Class Of Linear Differential Equations Research Articles

Two Schemes of Impulsive Runge–Kutta Methods for Linear Differential Equations with Delayed Impulses

In this paper, two different schemes of impulsive Runge–Kutta methods are constructed for a class of linear differential equations with delayed impulses. One scheme is convergent of order p if the corresponding Runge–Kutta method is p order. Another one in the general case is only convergent of order 1, but it is more concise and may suit for more complex differential equations with delayed impulses. Moreover, asymptotical stability conditions for the exact solution and numerical solutions are obtained, respectively. Finally, some numerical examples are provided to confirm the theoretical results.

Existence analysis of solutions to the linear Schrödinger Kirchhoff Poisson equation based on interrupted finite elements

Abstract In this paper, we firstly explore the existence of solutions to the following linear Schrödinger Kirchhoff Poisson equation with critical exponential growth on the full space □3 by using the discontinuous finite element (DG) as well as the principle of centralized compactness: { − ( a + b ∫ □ 3 | ∇ u | 2 ) Δ u + V ( x ) u + φ u - 1 2 u Δ ( u 2 ) = K ( x ) u p − 2 u , x ∈ □ 3 − Δ φ = u 2 , x ∈ □ 3 \left\{ {\matrix{{ - \left( {a + b\int_{{\square^3}} {{{\left| {\nabla u} \right|}^2}} } \right)\Delta u + V\left( x \right)u + \phi u - {1 \over 2}u\Delta \left( {{u^2}} \right) = K\left( x \right){u^{p - 2}}u,} & {x \in {\square^3}} \cr { - \Delta \phi = {u^2},} & {x \in {\square^3}} \cr } } \right. , x ∈. Secondly, we make reasonable assumptions on the V, K , f functions of the equation, and use the principle of variational division to firstly obtain the corresponding energy generalization of this equation, and then we prove the corresponding generalizations of the equation satisfy the (C) c conditions. Finally, the existence of the solution of the equation is obtained by numerical simulation and then by using the Yamaji Lemma. The results show that the error of the finite element solution of the linear Schrödinger Kirchhoff Poisson equation in the spatial direction P1 reaches the optimal estimation under the L 2 -parameter in an intermittent finite element numerical simulation environment, i.e., it is proved that there exist at least 1 and 3 positive solutions to the problem. The paper achieves rich research results which are informative for the solution of this class of linear differential equations.

Growth of Solutions of a Class of Linear Differential Equations Near a Singular Point

Growth of Solutions of a Class of Linear Differential Equations Near a Singular Point

BOUNDEDNESS OF SOLUTIONS OF A CLASS OF LINEAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER ON THE SEMIAXIS

In this paper based on a formula for one class of second order linear differential equations with variable coefficients were established sufficient conditions of exponential estimate of solutions for one class of second order linear differential equations on the semi axis.

On stability of linear differential equations with commensurate delayed arguments

The paper studies a class of linear differential equations with several delayed arguments formed by iterates of a given function. The main result of this paper improves the existing stability criteria and formulates an effective necessary and sufficient condition relating stability of the studied differential equations to stability of some auxiliary difference equations. In the case of a two-delay equation, this condition is presented explicitly in terms of the equation’s parameters. As an accompanying result, the asymptotic decay rate of the solutions is described as well.

The first positive root of the fundamental solution is an optimal oscillation bound for linear delay differential equations

We consider the nonautonomous linear delay differential equationx′(t)+p(t)x(t−1)=0,t≥t0, where p:[t0,∞)→R is a continuous function such that p(t)≥a>1/e for t≥t0. It is shown that the smallest positive constant ω=ω(a) such that every solution has at least one zero in [t0−1,t0+ω] coincides with the first positive root ζ(a) of the fundamental solution of the equation with constant coefficienty′(t)+ay(t−1)=0. A detailed analysis of the function ζ:(1/e,∞)→(1,∞) is given with the emphasis on the asymptotic description of ζ(a) as a→1/e+. As corollaries new oscillation criteria are obtained which improve some previous results in the literature. With appropriate modifications the method applies to a more general class of linear differential equations with time-dependent delay.

