Countable Systems Of Differential Equations Research Articles

Approximate Solution of Two Dimensional Disc-like Systems by One Dimensional Reduction: An Approach through the Green Function Formalism Using the Finite Elements Method

We present a comprehensive study for common second order PDE’s in two dimensional disc-like systems and show how their solution can be approximated by finding the Green function of an effective one dimensional system. After elaborating on the formalism, we propose to secure an exact solution via a Fourier expansion of the Green function, which entails solving an infinitely countable system of differential equations for the Green–Fourier modes that in the simplest case yields the source-free Green distribution. We present results on non separable systems—or such whose solution cannot be obtained by the usual variable separation technique—on both annulus and disc geometries, and show how the resulting one dimensional Fourier modes potentially generate a near-exact solution. Numerical solutions will be obtained via finite differentiation using Finite Difference Method (FDM) or Finite Element Method (FEM) with the three-point stencil approximation to derivatives. Comparing to known exact solutions, our results achieve an estimated numerical relative error below 10−6. Solutions show the well-known presence of peaks when r=r′ and a smooth behavior otherwise, for differential equations involving well-behaved functions. We also verified how the Green functions are symmetric under the presence of a “weight function”, which is guaranteed to exist in the presence of a curl-free vector field. Solutions of non-homogeneous differential equations are also shown using the Green formalism and showing consistent results.

On the Invariant Tori of Quasilinear Countable Systems of Differential Equations Defined on Infinite–Dimensional Tori

On the Invariant Tori of Quasilinear Countable Systems of Differential Equations Defined on Infinite–Dimensional Tori

On Oscillating Solutions of a Countable System of Differential Equations in the Resonance Case

On Oscillating Solutions of a Countable System of Differential Equations in the Resonance Case

NONLOCAL PROBLEM FOR A MIXED TYPE FOURTH-ORDER DIFFERENTIAL EQUATION WITH HILFER FRACTIONAL OPERATOR

In this paper, we consider a non-self-adjoint boundary value problem for a fourth-order differential equation of mixed type with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with spectral parameter in a negative rectangular domain. The mixed type differential equation under consideration is a fourth order differential equation with respect to the second variable. Regarding the first variable, this equation is a fractional differential equation in the positive part of the segment, and is a second-order differential equation with spectral parameter in the negative part of this segment. A rational method of solving a nonlocal problem with respect to the Hilfer operator is proposed. Using the spectral method of separation of variables, the solution of the problem is constructed in the form of Fourier series. Theorems on the existence and uniqueness of the problem are proved for regular values of the spectral parameter. For sufficiently large positive integers in unique determination of the integration constants in solving countable systems of differential equations, the problem of small denominators arises. Therefore, to justify the unique solvability of this problem, it is necessary to show the existence of values of the spectral parameter such that the quantity we need is separated from zero for sufficiently large \(n\). For irregular values of the spectral parameter, an infinite number of solutions in the form of Fourier series are constructed. Illustrative examples are provided.

Non-linear Volterra IDE, infinite systems and normal forms of ODE

We study the Volterra integro-differential equation in R n (⋆ ) d x d t = X ( t , x , ∫ 0 t K ( t , s ) g ( x ( s ) d s ) ) . We establish a connection between system (⋆) with a kernel of the form (⋆ ⋆) K ( t , s ) = ∑ j = 1 ∞ C j F j ( t ) G j ( s ) and a countable system of ordinary differential equations. Such a reduction allows use of results obtained earlier for the countable systems of differential equations in the study of integro-differential equations. In this paper we discuss problems related to the stability of systems (⋆) and (⋆⋆), as well as applications of the method of normal forms to solving some problems in the qualitative theory of integro-differential equations. In particular, it can be employed for the study of critical cases of stability and bifurcation problems in integro-differential equations.

Volterra integro-differential equations and infinite systems of ordinary differential equations

We establish a connection between finite-dimensional systems of integro-differential equations with the Hilbert-Schmidt kernel and ordinary differential equations in l 2 (countable systems of differential equations). Such a reduction allows use of results obtained earlier for the countable systems of differential equations in study of integro-differential equations. In particular, it can be employed for study of stability and solutions' constructions for integro-differential equations.

On invariant tori of countable systems of differential equations with delay

We obtain conditions for the existence and smoothness of invariant tori of countable systems of differential equations with delay.

Limit theorems in the theory of multipoint boundary-value problems

We present a reduction of a countable system of differential equations with countably-point boundary conditions to the case of a finite-dimensional multipoint boundary-value problem. We separately consider the case of a linear system.

A criterion of exponential dichotomy for a countable system of differential equations with quasiperiodic coefficients

We establish necesary and sufficient conditions for the exponential diehetemy with matrix projeseters of a countable system of differential equations with quasiperiodic coefficients.

Introduction of local coordinates for a countable system of differential equations in the neighborhood of an invariant manifold

We establish conditions of the existence of local coordinates for a countable system of differential equations in the neighborhood of an invariant manifold and present the form of this system in these coordinates.

On decomposability of countable systems of differential equations

Conditions under which there exists a change of variables that decomposes a countable system of differential equations are established for the entire real axis and a semiaxis. Similar problems are investigated for a countable system with pulse influence.

Exponential stability of an invariant torus of a nonlinear countable system of differential equations

Conditions are considered under which any trajectory from a small neighborhood of a torus of a nonlinear countable system of differential equations is attacted to a corresponding trajectory on the torus by an exponential law.

?Truncation? of countable systems of differential equations with impulse action

In the investigation of nonlinear partial differential equations ofhyperbolic type with the help of the averaging method [I] there arises a problem connected with the solution of a countable system of ordinary differential equations [1-3]. In the present article, we consider the analogous problem on the "truncation" (in the sense of [3]) of a countable system of differential equations with impulse action, to which we arrive by applying the averaging method to the study of hyperbolic systems with impulse action.

Question of reductibility of countable systems of differential equations with quasiperiodic coefficients

Question of reductibility of countable systems of differential equations with quasiperiodic coefficients

The existence of the invariant toroidal manifold of a countable system of differential equations with impulsive action

The existence of the invariant toroidal manifold of a countable system of differential equations with impulsive action

The existence and smoothness of invariant toroidal manifolds of nonlinear countable systems of differential equations

The existence and smoothness of invariant toroidal manifolds of nonlinear countable systems of differential equations

Periodic solutions of a countable system of differential equations with deviating neutral argument

Periodic solutions of a countable system of differential equations with deviating neutral argument