Periodic Boundary Value Problems Research Articles

On Approximate Solutions of Nonlinear Boundary-Value Problems by the Newton–Kantorovich Method

We establish necessary and sufficient conditions for the solvability of the nonlinear boundary-value problem in the critical case and develop a scheme for the construction of solutions of this problem. By using the Newton–Kantorovich method, we propose a new iterative scheme for the determination of solutions to the weakly nonlinear boundary-value problem for a system of ordinary differential equations in the critical case. As examples of application of the constructed iterative scheme, we obtain approximations to the solutions of a periodic boundary-value problem for the Duffing and Lienard equations. To check the accuracy of the established approximations to the solutions of the periodic boundary-value problem for the Duffing and Lienard equations, we use discrepancies in the original equations.

Convergence of collocation methods for solving periodic boundary value problems for renewal equations defined through finite‐dimensional boundary conditions

The problem of computing periodic solutions can be expressed as a boundary value problem and solved numerically via piecewise collocation. Here, we extend to renewal equations the corresponding method for retarded functional differential equations in (K. Engelborghs et al., SIAM J Sci Comput., 22 (2001), pp. 1593–1609). The theoretical proof of the convergence of the method has been recently provided in (A. Andò and D. Breda, SIAM J Numer Anal., 58 (2020), pp. 3010–3039) for retarded functional differential equations and in (A. Andò and D. Breda, submitted in 2021) for renewal equations and consists in both cases in applying the abstract framework in (S. Maset, Numer Math., 133 (2016), pp. 525–555) to a reformulation of the boundary value problem featuring an infinite-dimensional boundary condition. We show that, in the renewal case, the proof can also be carried out and even simplified when considering the standard formulation, defined by boundary conditions of finite dimension.

Nontrivial solutions for a second order periodic boundary value problem with the nonlinearity dependent on the derivative

In this paper, we study the existence of nontrivial solutions to the nonlinear differential equation with derivative term u′′(t)+a(t)u(t)=f(t,u(t),u′(t)),t∈[0,ω],u(0)=u(ω),u′(0)=u′(ω), where a: [0,ω]→R+(R+=[0,+∞)) is a continuous function with a(t)⁄≡0, and f: [0,ω]×R×R→R is continuous and may be sign-changing and unbounded from below. Without making any nonnegative assumption on the nonlinearity, by using the first eigenvalue corresponding to the relevant linear operator and the topological degree, the existence of nontrivial solutions to the above periodic boundary value problem is established in C1[0,ω] for the sublinear case. Moreover, two examples are given to demonstrate the validity of our main results.

Existence of positive solutions for periodic boundary value problems of second-order impulsive differential equation with derivative in the nonlinearity

Existence of positive solutions for periodic boundary value problems of second-order impulsive differential equation with derivative in the nonlinearity

An eigenfunction manifold generated by a family of periodic boundary value problems

An analytic and topological description is given of the manifold of periodic eigenfunctions generated by the space of one-dimensional stationary Schrödinger equations with periodic real potentials. Connections with results due to Neuman, Ince and Uhlenbeck are discussed. Bibliography: 11 titles.

New contributions for tripled fixed point methodologies via a generalized variational principle with applications

This manuscript was built to generalize Ekeland variational principle for mixed monotone functions in the setting of partially ordered complete metric spaces. The results obtained are applied to give different proofs for tripled fixed points of mixed monotone mappings in the mentioned space by using a variational technique. The results presented in our manuscripts generalize and expand many of the findings presented in the earlier period. For the sobriety and enhancement of our paper, two examples are given and the existence and uniqueness of the solution to a periodic boundary value problem are studied as applications.

On the existence for a class of periodic boundary value problems of nonlinear fractional hybrid differential equations

This article will study the existence result for periodic boundary value problems of nonlinear fractional hybrid differential equations CD0+γ[η(θ)−x(θ,η(θ))]=y(θ,η(θ)),0<θ<1,η(0)=η(1)=0, where 0<γ<1, CD0+γ is the Caputo derivative. By the Krasnoselskii’s fixed point theorem, the existence result for periodic boundary value problems of nonlinear fractional hybrid differential equations is obtained. An example is given to verify the existence theorem. The main results given in this article correct and revise the statement in Li et al. (2019).

Existence of periodic solutions and bifurcation points for generalized ordinary differential equations

The generalized ordinary differential equations (shortly GODEs), introduced by J. Kurzweil in 1957, encompass other types of equations. The first main result of this paper extends to GODEs some classical conditions on the existence of a periodic solution of a nonautonomous ODE. By means of the correspondence between impulse differential equations (shortly IDEs) and GODEs, we translate the result to IDEs. Instead of the classical hypotheses that the functions on the righthand side of an IDE are piecewise continuous, it is enough to require that they are integrable in the sense of Lebesgue, allowing such functions to have many discontinuities. Our second main result provides conditions for the existence of a bifurcation point with respect to the trivial solution of a periodic boundary value problem for a GODE depending upon a parameter, and, again, we apply such result to IDEs. The machinery employed to obtain the main results are the topological degree theory, tools from the theory of compact operators and an Arzelà-Ascoli-type theorem for regulated functions.

