Variation Of Constants Formula Research Articles

The Variation of Constants Formula in Lebesgue Spaces with Variable Exponents

The Variation of Constants Formula in Lebesgue Spaces with Variable Exponents

B-methods for the numerical solution of evolution problems with blow-up solutions part II: Splitting methods

B-methods are numerical methods which are especially tailored to solve non-linear partial differential equation that have blow up solutions. We have presented in Part I a systematic construction of B-methods based on the variation of constants formula. Here, we use splitting methods as a second way to construct B-methods, and we prove several special properties of such methods. We illustrate our analysis with numerical experiments.

A Numerical Scheme for Harmonic Stochastic Oscillators Based on Asymptotic Expansions

In this work, we provide a numerical method for discretizing linear stochastic oscillators with high constant frequencies driven by a nonlinear time-varying force and a random force. The presented method is constructed by starting from the variation of constants formula, in which highly oscillating integrals appear. To provide a suited discretisation of this type of integrals, we propose quadrature rules based on asymptotic expansions. Theoretical considerations and numerical experiments comparing the method with a standard approach on physical models are introduced.

A Unified View of DI- and ETD-FDTD Methods for Drude Media

A unified view of direct-integration (DI) and exponential-time-differencing (ETD) methods to incorporate Drude media, such as isotropic plasma and microwave graphene, into finite-difference time-domain (FDTD) simulators is provided. To this end, the Drude constitutive relation is expressed in integral form and the DI integrators are obtained by applying quadrature rules. Analogously, the ETD integrators are obtained by starting from the variation of constants formula and applying the same quadrature rules as in the DI case. This approach allows one to directly compare the two families of methods. In addition, the accuracy of each integrator is discussed and the stability condition of the resulting FDTD schemes is derived in exact closed form by applying the von Neumann method.

Results of Semigroup of Linear Operators Generating a Regular Weak*-continuous Semigroup

This paper present results of $\omega$-order preserving partial contraction mapping generating a regular weak*-continuous semigroup. We consider a semigroup on a Banach space $X$ and $B:X^\odot\rightarrow X^*$ is bounded, then the intertwining formula was used to define a semigroup $T^B(t)$ on $X^*$ which extends the perturbed semigroup $T^B_0(t)$ on $X^\odot$ using the variation of constants formula. We also investigated a certain class of weak*-continuous semigroups on dual space $X^*$ which contains both adjoint semigroups and their perturbations by operators $B:X^\odot\rightarrow X^*$.

An algorithm for solving linear nonhomogeneous quaternion-valued differential equations and some open problems

<p style='text-indent:20px;'>Quaternion-valued differential equations (QDEs) is a new kind of differential equations. In this paper, an algorithm was presented for solving linear nonhomogeneous quaternionic-valued differential equations. The variation of constants formula was established for the nonhomogeneous quaternionic-valued differential equations. Moreover, several examples showed the feasibility of our algorithm. Finally, some open problems end this paper.</p>

Center manifolds for ill-posed stochastic evolution equations

<p style='text-indent:20px;'>The aim of this paper is to develop a center manifold theory for a class of stochastic partial differential equations with a non-dense domain through the Lyapunov-Perron method. We construct a novel variation of constants formula by the resolvent operator to formulate the integrated solutions. Moreover, we impose an additional condition involving a non-decreasing map to deduce the required estimate since Young's convolution inequality is not applicable. As an application, we present a stochastic parabolic equation to illustrate the obtained results.</p>

Delayed analogue of three‐parameter Mittag‐Leffler functions and their applications to Caputo‐type fractional time delay differential equations

In this paper, we consider a Cauchy problem for a Caputo‐type time delay linear system of fractional differential equations with permutable matrices. First, we provide a new representation of solutions to linear homogeneous fractional differential equations using the Laplace integral transform and variation of constants formula via a newly defined delayed Mittag‐Leffler type matrix function introduced through a three‐parameter Mittag‐Leffler function. Second, with the help of a delayed perturbation of a Mittag‐Leffler type matrix function, we attain an explicit formula for solutions to a linear nonhomogeneous time delay fractional order system using the superposition principle. Furthermore, we prove the existence and uniqueness of solutions to nonlinear fractional delay differential equations using the contraction mapping principle. Finally, we present an example to illustrate the applicability of our results.

Variation of constants formula and exponential dichotomy for nonautonomous non-densely defined Cauchy problems

AbstractIn this paper, we extend to the non-Hille–Yosida case a variation of constants formula for a nonautonomous and nonhomogeneous Cauchy problems first obtained by Gühring and Räbiger. By using this variation of constants formula, we derive a necessary and sufficient condition for the existence of an exponential dichotomy for the evolution family generated by the associated nonautonomous homogeneous problem. We also prove a persistence result of the exponential dichotomy for small perturbations. Finally, we illustrate our results by considering two examples. The first example is a parabolic equation with nonlocal and nonautonomous boundary conditions, and the second example is an age-structured model that is a hyperbolic equation.