Approximation of Families of Characteristics of Energy Objects in the Class of Linear Differential Equations

A method is proposed for constructing a fast-computable mathematical model of a family of energy characteristics based on its comparison with a family of integral curves in the class of linear differential equations. The method uses an affine transformation and takes into account the two-level structure of the mathematical model of a family of characteristics (differential operator and boundary conditions). It allows reducing the number of approximating summands and increasing the adequacy by including additional information of a systemic nature about the object under study.

The hyper order and fixed points of solutions of a class of linear differential equations

In this paper, we precise the hyper order of solutions for a class of higher-order linear differential equations and investigate the exponents of convergence of the fixed points of solutions and their first derivatives for the second-order case. These results generalize those of Nan Li and Lianzhong Yang and of Chen and Shon.

Adapting the Laplace transform to create solitary solutions for the nonlinear time-fractional dispersive PDEs via a new approach

It is known that the Laplace transform method is used to solve only a finite class of linear differential equations. In this paper, we suggest a new method that relies on a new fractional expansion in the Laplace transform space and residual power series method to construct exact solitary solutions to the nonlinear time-fractional dispersive PDEs. The main idea of this method is to convert the original fractional differential equation into the Laplace space, and then, solve the newly obtained equation using a new approach of the residual power series method. Moreover, four important and attractive applications are also presented and discussed based on our proposed method to reveal the efficiency, effectiveness, and applicability of this method for constructing solitary solutions to the target equations. The results confirmed that the proposed method is a simple, understandable, and very fast method to obtain solitary solutions compared with other methods such as homotopy perturbation method, variation iteration method, sine–cosine method, and residual power series method.

On some classes of linear difference equations with convergent solutions

We present some classes of homogeneous linear difference equations with constant coefficients such that their associated characteristic polynomials have multiple zeros, and all solutions to the equations are convergent. The cases of the equations of third and fourth order are considered in detail, whereas in the case of equations of arbitrary order bigger than four, we present a subclass of such equations. We also show that the problem of calculating limits of solutions to the equations, when all the zeros of the characteristic polynomials are simple, is essentially folklore.

New formula for solution of one class of linear differential equations of the second order with the variable coefficients

Exact solutions for linear and nonlinear differential equations play an important rolein theoretical and practical research. In particular many works have been devoted tofinding a formula for solving second order linear differential equations with variablecoefficients. In this paper we obtained the formula for the common solution of thelinear differential equation of the second order with the variable coefficients in themore common case. We also obtained the new formula for the solution of the Cauchyproblem for the linear differential equation of the second order with the variablecoefficients.Examples illustrating the application of the obtained formula for solvingsecond-order linear differential equations are given.Key words: The linear differential equation, the second order, the variablecoefficients,the new formula for the common solution, Cauchy problem, examples.

On strongly irregular periodic solutions of the linear nonhomogeneous discrete equation of the first order

As is proved earlier (the Massera theorem), the first-order scalar periodic ordinary differential equation does not have strongly irregular periodic solutions (solutions with a period incommensurable with the period of the equation). For difference equations with discrete time, strong irregularity means that the equation period and the period of its solution are relatively prime numbers. It is known that in the case of discrete equations, the mentioned result has no complete analog.The purpose of this paper is to investigate the possibility of realizing an analog of the Massera theorem for certain classes of difference equations. To do this, we consider the class of linear difference equations. It is proved that a linear nonhomogeneous non-stationary periodic discrete equation of the first order does not have strongly irregular non-stationary periodic solutions.

Fractional extreme distributions

We consider three classes of linear differential equations on distribution functions, with a fractional order $\alpha \in [0,1]$. The integer case $\alpha =1$ corresponds to the three classical extreme families. In general, we show that there is a unique distribution function solving these equations, whose underlying random variable is expressed in terms of an exponential random variable and an integral transform of an independent $\alpha $-stable subordinator. From the analytical viewpoint, this distribution is in one-to-one correspondence with a Kilbas-Saigo function for the Weibull and Fréchet cases, and with a Le Roy function for the Gumbel case.

