Linear Homogeneous Differential Equations Research Articles

First Order Linear Homogeneous Fuzzy Ordinary Differential Equation Based on Lagrange Multiplier Method

In this paper the First Order Linear Fuzzy Ordinary Differential Equations are described. Here coefficients and /or initial condition of said differential equation are taken as the Generalized Triangular Fuzzy Numbers (GTFNs).The solution procedure of this Fuzzy Differential Equation is developed by Lagrange Multiplier Method. An imprecise barometric pressure problem is described.

The Similar Structure Method for Solving the Model of Fractal Dual-Porosity Reservoir

This paper proposes a similar structure method (SSM) to solve the boundary value problem of the extended modified Bessel equation. The method could efficiently solve a second-order linear homogeneous differential equation’s boundary value problem and obtain its solutions’ similar structure. A mathematics model is set up on the dual-porosity media, in which the influence of fractal dimension, spherical flow, wellbore storage, and skin factor is taken into cosideration. Researches in the model found that it was a special type of the extended modified Bessel equation in Laplace space. Then, the formation pressure and wellbore pressure under three types of outer boundaries (infinite, constant pressure, and closed) are obtained via SSM in Laplace space. Combining SSM with the Stehfest algorithm, we propose the similar structure method algorithm (SSMA) which can be used to calculate wellbore pressure and pressure derivative of reservoir seepage models clearly. Type curves of fractal dual-porosity spherical flow are plotted by SSMA. The presented algorithm promotes the development of well test analysis software.

The Hyperorder of Solutions of Second-Order Linear Differential Equations

We prove that the hyperorder of every nontrivial solution of homogenous linear differential equations of type and nonhomogeneous equation of type is one, where are entire functions of order less than one, improving the previous results of Chen, Wang, and Laine.

Estimates for exponential sums. Applications

We obtain estimates of exponential (in particular, trigonometric) sums in terms of rational functions. Examples of sharp inequalities are given. These inequalities are used for estimating solutions to linear homogeneous differential equations with constant coefficients. The main results are based on the estimates of the moduli of the derivatives of rational functions by variable majorants (comparison functions) of a special form. Bibliography: 9 titles.

On the dimension of the solution set to the homogeneous linear functional differential equation of the first order

Consider the homogeneous equation $$u'(t) = l(u)(t){\rm{ for a}}{\rm{.e}}{\rm{. }}t \in [a,b]$$ where l: C([a, b];ℝ) → L([a, b];ℝ) is a linear bounded operator. The efficient conditions guaranteeing that the solution set to the equation considered is one-dimensional, generated by a positive monotone function, are established. The results obtained are applied to get new efficient conditions sufficient for the solvability of a class of boundary value problems for first order linear functional differential equations.

The Separation of Zeros of Solutions of Higher Order Linear Differential Equations With Entire Coefficients

Suppose that a homogeneous linear differential equation has entire coefficients of finite order and a fundamental set of solutions each having zeros with finite exponent of convergence. Upper bounds are given for the number of zeros of these solutions in small discs in a neighbourhood of infinity.

Zeros of entire solutions to complex linear difference equations

In this article, we study the complex oscillation problems of entire solutions to homogeneous and nonhomogeneous linear difference equations, and obtain some relations of the exponent of convergence of zeros and the order of growth of entire solutions to complex linear difference equations.

On the irreversible thermodynamics of an electron gas in the vicinity of a suddenly moving rigid plate

The Rayleigh flow problem is studied in the framework of the Bhatnager-Gross-Krook (BGK) approximation to the Boltzmann equation supplemented by Maxwell’s equations. The behavior of a rarefied electron gas extracted from noble gases is investigated, considering the method of moments with a two-sided distribution function of Liu and Lees type. Exact solutions of the fourth-order linear homogeneous differential equations, based on the method of the separation of variables for both the mean flow velocity and the shear stress are obtained in terms of the coordinate y and time t with suitable boundary conditions. The advantage of dealing with the Boltzmann equation permits to consider the principles of the irreversible thermodynamics near-equilibrium, by substituting the calculated distribution functions in the formulae of Boltzmann’s H-theorem to evaluate entropy and entropy production. The different contributions of the forces exerted on the system change its internal energies. They are expressed via the Gibbs formula within which the diamagnetic properties of the system are predicted.

GENERAL SOLUTIONS OF SECOND ORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS

GENERAL SOLUTIONS OF SECOND ORDER LINEAR HOMOGENEOUS DIFFERENTIAL EQUATIONS

On some linear difference equations and inequalities with continuous argument

We study some properties of solutions of linear difference equations and inequalities with continuous argument in terms of the location of the zeroes of the characteristic polynomial of the associated homogeneous linear difference equation with constant coefficients.

System of First Order Linear Homogeneous Differential Equations (FLHE) and Riccati Equation

We present a simple method of solving a system of two first order linear homogeneous differential equations with variable co-efficients and demonstrate a correspondence of the system of equations with non-linear Riccati equation. By using appropriate substitution, this non-linear equation is converted into second order linear homogeneous differential equation, the solution of which is used to find the solution of the system of equations. It is seen that different sets of the variable co-efficients of the system of equations correspond to second order linear homogeneous differential equations of different nature. We have illustrated the method with one example. We have shown that the method can be applied to obtain the approximate analytical solutions of the spin dependent coupled Altarelli–Parisi evolution equation in Quantum Chromo Dynamics under plausible conditions.