Explicit estimates and limit formulae for the solutions of linear delay functional differential systems with nonnegative Volterra type operators

Abstract In this paper we discuss the asymptotic properties of solutions of special inhomogeneous linear delay functional differential systems generated by nonnegative Volterra type operators. In a recent paper of us a variation of constants formula has been given for such systems, and this is particularly suited to handling the studied problems. First, we investigate the boundedness of the solutions. Then some estimates are given for the upper and lower limits of the solutions. By using this, we study the existence of the limit of the solutions, and obtain a formula for the limit. The applicability of our results is illustrated by studying the dependence of limits of the solutions on the choice of inhomogeneities, synchronisation, and Lotka-Volterra type delay functional differential systems.

Global attractivity and asymptotic stability of mixed‐order fractional systems

This study investigates the asymptotic properties of mixed-order fractional systems. By using the variation of constants formula, properties of real Mittag-Leffler functions, and Banach fixed-point theorem, the authors first propose an explicit criterion guaranteeing global attractivity for a class of mixed-order linear fractional systems. The criterion is easy to check requiring the system's matrix to be strictly diagonally dominant (C1) and elements on its main diagonal to be negative (C2). The authors then show the asymptotic stability of the trivial solution to a non-linear mixed-order fractional system linearised along with its equilibrium point such that its linear part satisfies the conditions (C1) and (C2). Two numerical examples with simulations are given to illustrate the effectiveness of the results over existing ones in the literature.

Fractionalization of a Class of Semi-Linear Differential Equations

The dynamics of a fractionalized semi-linear scalar differential equation is considered with a Caputo fractional derivative. By using a symbolic operational method, a fractional order initial value problem is converted into an equivalent Volterra integral equation of second kind. A brief discussion is included to show that the fractional order derivatives and integrals incorporate a fading memory (also known as long memory) and that the order of the fractional derivative can be considered to be an index of memory. A variation of constants formula is established for the fractionalized version and it is shown by using the Fourier integral theorem that this formula reduces to that of the integer order differential equation as the fractional order approaches an integer. The global existence of a unique solution and the global Mittag-Leffler stability of an equilibrium are established by exploiting the complete monotonicity of one and two parameter Mittag-Leffler functions. The method and the analysis employed in this article can be used for the study of more general systems of fractional order differential equations.

Vector differential equations

The solution of a scalar differential equation is given by the variation of constants formula in terms of any fundamental matrix solution of its homogeneous form and its inverse. We extend this to vector differential equations (both linear and non-linear). It is important to make this inverse as explicit as possible. We do this for vector differential equations with constant matrix coefficients. In addition, we also give new results for the well studied case of a scalar differential equation with constant coefficients.

Second Order Abstract Neutral Functional Differential Equations

In this paper we are concerned with a class of second order abstract neutral functional differential equations with finite delay in a Banach space. We establish the existence of mild and classical solutions for the nonlinear equation, and we show that the map defined by the mild solutions of the linear equation is a strongly continuous semigroup of bounded linear operators on an appropriate space. We use this semigroup to establish a variation of constants formula to solve the inhomogeneous linear equation.

BOUNDEDNESS IN NONLINEAR PERTURBED FUNCTIONAL DIFFERENTIAL SYSTEMS

In this paper, we investigate bounds for solutions of nonlinear perturbed functional difierential systems. Integral inequalities play a vital role in the study of boundedness and other qualitative properties of solutions of difierential equations. The method incorporating integral inequalities takes an important place among the methods developed for the qualitative analysis of solutions to linear and nonlinear system of difierential equations. As is traditional in a pertubation theory of nonlinear difierential equations, the behavior of solutions of a perturbed system is determined in terms of the behavior of solutions of an unperturbed system. There are three useful methods for showing the qualitative behavior of the solutions of perturbed nonlinear system : Lyapunov's second method, the use of integral inequalities , and the method of variation of constants formula. In the presence the method of integral inequalities is as e-cient as the direct Lyapunov's method. Pinto (15,16) introduced h-stability (hS) with the intention of obtain- ing results about stability for a weakly stable system (at least, weaker than those given exponential asymptotic stability) under some pertur- bations. That is, Pinto extended the study of exponential asymptotic stability to a variety of reasonable systems called h-systems. Using this notion, Choi and Ryu(3,4) investigated bounds of solutions for nonlinear perturbed systems and nonlinear functional difierential systems. Also, Goo et al.(8,11) studied the boundedness of solutions for nonlinear func- tional perturbed systems.

