Integro-differential Boundary Value Problem Research Articles

Asymptotics of Regular and Irregular Solutions in Chains of Coupled van der Pol Equations

Chains of coupled van der Pol equations are considered. The main assumption that motivates the use of special asymptotic methods is that the number of elements in the chain is sufficiently large. This allows moving from a discrete system of equations to the use of a continuity argument and obtaining an integro-differential boundary value problem as the initial model. In the study of the behaviour of all its solutions in a neighbourhood of the equilibrium state, infinite-dimensional critical cases arise in the problem of the stability of solutions. The main results include the construction of special families of quasi-normal forms, namely non-linear boundary value problems of either Schrödinger or Ginzburg–Landau type. Their solutions make it possible to determine the main terms of the asymptotic expansion of both regular and irregular solutions to the original system. The main goal is the study of chains with diffusion- and advective-type couplings, as well as fully connected chains.

Nonlinear Integrodifferential Boundary-Value Problems with Deviating Argument Unsolved with Respect to the Derivative

Nonlinear Integrodifferential Boundary-Value Problems with Deviating Argument Unsolved with Respect to the Derivative

Nonlinear Integrodifferential Boundary-Value Problems Unsolvable with Respect to the Derivative

Nonlinear Integrodifferential Boundary-Value Problems Unsolvable with Respect to the Derivative

Quasinormal Forms for Chains of Coupled Logistic Equations with Delay

In this paper, chains of coupled logistic equations with delay are considered, and the local dynamics of these chains is investigated. A basic assumption is that the number of elements in the chain is large enough. This implies that the study of the original systems can be reduced to the study of a distributed integro–differential boundary value problem that is continuous with respect to the spatial variable. Three types of couplings of greatest interest are considered: diffusion, unidirectional, and fully connected. It is shown that the critical cases in the stability of the equilibrium state have an infinite dimension: infinitely many roots of the characteristic equation tend to the imaginary axis as the small parameter tends to zero, which characterizes the inverse of the number of elements of the chain. In the study of local dynamics in cases close to critical, analogues of normal forms are constructed, namely quasinormal forms, which are boundary value problems of Ginzburg–Landau type or, as in the case of fully connected systems, special nonlinear integro–differential equations. It is shown that the nonlocal solutions of the obtained quasinormal forms determine the principal terms of the asymptotics of solutions to the original problem from a small neighborhood of the equilibrium state.

Oscillations of a partially wetting bubble

We study the linear stability of a compressible sessile bubble in an ambient fluid that partially wets a planar solid support, where the gas is assumed to be an ideal gas that obeys the adiabatic law. The frequency spectrum is computed from an integrodifferential boundary value problem and depends upon the wetting conditions through the static contact angle$\alpha$, the dimensionless equilibrium bubble pressure$\varPi$, and the contact-line dynamics that we assume to be either (i) pinned or (ii) freely moving with fixed contact angle. Corresponding mode shapes are defined by the polar-azimuthal mode number pair$[k,\ell ]$with$k+\ell =\mathbb {Z}^{+}_{even}$. We report instabilities to the (i)$[0,0]$breathing mode associated with volume change, and (ii)$[1,1]$mode that is linked to horizontal centre-of-mass motion of the bubble. Stability diagrams and instability growth rates are computed, and the respective instability mechanisms are revealed through an energy analysis. The zonal$\ell =0$modes are associated with volume change, and we show that there is a complex dependence between the classical volume and shape change modes for wetting conditions that differ from neutral wetting$\alpha =90^\circ$. Finally, we show how the classical frequency degeneracy for the Rayleigh–Lamb modes of the free bubble splits for the azimuthal modes$\ell \neq 0,1$.

An Existence Result for ψ-Hilfer Fractional Integro-Differential Hybrid Three-Point Boundary Value Problems

In this research work, we study a new class of ψ-Hilfer hybrid fractional integro-differential boundary value problems with three-point boundary conditions. An existence result is established by using a generalization of Krasnosel’skiĭ’s fixed point theorem. An example illustrating the main result is also constructed.

