Periodic Boundary Value Problems Research Articles

Periodic Boundary-Value Problem for a Rayleigh-Type Equation Unsolved with Respect to the Derivative

Periodic Boundary-Value Problem for a Rayleigh-Type Equation Unsolved with Respect to the Derivative

An FFT based adaptive grid framework to represent non-singular dislocations

Non-singular theories aim at regularizing the unrealistic stress singularity around a dislocation line predicted by elasticity by spreading the Burgers vector around the core. The use of these approaches in discrete dislocation models or field dislocation mechanics require very fine discretization around the dislocation lines. FFT solvers commonly used in these problems rely on a regular grid which enforces the use of very fine discretizations or very small domains. In this work, a methodology is proposed to solve the boundary value problems associated with dislocation modeling using FFT with an adaptive grid which is refined around dislocation cores. The framework introduces a regular domain to compute Fourier derivatives mapped to a physical domain with more points concentrated the areas of interest. The linear differential operators involved are obtained by the product of the Fourier derivatives in the regular space and the Jacobian of the map defining the transformation between computational and physical domains. The periodic boundary value problem involved is transformed in a linear system that can be efficiently solved using Krylov solvers. The method for adaptive FFT grids is fully general and can be applied to any other micromechanical problem. It is demonstrated that the method allows to use several grid points within the core region of 2D and 3D dislocations even using coarse discretizations, allowing to resolve the non-singular stress fields within this region and also strongly reducing the numerical noise. Moreover, the method allows to preserve the accuracy of the results in fine meshes by reducing the grid spacing up to four times.

Fractional periodic boundary value and Cauchy problems with Hilfer–Prabhakar operator

We introduce a successive approximations method to study one fractional periodic boundary value problem of the Hilfer-Prabhakar type. The problem is associated to the corresponding Cauchy problem, whose solution depends on an unknown initial value. To find this value we numerically solve the so-called ’determining system’ of algebraic or transcendental equations. As a result, we determine an approximate solution of the studied problem, written in a closed form. Finally, we evaluate efficiency of our method on a nonlinear numerical example.

Multiplicity of Solutions for Discrete 2n-TH Order Periodic Boundary Value Problem with φp-Laplacian

The purpose of this paper is to investigate the existence and multiplicity of nontrivial solutions with the φp-Laplacian for the discrete 2n-th order periodic boundary value issue. To support these conclusions, we have employed variational techniques and contemporary critical point theory. A few new findings are expanded upon and enhanced. We give an example to show how our key findings can be applied.

Dynamics of vertical axis washing machines with uncertain unbalance

A washing machine is a household appliance that has an interesting and complex dynamical behavior, which can be well described by a set of nonlinear differential equations. When analyzing the dynamics of a washing machine, the steady state motion (periodic solution) is an important response to consider and can be evaluated as a solution of a periodic boundary-value problem. The unbalance generated by the unevenly distribution of clothes during centrifugation is highly random and, therefore, a stochastic model is necessary to take this characteristic into account. The novelty of this paper consists in the analysis of a washing machine dynamics considering the uncertainty in the unbalance. Therefore, a stochastic model is proposed for the dynamics of a washing machine. The steady state solutions are calculated using the shooting method combined with a sequential continuation to evaluate it across all the spin speeds of the machine. The probability distributions of the washing machine vibration at those different spin speeds are approximated using Monte Carlo simulations. The impact of the random unbalance in the vibration amplitude of the washing machine is also investigated and consists in a key input for the fatigue design of many components.

Tripled Fixed Point Approaches and Hyers-Ulam Stability With Applications

In this paper, we present tripling fixed point results for extended contractive mappings in the context of a generalized metric space. Many publications in the literature are improved, unified, and generalized by our theoretical results. Furthermore, the Ulam-Hyers stability problem for the tripled fixed point problem in vector-valued metric spaces has been examined as a stability analysis for fixed point approaches. Finally, as a type of application to support our research, the theoretical conclusions are used to explore the existence and uniqueness of solutions to a periodic boundary value problem.

On Graphical Symmetric Spaces, Fixed-Point Theorems and the Existence of Positive Solution of Fractional Periodic Boundary Value Problems

The rationale of this work is to introduce the notion of graphical symmetric spaces and some fixed-point results are proved for H-(ϑ,φ)-contractions in this setting. The idea of graphical symmetric spaces generalizes various spaces equipped with a function which characterizes the distance between two points of the space. Some topological properties of graphical symmetric spaces are discussed. Some fixed-point results for the mappings defined on graphical symmetric spaces are proved. The fixed-point results of this paper generalize and extend several fixed-point results in this new setting. The main results of this paper are applied to obtain the positive solutions of fractional periodic boundary value problems.