Asymptotic Integration of Certain Differential Equations in Banach Space

In this work, we investigate the problem of constructing asymptotic representations for weak solutions of a certain class of linear differential equations in the Banach space as an independent variable tends to infinity. We consider the class of equations that represent a perturbation of a linear autonomous equation, in general, with an unbounded operator. The perturbation takes the form of a family of bounded operators that, in a sense, oscillatorally decreases at infinity. It is assumed that the unperturbed equation satisfies the standard requirements of the center manifold theory. The essence of the proposed asymptotic integration method is to prove the existence of a center-like manifold (a critical manifold) for the initial equation. This manifold is positively invariant with respect to the initial equation and attracts all trajectories of the weak solutions. The dynamics of the initial equation on the critical manifold is described by the finite-dimensional system of ordinary differential equations. The asymptotics of the fundamental matrix of this system can be constructed by using the method developed by P.N. Nesterov for asymptotic integration of systems with oscillatory decreasing coefficients. We illustrate the proposed technique by constructing the asymptotic representations for solutions of the perturbed heat equation.

Liouville Formula for Some Classes of Differential Equations with the Hukuhara Derivative

We use the vector Lyapunov function method and the comparison method in conjunction with methods of convex geometry and the theory of mixed volumes to obtain analogs of the Liouville formula for some classes of linear differential equations with the Hukuhara derivative defined on the space of convex compact subsets of the plane. A continual generalization of the construction of a vector Lyapunov function is proposed.

Applying the random variable transformation method to solve a class of random linear differential equation with discrete delay

We randomize the following class of linear differential equations with delay, xτ′(t)=axτ(t)+bxτ(t−τ),t > 0, and initial condition, xτ(t)=g(t),−τ≤t≤0, by assuming that coefficients a and b are random variables and the initial condition g(t) is a stochastic process. We consider two cases, depending on the functional form of the stochastic process g(t), and then we solve, from a probabilistic point of view, both random initial value problems by determining explicit expressions to the first probability density function, f(x, t; τ), of the corresponding solution stochastic processes. Afterwards, we establish sufficient conditions on the involved random input parameters in order to guarantee that f(x, t; τ) converges, as τ→0+, to the first probability density function, say f(x, t), of the corresponding associated random linear problem without delay (τ=0). The paper concludes with several numerical experiments illustrating our theoretical findings.

Solution Area for a Class of Linear Differential Equations with Hukuhara Derivative

A formula for the solution area for a class of linear differential equations with Hukuhara derivative is obtained.

Асимптотика решений одного класса линейных дифференциальных уравнений

Асимптотика решений одного класса линейных дифференциальных уравнений

STRONGLY IRREGULAR PERIODIC SOLUTIONS OF THE FIRST-ORDER LINEAR HOMOGENEOUS DISCRETE EQUATION

In 1950 J. Massera proved that a fi rst-order scalar periodic ordinary differential equation has no strongly ira proved that a first-order scalar periodic ordinary differential equation has no strongly irregular periodic solutions, that is, such solutions whose period of solution is incommensurable with the period of equation. For difference equations with discrete time, strong irregularity means that the period of the equation and the period of its solution are relatively prime numbers. It is known that in the case of discrete equations, the above result of J. Massera has no complete analog.The purpose of this article is to investigate the possibility to realize Massera’s theorem for certain classes of difference equations. To do this, we consider the class of linear difference equations. It is proved that a first-order linear homogeneous non-stationary periodic discrete equation has no strongly irregular non-stationary periodic solutions.

Asymptotic Integration of Certain Differential Equations in Banach Space

We investigate the problem of constructing the asymptotics for weak solutions of certain class of linear differential equations in the Banach space as the independent variable tends to infinity. The studied class of equations is the perturbation of linear autonomous equation, generally speaking, with an unbounded operator. The perturbation takes the form of the family of the bounded operators that, in a sense, decreases oscillatory at infinity. The unperturbed equation satisfies the standard requirements of the center manifold theory. The essence of the proposed asymptotic integration method is to prove the existence for the initial equation of the center-like manifold (critical manifold). This manifold is positively invariant with respect to the initial equation and attracts all the trajectories of the weak solutions. The dynamics of the initial equation on the critical manifold is described by the finite-dimensional ordinary differential system. The asymptotics for the fundamental matrix of this system may be constructed by using the method proposed by the author for asymptotic integration of the systems with oscillatory decreasing coefficients. We illustrate the suggested technique by constructing the asymptotic formulas for solutions of the perturbed heat equation.