Existence Results for ψ -Hilfer Fractional Integro-Differential Hybrid Boundary Value Problems for Differential Equations and Inclusions

In this research work, we study a new class of ψ -Hilfer hybrid fractional integro-differential boundary value problems with nonlocal boundary conditions. Existence results are established for single and multivalued cases, by using suitable fixed-point theorems for the product of two single or multivalued operators. Examples illustrating the main results are also constructed.

ASYMPTOTIC EXPANSION OF SOLUTION OF BVP WITH INITIAL JUMPS FOR SINGULARLY PERTURBED INTEGRO-DIFFERENTIAL EQUATION

We consider the undivided boundary value problem for singularly perturbed linear integro-differential equations of th order with Fredholm’s integral terms have an initial jump of the phenomenon th order on the left end of the segment. Defined regular and boundary layer members of the asymptotic expansions of the solutions. Regular members of the asymptotics constructed in the form of integro-differential equations, which different from the usual unperturbed equations by the presence of additional term called initial jump of integral terms. The values of the initial jumps are defined. Boundary conditions for the regular members of the asymptotics also contain additional term, called an initial jump of the derivatives th order. Thus, for the determination of the regular members of the asymptotics obtained boundary value problems for linear integro-differential with additional parameter. To determine the boundary layer of the asymptotic expansion of solution we obtained initial problems for homogeneous an inhomogeneous ordinary differential equations with constant coefficients. Exponential estimates for boundary layer terms of the asymptotics are obtained. A theorem of existence and uniqueness and the asymptotic representations of solution with an estimate of the remainder term of the asymptotics. It is found that the constructed asymptotics approximation to the solution of the original singularly perturbed integro-differential boundary value problem is uniformly throughout the considered interval.

Some Existence and Dependence Criteria of Solutions to a Fractional Integro-Differential Boundary Value Problem via the Generalized Gronwall Inequality

The main intention of the present research study is focused on the analysis of a Caputo fractional integro-differential boundary problem (CFBVP) in which the right-hand side of supposed differential equation is represented as a sum of two nonlinear terms. Under the integro-derivative boundary conditions, we extract an equivalent integral equation and then define new operators based on it. With the help of three distinct fixed-point theorems attributed to Krasnosel’skiĭ, Leray–Schauder, and Banach, we investigate desired uniqueness and existence results. Additionally, the dependence criterion of solutions for this CFBVP is checked via the generalized version of the Gronwall inequality. Next, three simulative examples are designed to examine our findings based on the procedures applied in the theorems.

Uniform Approximation to the Solution of a Singularly Perturbed Boundary Value Problem with an Initial Jump

In this study, a third-order linear integro-differential equation with a small parameter at two higher derivatives was considered. An asymptotic expansion of the solution to the boundary value problem for the considered equation is constructed by considering the phenomenon of an initial jump of the second degree zeroth order on the left end of a given segment. The asymptotics of the solution has been sought in the form of a sum of the regular part and the part of the boundary layer. The terms of the regular part are defined as solutions of integro-differential boundary value problems, in which the equations and boundary conditions contain additional terms, called the initial jumps of the integral terms and solutions. Boundary layer terms are defined as solutions of third-order differential equations with initial conditions. A theorem on the existence, uniqueness, and asymptotic representation of a solution is presented along with an asymptotic estimate of the remainder term of the asymptotics. The purpose of this study is to construct a uniform asymptotic approximation to the solution to the original boundary value problem over the entire considered segment.