On Approximate Solutions of a Nonlinear Periodic Boundary-Value Problem with Switching in the Case of Parametric Resonance by the Newton–Kantorovich Method

On Approximate Solutions of a Nonlinear Periodic Boundary-Value Problem with Switching in the Case of Parametric Resonance by the Newton–Kantorovich Method

Iterative Schemes for Periodic Boundary-Value Problems with Switchings

Iterative Schemes for Periodic Boundary-Value Problems with Switchings

Multiple solutions to a quasilinear periodic boundary value problem with impulsive effects

The authors investigate the multiplicity of solutions to a quasilinear periodic boundary value problem with impulsive effects. They use variational methods and some critical points theorems for smooth functionals, due to Ricceri, that are defined on reflexive Banach spaces. They obtain the existence of at least three solutions to the problem. The applicability of the results is illustrated with an example.

On the reduction of an autonomous nonlinear boundary value problem, unsolved with respect to the derivative, to the first-order critical case

Constructive solvability conditions and a scheme for constructing solutions of an autonomous nonlinear boundary value problem unsolved with respect to the derivative have been found. A convergent iterative scheme for finding approximations to the solutions of a nonlinear autonomous nonlinear boundary value problem unsolved with respect to the derivative was constructed by reducing the problem to the first-order critical case. As an example of application of the constructed iterative scheme, approximations to the solutions of a periodic boundary value problem for the autonomous Lotka-Volterra equation were determined.

Boundary value problems of quaternion-valued differential equations: solvability and Green’s function

This paper is associated with Sturm–Liouville type boundary value problems and periodic boundary value problems for quaternion-valued differential equations (QDEs). Employing the theory of quaternionic matrices, we prove the conditions for the solvability of the linear boundary valued problem and find Green’s function.

Almost Boyd-Wong Type Contractions under Binary Relations with Applications to Boundary Value Problems

This article is devoted to investigating the fixed point theorems for a new contracitivity contraction, which combines the idea involved in Boyd-Wong contractions, strict almost contractions and relational contractions. Our results improve and expand existing fixed point theorems of literature. In process, we deduce a metrical fixed point theorem for strict almost Boyd-Wong contractions. To demonstrate the credibility of our results, we present a number of a few examples. Applying our findings, we find a unique solution to a particular periodic boundary value problem.

Positive Solutions for Periodic Boundary Value Problems of Fractional Differential Equations with Sign-Changing Nonlinearity and Green’s Function

In this paper, a class of nonlinear fractional differential equations with periodic boundary condition is investigated. Although the nonlinearity of the equation and the Green’s function are sign-changing, the results of the existence and nonexistence of positive solutions are obtained by using the Schaefer’s fixed-point theorem. Finally, two examples are given to illustrate the main results.

Multiplicity of solutions for discrete 2n-th order periodic boundary value problem

Multiplicity of solutions for discrete 2n-th order periodic boundary value problem

Solvability, Approximation and Stability of Periodic Boundary Value Problem for a Nonlinear Hadamard Fractional Differential Equation with p-Laplacian

The fractional order p-Laplacian differential equation model is a powerful tool for describing turbulent problems in porous viscoelastic media. The study of such models helps to reveal the dynamic behavior of turbulence. Therefore, this article is mainly concerned with the periodic boundary value problem (BVP) for a class of nonlinear Hadamard fractional differential equation with p-Laplacian operator. By virtue of an important fixed point theorem on a complete metric space with two distances, we study the solvability and approximation of this BVP. Based on nonlinear analysis methods, we further discuss the generalized Ulam-Hyers (GUH) stability of this problem. Eventually, we supply two example and simulations to verify the correctness and availability of our main results. Compared to many previous studies, our approach enables the solution of the system to exist in metric space rather than normed space. In summary, we obtain some sufficient conditions for the existence, uniqueness, and stability of solutions in the metric space.

Singular and Nonsingular Third-Order Periodic Boundary Value Problems with Indefinite Weight

Singular and Nonsingular Third-Order Periodic Boundary Value Problems with Indefinite Weight

Periodic Boundary Value Problem for Impulsive Evolution Equations with Noncompact Semigroup

Periodic Boundary Value Problem for Impulsive Evolution Equations with Noncompact Semigroup

Random Solutions for Generalized Caputo Periodic and Non-Local Boundary Value Problems

In this article, we present some results on the existence and uniqueness of random solutions to a non-linear implicit fractional differential equation involving the generalized Caputo fractional derivative operator and supplemented with non-local and periodic boundary conditions. We make use of the fixed point theorems due to Banach and Krasnoselskii to derive the desired results. Examples illustrating the obtained results are also presented.

Non-classical periodic boundary value problems with impulsive conditions

This study investigates some spectral properties of a new type of periodic Sturm-Liouville problem. The problem under consideration differs from the classical ones in that the differential equation is given on two disjoint segments that have a common end, and two additional interaction conditions are imposed on this common end (such interaction conditions are called various names, including transmission conditions, jump conditions, interface conditions, impulsive conditions, etc.). At first, we proved that all eigenvalues are real and there is a corresponding real-valued eigenfunction for each eigenvalue. Then we showed that two eigenfunctions corresponding to different eigenvalues are orthogonal. We also defined some left and right-hand solutions, in terms of which we constructed a new transfer characteristic function. Finally, we have defined asymptotic formulas for the transfer characteristic functions and also for the eigenvalues. The results obtained are a generalization of similar results of the classical Sturm-Liouville theory.