On the solvability of a nonlinear difference boundary value problem in the case of parametric resonance

We construct necessary and sufficient conditions for the existence of solution of seminonlinear boundary value problem for a parametric excitation system of difference equations. The convergent iteration algorithms for the construction of the solutions of the semi-nonlinear boundary value problem for a parametric excitation system difference equations in the critical case have been found. The investigation of periodic and Noetherian boundary-value problems in the critical cases is traditionally performed under the assumption that the differential equation and boundary conditions are known and fixed. As a rule, the study of periodic problems in the case of parametric resonance is reduced to the investigation of the problems of stability. At the same time, due to numerous applications in electronics, geodesy, plasma theory, nonlinear optics, mechanics, and machine-building, the analysis of periodic boundary-value problems in the case of parametric resonance requires not only to find the solutions but also to determine the eigenfunctions of the corresponding differenсе equation. The investigation of autonomous Noetherian boundary-value problems is also reduced to the study of Noetherian boundary-value problems in the case of parametric resonance because the change of the independent variable in the critical case gives a nonautonomous boundary-value problem with an additional unknown quantity. The aim of the present paper is to construct the solutions of Noetherian boundary-value problems in the case of parametric resonance whose solvability is guaranteed by the corresponding choice of the eigenfunction of the analyzed boundary-value problem. The applied classification of Noetherian boundary-value prob\-lems in the case of parametric resonance depending on the simplicity or multiplicity of roots of the equation for generating constants noticeably differs from a similar classification of periodic problems in the case of parametric resonance and corresponds to the general classification of periodic and Noetherian boundary-value problems. The equation for generating constants obtained for the Noetherian boundary-value problems in the case of parametric resonance strongly differs from the conventional equation for generating constants in the absence of parametric resonance by the dependence both of the equation and of its roots on a small parameter, which leads to noticeable corrections of the approximate solutions as compared with the approximations obtained by the Poincare method. Using the convergent iteration algorithms we expand solution of seminonlinear two-point boundary value problem for a parametric excitation Mathieu type difference equation in the neighborhood of the generating solution. Estimates for the value of residual of the solutions of the seminonlinear two-point boundary value problem for a parametric excitation Mathieu type difference equation are found.

A Relation-Theoretic Matkowski-Type Theorem in Symmetric Spaces

In this paper, we present a fixed-point theorem in R-complete regular symmetric spaces endowed with a locally T-transitive binary relation R using comparison functions that generalizes several relevant existing results. In addition, we adopt an example to substantiate the genuineness of our newly proved result. Finally, as an application of our main result, we establish the existence and uniqueness of a solution of a periodic boundary value problem.

Periodic boundary value problem for second-order differential equations from geophysical fluid flows

In this paper, we study a new model for a jet component of the Antarctic Circumpolar Current. In the case of vorticity with perturbation, we present the existence results for positive solutions to periodic boundary value problems for general nonlinear, weak nonlinearity and semilinear terms. A computational method is established and an approximate scheme of solution is also given.

A new finite difference method with optimal phase and stability properties for problems in chemistry

In the present paper we introduce a new FD (Finite Difference) method with eliminated phase–lag and its derivatives up to order six, with also optimal stability properties, for periodic initial or boundary value problems in Chemistry.

Evolutionary computing for nonlinear singular boundary value problems using neural network, genetic algorithm and active-set algorithm

In this numerical study, a class of nonlinear singular boundary value problem is solved by implementation of a novel meta-heuristic computing tool based on the artificial neural networks (ANNs) modeling of system and the optimization of decision variable of ANNs through the combined strength of global search via genetic algorithms (GA) and local search ability of active-set algorithm (ASA), i.e., ANN–GA–ASA. The proposed intelligent computing solver ANN–GA–ASA exploits the input, hidden, and output layers’ structure of ANNs. This is to represent the differential model in the nonlinear singular second-order periodic boundary value problems, which are connected to form an error-based objective function (OF) and optimize the OF by the integrated heuristics of GA–ASA. The purpose to present this research is to associate the operational legacy of neural networks and to challenge such kinds of inspiring models. Two different examples of the singular periodic model have been investigated to observe the robustness, proficiency and stability of the ANN–GA–ASA. The proposed outcomes of ANN–GA–ASA are compared with reference to true results so as to establish the value of the designed scheme. Exhaustive comparison has been made and presented between the Log-sigmoidal ANNs results and the radial basis ANNs outcomes. The reliability of the results obtained is endorsed by using both types of networks as well as the value of designed schemes.

Invariant varieties of the periodic boundary value problem of the nonlocal Ginzburg–Landau equation

One of the variants of the nonlocal Ginzburg–Landau equation is considered. This equation arises in the mathematical modeling of physical phenomena, such as ferromagnetism. For the corresponding initial boundary value problem in the case of periodic boundary conditions, current problems of the theory of infinite‐dimensional dynamical systems are considered. The question of the existence of smooth global solutions and a global attractor is studied. Its dimension and structure are determined. All the solutions of the nonlinear initial boundary value problem are found in explicit form. In particular, it was established that all the solutions belonging to the global attractor will be periodic or quasiperiodic functions of the temporal variable.

