and a particular solution. Motivated by the method of fundamental solutions for solving homogeneous equations, we propose a similar approach using the method of approximate particular solutions for solving linear inhomogeneous differential equations without the need of finding the homogeneous solution. This leads to a much simpler numerical scheme with similar accuracy to the traditional approach. To demonstrate the simplicity of the new approach, three numerical examples are given with excellent results. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 506–522, 2012

General Inhomogeneous Discrete Linear Partial Differential Equations with Constant Coefficients on the Whole Spaces

In this paper we shall introduce new constructions of approximate solutions of general linear partial differential equations with constant coefficients on the whole spaces, and establish fundamental estimates of the solutions depending on the inhomogeneous terms. This will be done by combining general ideas of the Tikhonov regularization and discretization of bounded linear operator equations on reproducing kernel Hilbert spaces. Furthermore, we will provide approximate solutions for the related inverse source problems.

Solving Inhomogeneous Linear Partial Differential Equations

Solving Inhomogeneous Linear Partial Differential Equations

On holomorphic solutions of some inhomogeneous linear differential equations in a banach space over a non-archimedean field

Let A be a closed linear operator on a Banach space \( \mathfrak{B} \) over the field Ω of complex p-adic numbers having an inverse operator defined on the whole \( \mathfrak{B} \), and f be a locally holomorphic at 0 \( \mathfrak{B} \)-valued vector function. The problem of existence and uniqueness of a locally holomorphic at 0 solution of the differential equation y(m) − Ay = f is considered in this paper. In particular, it is shown that this problem is solvable under the condition \( \mathop {\lim }\limits_{n \to \infty } \sqrt[n]{{\left\| {A^{ - n} } \right\|}} \) = 0. It is proved also that if the vector-function f is entire, then there exists a unique entire solution of this equation. Moreover, the necessary and sufficient conditions for the Cauchy problem for such an equation to be correctly posed in the class of locally holomorphic functions are presented.

Dressed electrostatic solitary waves in quantum dusty pair plasmas

Quantum-hydrodynamics model is applied to investigate the nonlinear propagation of electrostatic solitary excitations in a quantum dusty pair plasma. A Korteweg de Vries evolution equation is obtained using reductive perturbation technique and the higher-nonlinearity effects are derived by solving the linear inhomogeneous differential equation analytically using Kodama–Taniuti renormalizing method. The possibility of propagation of bright- and dark-type solitary excitations is examined. It is shown that a critical value of quantum diffraction parameter H exists, on either side of which, only one type of solitary propagation is possible. It is also found that unlike for the first-order amplitude component, the variation of H parameter dominantly affects the soliton amplitude in higher-order approximation. The effect of fractional quantum number density on compressive and rarefactive soliton dynamics is also discussed.

The method of particular solutions for solving scalar wave equations

AbstractA standard approach for solving 2D scalar wave propagation problems is presented with the use of the method of particular solutions (MPS). Motivated by the method of fundamental solutions for solving homogeneous equations, we propose a similar approach using the MPS for solving linear inhomogeneous differential equations without the need of finding the homogeneous solution. The wave equations are analyzed in the Laplace transformed domain and the Durbin inversion method is used to determine the solutions in the time domain. To demonstrate the effectiveness and simplicity of the new approach, three numerical examples have been given with excellent accuracy. Copyright © 2009 John Wiley & Sons, Ltd.

An approximation property of exponential functions

We solve the inhomogeneous linear first order differential equations of the form y′(x) − λy(x) = Σm=0∞am(x − c)m, and prove an approximation property of exponential functions. More precisely, we prove the local Hyers-Ulam stability of linear first order differential equations of the form y′(x) = λy(x) in a special class of analytic functions.

Existence of a limit distribution of a solution of a linear inhomogeneous stochastic differential equation

We find conditions for the existence of a limit distribution (as t → ∞) of a vector process ξ defined in R+ and determined by an inhomogeneous stochastic differential equation ξ(t) = ξ(0)−ξ◦α+f∗ν+g∗μ, where α is a nonrandom continuous increasing function, ν and μ are independent Poisson and centered Poisson measures, respectively.

Modified prony algorithm for an inhomogeneous linear ordinary differential equation with constant coefficients

Modified prony algorithm for an inhomogeneous linear ordinary differential equation with constant coefficients

A local-global problem for linear differential equations

An inhomogeneous linear differential equation Ly = f over a global differential field can have a formal solution for each place without having a global solution. The vector space lgl(L) measures this phenomenon. This space is interpreted in terms of cohomology of linear algebraic groups and is computed for abelian differential equations and for regular singular equations. An analogue of Artin reciprocity for abelian differential equations is given. Malgrange’s work on irregularity is reproved in terms cohomology of linear algebraic groups.