Thermal-induced snap-through buckling of simply-supported functionally graded beams

The instability of functionally graded material (FGM) structures is one of major threats to their service safety in widely engineering applications. This paper aims to clarify a long-standing controversy on the type of thermal instability of simply-supported FGM beams. Firstbased on the Euler-Bernoulli beam theory and von Karman geometric nonlinearitya nonlinear governing equation of simply-supported FGM beams under uniform thermal loads by Zhang’s two-variable method is formulated. Secondan approximate analytic solution to the nonlinear integro-differential boundary value problem with a thermal-induced inhomogeneous force boundary condition is obtained by using a semi-inverse method when the coordinate axis is relocated to the bending axis (physical neutral plane), and then the analytical predictions are verified by the differential quadrature method (DQM). Finallybased on the free energy theoremit is revealed that the symmetry breaking caused by the material inhomogeneity can make the simply-supported FGM beam under uniform thermal loads occur snap-through postbuckling only in odd modes; furthermorethe nonlinear critical load of thermal buckling varies non-monotonically with the functional gradient index due to the stretching-bending coupling effect. These results are expected to provide new ideas and references for the design and regulation of FGM structures.

On Caputo–Riemann–Liouville Type Fractional Integro-Differential Equations with Multi-Point Sub-Strip Boundary Conditions

In this paper, we derive existence and uniqueness results for a nonlinear Caputo–Riemann–Liouville type fractional integro-differential boundary value problem with multi-point sub-strip boundary conditions, via Banach and Krasnosel’skii⏝’s fixed point theorems. Examples are included for the illustration of the obtained results.

Solving Integro-Differential Boundary Value Problems Using Sinc-Derivative Collocation

In this paper, the sinc-derivative collocation approach is used to solve second order integro-differential boundary value problems. While the derivative of the unknown variables is interpolated using sinc numerical methods, the desired solution and the integral terms are evaluated through numerical integration and all higher order derivatives are approximated through successive numerical differentiation. Suitable transformations are used to reduce non-homogeneous boundary conditions to homogeneous. Comparison of the proposed method with different approaches that were previously considered in the literature is carried out in order to test its accuracy and efficiency. The results show that the sinc-derivative collocation method performs well.

Asymptotic Behavior of the Solutions of Boundary-Value Problems for Singularly Perturbed Integrodifferential Equations

We study the asymptotic behavior of the solutions of a boundary-value problem with boundary jumps for linear integrodifferential equations of the third order with small parameters at the two higher derivatives. The asymptotic convergence of the solution of a singularly perturbed integrodifferential boundary-value problem to the solution of the corresponding modified degenerate boundary-value problem is proved.

Multidimensional Manifold Continuation for Adaptive Boundary-Value Problems

Abstract Parameter continuation of finitely parameterized, approximate solutions to integro-differential boundary-value problems typically involves regular adaptive updates to the number and meaning of the unknowns and/or the associated constraints. Different continuation steps produce solutions with different discretizations or to formally different sets of equations. Existing general-purpose, multidimensional continuation algorithms fail to account for such differences without significant additional coding and are therefore prone to redundant coverage of the set of solutions. We describe a new algorithm, implemented in the software package coco, which overcomes this problem by characterizing the solution set in an invariant, finite dimensional, projected geometry rather than in the space of unknowns corresponding to any particular discretization. It is in this geometry that distances between solutions and angles between tangent spaces are quantified and used to construct possible directions of outward expansion. A pointwise lift identifies such directions in the projected geometry with directions of continuation in the full set of unknowns, used by a nonlinear predictor-corrector algorithm to expand into uncharted parts of the solution set. Several benchmark problems from the analysis of periodic orbits in autonomous dynamical systems are used to illustrate the theory.

Asymptotic Convergence of the Solution of a Singularly Perturbed Integro-Differential Boundary Value Problem

In this study, the asymptotic behavior of the solutions to a boundary value problem for a third-order linear integro-differential equation with a small parameter at the two higher derivatives has been examined, under the condition that the roots of the additional characteristic equation are negative. Via the scheme of methods and algorithms pertaining to the qualitative study of singularly perturbed problems with initial jumps, a fundamental system of solutions, the Cauchy function, and the boundary functions of a homogeneous singularly perturbed differential equation are constructed. Analytical formulae for the solutions and asymptotic estimates of the singularly perturbed problem are obtained. Furthermore, a modified degenerate boundary value problem has been constructed, and it was stated that the solution of the original singularly perturbed boundary value problem tends to this modified problem’s solution.