Locally exact asymptotic homogenization of viscoelastic composites under anti-plane shear loading

The recently developed elasticity-based locally exact asymptotic homogenization theory is extended to accommodate non-ageing linearly viscoelastic composites under anti-plane shear loading. The theory is based on transforming the problem to the Laplace domain using the elastic-viscoelastic correspondence principle, solving it in the transformed domain and transforming it back to time domain using an efficient and accurate inversion technique. In the Laplace domain, the microscale unit cell interior problems are solved exactly up to the third order through Fourier series expansions, whereas the inseparable exterior periodic boundary value problems are tackled by asymptotic extensions of a balanced variational principle. Numerical examples of microscale solutions are presented and compared with the finite element method in support of the theory's accuracy. The effectiveness of the theory is assessed through a solution of a viscoelastic composite structural problem with finite-sized microstructures, illustrating good comparison with the results of a direct numerical solution. The theory is also applied to the microscale analysis of composites reinforced by viscoelastic fibers with different relaxation rates rarely reported in the literature. The analytical solution methodology in the Laplace domain is unique among the existing higher-order asymptotic homogenization methods employed in the solution of viscoelastic composite structural problems with non-vanishing microstructural scales. It serves as a gold standard for validating strictly numerical approaches.

Action potential propagation and block in a model of atrial tissue with myocyte-fibroblast coupling.

The electrical coupling between myocytes and fibroblasts and the spacial distribution of fibroblasts within myocardial tissues are significant factors in triggering and sustaining cardiac arrhythmias, but their roles are poorly understood. This article describes both direct numerical simulations and an asymptotic theory of propagation and block of electrical excitation in a model of atrial tissue with myocyte-fibroblast coupling. In particular, three idealized fibroblast distributions are introduced: uniform distribution, fibroblast barrier and myocyte strait-all believed to be constituent blocks of realistic fibroblast distributions. Primary action potential biomarkers including conduction velocity, peak potential and triangulation index are estimated from direct simulations in all cases. Propagation block is found to occur at certain critical values of the parameters defining each idealized fibroblast distribution, and these critical values are accurately determined. An asymptotic theory proposed earlier is extended and applied to the case of a uniform fibroblast distribution. Biomarker values are obtained from hybrid analytical-numerical solutions of coupled fast-time and slow-time periodic boundary value problems and compare well to direct numerical simulations. The boundary of absolute refractoriness is determined solely by the fast-time problem and is found to depend on the values of the myocyte potential and on the slow inactivation variable of the sodium current ahead of the propagating pulse. In turn, these quantities are estimated from the slow-time problem using a regular perturbation expansion to find the steady state of the coupled myocyte-fibroblast kinetics. The asymptotic theory gives a simple analytical expression that captures with remarkable accuracy the block of propagation in the presence of fibroblasts.

Двусторонние оценки решений краевых задач для неявных дифференциальных уравнений

We consider a two-point (including periodic) boundary value problem for the following system of differential equations that are not resolved with respect to the derivative of the desired function: f_i (t,x,x ̇,(x_i ) ̇ )=0,i= (1,n) ̅. Here, for any i= (1,n) ̅ the function f_i:[0,1]×R^n×R^n×R→R is measurable in the first argument, continuous in the last argument, right-continuous, and satisfies the special condition of monotonicity in each component of the second and third arguments. Assertions about the existence and two-sided estimates of solutions (of the type of Chaplygin’s theorem on differential inequality) are obtained. Conditions for the existence of the largest and the smallest (with respect to a special order) solution are also obtained. The study is based on results on abstract equations with mappings acting from a partially ordered space to an arbitrary set (see [S. Benarab, Z.T. Zhukovskaya, E.S. Zhukovskiy, S.E. Zhukovskiy. On functional and differential inequalities and their applications to control problems // Differential Equations, 2020, 56:11, 1440–1451]).

Dispersive Quantization of 2D Linear Dispersive Equations

The dispersive quantization of the 2D linear KdV equation and the 2D linear Schrödinger equation were studied over a bounded rectangle domain in the plane. The research shows that, for the KdV equation, if the period ratio is a rational number, at the rational moments, the solution to the periodic initial boundary value problem will be the linear combination of the initial value conditions; whereas, at the irrational moments, the solution will be continuous and nondifferentiable, and exhibit a fractallike profile. The same is true for the 2D linear Schrödinger equation.

带有 p-Laplace 算子的离散混合周期边值问题正解的存在性

带有 p-Laplace 算子的离散混合周期边值问题正解的存在性

An existence result for a periodic boundary value problem of fractional semilinear differential equations in a Banach space

An existence result for a periodic boundary value problem of fractional semilinear differential equations in a Banach space