The virial series for a gas of particles with uniformly repulsive pairwise interaction and its relation with the approach to the mean field

The pressure of a gas of particles with a uniformly repulsive pair interaction in a finite container is shown to satisfy (exactly as a formal object) a “viscous” Hamilton–Jacobi (H–J) equation whose solution in power series is recursively given by the variation of constants formula. We investigate the solution of the H–J and of its Legendre transform equation by the Cauchy-majorant method and provide a lower bound to the radius of convergence on the virial series of the fluid which goes beyond the threshold established by Lagrange’s inversion formula. Such results obtained in (On the virial series for a gas of particles with uniformly repulsive pairwise interactions (2014) Preprint) are reviewed and regarded as the first step towards the solution of a problem posed by Kac, Uhlenbeck and Hemmer (J. Math. Phys. 4 (1963) 216–228), questioning on the relation of the approach to the mean field theory with Ursell–Mayer expansion.

Nonlinear Variation of Constants Formula for Differential Equations with State-Dependent Delays

In this paper we consider a class of differential equations with state-dependent delays. We show differentiability of the solution with respect to the initial function and the initial time for each fixed time value assuming that the state-dependent time lag function is piecewise monotone increasing. Based on these results, we prove a nonlinear variation of constants formula for differential equations with state-dependent delay. As an application, we discuss asymptotic properties of perturbed nonlinear differential equations with state-dependent delays.

BOUNDEDNESS IN PERTURBED FUNCTIONAL DIFFERENTIAL SYSTEMS

In this paper, we investigate bounds for solutions of the the perturbed functional difierential systems. As is traditional in a pertubation theory of nonlinear difierential sys- tems, the behavior of solutions of a perturbed system is determined in terms of the behavior of solutions of an unperturbed system. Among useful methods for investigating the qualitative behavior of the solu- tions of perturbed nonlinear system of difierential systems, there are the method of variation of constants formula, Lyapunov' second method, and the use of inequalities. The theory of integral inequalities can be employed to study various properties of nonlinear difierential systems. In the presence the method of integral inequalities is as e-cient as the direct Lyapunov's method. Pinto(13,14) introduced h-stability(hS) which is an important exten- tion of the notions of exponential asymptotic stability and uniform Lip- schitz stability. Also, he obtained some properties about asymptotic be- havior of solutions of perturbed h-systems, some general results about asymptotic integration and gave some important examples in (14). He obtained a general variational h-stability and some properties about as- ymptotic behavior of solutions of difierential systems called h-systems. Also, he studied some general results about asymptotic integration and gave some important examples in (13). Choi and Ryu (3), Choi, Koo(5), and Choi et al. (4) investigated bounds of solutions for the perturbed

BOUNDEDNESS IN NONLINEAR PERTURBED DIFFERENTIAL SYSTEMS

In this paper, we investigate bounds for solutions of nonlinear perturbed differential systems. The behavior of solutions of a perturbed system is determined in terms of the behavior of solutions of an unperturbed system. There are three useful methods for showing the qualitative behavior of the solutions of perturbed non- linear system : Lyapunov's second method, the use of integral inequalities , and the method of variation of constants formula. The method incorporating in- tegral inequalities takes an important place among the methods developed for the qualitative analysis of solutions to linear and nonlinear system of differential equations. In the presence the method of integral inequalities is as efficient as the direct Lyapunov's method. The notion of h-stability (hS) was introduced by Pinto (15,16) with the in- tention of obtaining results about stability for a weakly stable system (at least, weaker than those given exponential asymptotic stability) under some pertur- bations. That is, Pinto extended the study of exponential asymptotic stability to a variety of reasonable systems called h-systems. Using this notion, Choi and Ryu (3,5) investigated bounds of solutions for nonlinear perturbed systems and nonlinear functional differential systems. Also, Goo et al. (8) studied the boundedness of solutions for nonlinear perturbed systems. In this paper, we obtain some results on boundedness of solutions of nonlinear perturbed differential systems under suitable conditions on perturbed term. To do this we need some integral inequalities.

Pseudo almost periodic solutions of infinite class for some functional differential equations

The aim of this work is to investigate the existence and uniqueness of pseudo almost periodic solutions for some neutral partial functional differential equations in a Banach space when the delay is distributed using the variation of constants formula and the spectral decomposition of the phase space developed in Adimy et al. [M. Adimy, K. Ezzinbi, and A. Ouhinou, Variation of constants formula and almost periodic solutions for some partial functional differential equations with infinite delay, J. Math. Anal. Appl. 317(2) (2006), pp. 668–689]. Here, we assume that the undelayed part is not necessarily densely defined and satisfies the well-known Hille–Yosida condition, the delayed part is assumed to be pseudo almost periodic with respect to the first argument and Lipschitz continuous with respect to the second argument.