NONSINGULAR INTEGRO-DIFFERENTIAL BOUNDARY VALUE PROBLEM NOT SOLVED WITH RESPECT TO THE DERIVATIVE

The study of the differential-algebraic boundary value problems was established in the papers of K. Weierstrass, M.M. Lusin and F.R. Gantmacher. Works of S. Campbell, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, M.O. Perestyuk, V.P. Yakovets, O.A. Boi- chuk, A. Ilchmann and T. Reis are devoted to the systematic study of differential-algebraic boundary value problems. At the same time, the study of differential-algebraic boundary-value problems is closely related to the study of linear boundary-value problems for ordinary di- fferential equations, initiated in the works of A. Poincare, A.M. Lyapunov, M.M. Krylov, N.N. Bogolyubov, I.G. Malkin, A.D. Myshkis, E.A. Grebenikov, Yu.A. Ryabov, Yu.A. Mitropolsky, I.T. Kiguradze, A.M. Samoilenko, M.O. Perestyuk and O.A. Boichuk. The study of the linear differential-algebraic boundary value problems is connected with numerous applications of corresponding mathematical models in the theory of nonlinear osci- llations, mechanics, biology, radio engineering, the theory of the motion stability. Thus, the actual problem is the transfer of the results obtained in the articles and monographs of S. Campbell, A.M. Samoilenko and O.A. Boichuk on the linear boundary value problems for the integro-differential boundary value problem not solved with respect to the derivative, in parti- cular, finding the necessary and sufficient conditions of the existence of the desired solutions of the linear integro-differential boundary value problem not solved with respect to the derivative. In this article we found the conditions of the existence and constructive scheme for finding the solutions of the linear Noetherian integro-differential boundary value problem not solved with respect to the derivative. The proposed scheme of the research of the nonlinear Noetherian integro-differential boundary value problem not solved with respect to the derivative in the critical case in this article can be transferred to the seminonlinear integro-differential boundary value problem not solved with respect to the derivative.

On some functionals of the first passage times in jump models of stochastic volatility

We compute some functionals related to the generalized joint Laplace transforms of the first times at which two-dimensional jump processes exit half strips. It is assumed that the state space components are driven by Cox processes with both independent and common (positive) exponential jump components. The method of proof is based on the solutions of the equivalent partial integro-differential boundary-value problems for the associated value functions. The results are illustrated on several two-dimensional jump models of stochastic volatility which are based on non-affine analogs of certain mean-reverting or diverting diffusion processes representing closed-form solutions of the appropriate stochastic differential equations.

Dynamical behavior of integro-differential boundary value problem arising in nano-structures via Cellular Nanoscale Network approach

Piezoelectrical solid with nano-holes under time-harmonic anti-plane load is studied in the paper. The model is defined by the system of two partial differential equations and the boundary conditions which define the generalized stress. We reduce the model into an integro-differential boundary value problem and construct Cellular Nanoscale Network (CNN) architecture. Dynamical behavior of the obtained CNN model is studied by means of describing function technique. The simulations are provided which illustrate the theoretical results. The results of this paper are applicable in the field of non-destructive testing and fracture mechanics of multi-functional materials and structural elements based on them.

Staged Construction of Adjoints for Constrained Optimization of Integro-Differential Boundary-Value Problems

The objectives of this paper are threefold. First, the paper defines a staged construction paradigm for the necessary conditions for locally optimal solutions to integro-differential boundary-value problems that mirrors the natural decomposition of the original constraints. Second, it details a library of realizations of the zero functions, monitor functions, and contributions to the adjoint equations associated with each stage of problem construction. Third, it illustrates the application of this framework to multisegment trajectory problems, for example, in the analysis of spacecraft optimal control problems. The proposed construction paradigm generalizes a formulation for the construction of integro-differential-algebraic continuation problems in the software package COCO to also include automatically generated adjoint terms. The implementation enables arbitrarily many levels of nested construction with larger problems assembled from smaller ones, and multiple instances of a problem class combined insi...