Differential homomorphisms of linear homogeneous systems of differential equations

Linear differential operators with complex-valued infinitely differentiable coefficients, linear homogeneous systems of differential equations, and modules over algebras of scalar linear differential operators are considered. Linear differential changes of variables and homomorphisms of special quotient modules (differential homomorphisms) generated by these changes are studied. In terms of differential homomorphisms, relationships between Maxwell equations and equations of electromagnetic potential and between Dirac equations and the Klein-Gordon system of independent equations are described. It is proved that all ordinary nondegenerate linear homogeneous differential equations of some common order and the homogeneous normal systems of the same common order are differentially isomorphic.

On Maximum Principles for -Metaharmonic Equations

We study homogeneous linear elliptic partial differential equations of even order. Several maximum principle results are deduced for such equations as well as a priori bounds for certain boundary value problems.

Solutions of Euler-type homogeneous differential equations with finite number of fractional derivatives

Linear homogeneous differential equations with finite number of left-hand sided Liouville fractional derivatives, being analogues of Euler-type ordinary differential equations, are investigated. Using the direct and inverse Mellin transforms, the residue theory and properties of fractional derivatives and Euler Γ and psi functions, general solutions of the considered equations are established. The corresponding results are deduced for Euler-type ordinary differential equations.

On the set of zero coefficients of a function satisfying a linear differential equation

AbstractLet K be a field of characteristic zero and suppose that f: → K satisfies a recurrence of the form \[ f(n) = \sum_{i=1}^d P_i(n) f(n-i), \] for n sufficiently large, where P1(z),. . .,Pd(z) are polynomials in K[z]. Given that Pd(z) is a nonzero constant polynomial, we show that the set of n ∈ for which f(n) = 0 is a union of finitely many arithmetic progressions and a finite set. This generalizes the Skolem–Mahler–Lech theorem, which assumes that f(n) satisfies a linear recurrence. We discuss examples and connections to the set of zero coefficients of a power series satisfying a homogeneous linear differential equation with rational function coefficients.

Study of Solutions to Some Functional Differential Equations with Piecewise Constant Arguments

We provide optimal conditions for the existence and uniqueness of solutions to a nonlocal boundary value problem for a class of linear homogeneous second‐order functional differential equations with piecewise constant arguments. The nonlocal boundary conditions include terms of the state function and the derivative of the state function. A similar nonhomogeneous problem is also discussed.

Functional Differential and Difference Equations with Applications

and Applied Analysis 3 solutions to a class of nonlocal boundary value problems for linear homogeneous secondorder functional differential equations with piecewise constant arguments are obtained. The last but not the least, this issue features a number of publications that report recent progress in the analysis of problems arising in various applications. In particular, dynamics of delayed neural network models consisting of two neurons with inertial coupling were studied, properties of a stochastic delay logistic model under regime switching were explored, and analysis of the permanence and extinction of a single species with contraception and feedback controls was conducted. Other applied problems addressed in this special issue regard Cohen-Grossberg BAM neural networks with time-varying delays, adaptive observer-based fault estimation for stochastic Markovian jumping systems, hematopoiesis models, Lotka-Volterra systems, and finite-time attractivity for diagonally dominant systems with off-diagonal delays. It is certainly impossible to provide in this short editorial note a more comprehensive description for all articles in the collection. However, the team of the Guest Editors believes that results included in this volume represent significant contemporary trends in the qualitative theory of ordinary, functional, partial, impulsive, dynamic, stochastic differential, and difference equations and in applications. We hope that this special issue will serve as a source of inspiration for researchers working in related areas providing specialists with a wealth of new ideas, techniques, and unsolved problems. Josef Diblik Elena Braverman

A Linear Homogeneous Partial Differential Equation with Entire Solutions Represented by Laguerre Polynomials

We study a homogeneous partial differential equation and get its entire solutions represented in convergent series of Laguerre polynomials. Moreover, the formulae of the order and type of the solutions are established.

A global Holmgren theorem for multidimensional hyperbolic partial differential equations

Holmgren's theorem guarantees unique continuation across non-characteristic surfaces of class C 1 for solutions to homogeneous linear partial differential equations with analytic coefficients. Based on this result a global uniqueness theorem for solutions to hyperbolic differential equations with analytic coefficients in a space-time cylinder Q = (0, T) × Ω is established. Zero Cauchy data on a part of the lateral C 1-boundary will force every solution to a homogeneous hyperbolic PDE to vanish at half time T/2 provided that T > T 0. The minimal time T 0 depends on the geometry of the slowness surface of the hyperbolic operator, and can be determined explicitly as demonstrated by several examples including the system of transversely isotropic elasticity and Maxwell's equations. Furthermore, it is shown that this global uniqueness result holds for hyperbolic operators with C 1-coefficients as long as they satisfy the conclusion of Holmgren's Theorem.

The Szego matrix recurrence and its associated linear non-autonomous area-preserving map

A  change  to  the  Szegö matrix  recurrence  relation,  satisfied  by  orthonormal  poly- nomials on the unit circle, gives rise to a linear map by the action of matrices belonging to the group SU (1; 1). The companion factorization of such matrices, via 2nd-order linear homogeneous difference equations, provides a compact representation of the orthogonal polynomial on the circle. Moreover, an isomorphism SU (1; 1) ≃ SL(2; R) enables the introduction of a linear non-autonomous area-preserving  map.   This  dynamical  system  has  counterparts  in  those  from  the  complex Szegö recurrence relation, and some basic results are outlined.