Infinite Boundary Value Problems Research Articles

Bifurcation analysis of piecewise-smooth engineering systems with delays through numeric continuation of periodic orbits

This paper presents a numeric continuation framework for periodic orbits of piecewise-smooth and hybrid dynamical systems with fixed point delays. For the numeric solution of the corresponding infinite dimensional multi-point boundary value problem, a novel discretization and interpolation scheme is developed employing Chebyshev polynomial based spectral collocation techniques. The same approach is employed for the formulation of the corresponding monodromy matrix enabling stability analysis on the found periodic orbits. Special care is attributed to the accurate detection of discontinuity induced bifurcations such as grazing and sliding, and the implemented pseudo-arclength framework is adapted to allow two parameter continuation of these critical points. The capabilities of the developed algorithms are demonstrated on a set of delayed piecewise-smooth and hybrid dynamical systems, showcasing potential engineering applications from the fields of control theory, traffic dynamics modelling, and machine tool vibrations. Finally, a detailed tutorial is attached in the appendix to accompany the open-source release of the developed codebase.

A New Semi Analytical Method for Solving Some Non-Linear Infinite Boundary Value Problems in Physical Sciences

A New Semi Analytical Method for Solving Some Non-Linear Infinite Boundary Value Problems in Physical Sciences

On Monge-Ampère equations with nonlinear gradient terms – Infinite boundary value problems

The paper investigates the existence, asymptotic boundary behavior and uniqueness of convex solutions to the Monge-Ampère equation that take infinite values on the boundary of smooth and bounded open convex sets in R n .

Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem

<p style='text-indent:20px;'>We prove gradient boundary blow up rates for ergodic functions in bounded domains related to fully nonlinear degenerate/singular elliptic operators. As a consequence, we deduce the uniqueness, up to constants, of the ergodic functions. The results are obtained by means of a Liouville type classification theorem in half-spaces for infinite boundary value problems related to fully nonlinear, uniformly elliptic operators.

Positive Solution for the Integral and Infinite Point Boundary Value Problem for Fractional-Order Differential Equation Involving a Generalized ϕ-Laplacian Operator

In this paper, we establish the existence of nontrivial positive solution to the following integral-infinite point boundary-value problem involving ϕ-Laplacian operator D0+αϕx,D0+βux+fx,ux=0,x∈0,1,D0+σu0=D0+βu0=0,u1=∫01 gtutdt+∑n=1n=+∞ αnuηn, where ϕ:0,1×R→R is a continuous function and D0+p is the Riemann-Liouville derivative for p∈α,β,σ. By using some properties of fixed point index, we obtain the existence results and give an example at last.

Existence of solutions for fractional differential equations with infinite point boundary conditions at resonance

In this paper, by using Mawhin’s continuation theorem, we establish some sufficient conditions for the existence of at least one solution for a class of fractional infinite point boundary value problem at resonance. Moreover, an example is given to illustrate our results.

Large solutions to non-divergence structure semilinear elliptic equations with inhomogeneous term

AbstractMotivated by the work [9], in this paper we investigate the infinite boundary value problem associated with the semilinear PDE{Lu=f(u)+h(x)}on bounded smooth domains{\Omega\subseteq\mathbb{R}^{n}}, whereLis a non-divergence structure uniformly elliptic operator with singular lower-order terms. In the equation,fis a continuous non-decreasing function that satisfies the Keller–Osserman condition, whilehis a continuous function in Ω that may change sign, and which may be unbounded on Ω. Our purpose is two-fold. First we study some sufficient conditions onfandhthat would ensure existence of boundary blow-up solutions of the above equation, in which we allow the lower-order coefficients to be singular on the boundary. The second objective is to provide sufficient conditions onfandhfor the uniqueness of boundary blow-up solutions. However, to obtain uniqueness, we need the lower-order coefficients ofLto be bounded in Ω, but we still allowhto be unbounded on Ω.

Large solutions for non-divergence structure equations with singular lower order terms

In this paper we investigate infinite boundary value problems associated with the semi-linear PDE Lu=k(x)f(u) on a bounded smooth domain Ω⊂Rn, where L is a non-divergence structure, uniformly elliptic operator with singular lower order terms. The weight k is a continuous non-negative function and f is a continuous nondecreasing function that satisfies the Keller–Osserman condition. We study a sufficient condition on k that ensures existence of a large solution u. In case the lower order terms of L are bounded, under further assumptions on f and k we establish asymptotic bounds of solutions u near the boundary ∂Ω and, as a consequence a uniqueness result.

Abstract elliptic equations with integral boundary conditons

This paper focuses on nonlocal integral boundary value problems for elliptic differential-operator equations. Here given conditions guarantee that maximal regularity and Fredholmness in Lp spaces. These results are applied to the Cauchy problem for abstract parabolic equations, its infinite systems and boundary value problems for anisotropic partial differential equations in mixed Lp norm

Existence of two positive solutions for a class of third-order impulsive singular integro-differential equations on the half-line in Banach spaces

In this paper, we discuss the existence of two positive solutions for an infinite boundary value problem of third-order impulsive singular integro-differential equations on the half-line in Banach spaces by means of the fixed point theorem of cone expansion and compression with norm type.

尤拉方程在半无界区域的边值问题的基本解

本文把永久美式期权确定最佳实施边界的问题归结为尤拉方程在半无界区域的边值问题的基本解来研究。获得了基本解的表达式，同时确定了基本解的奇异点。证明了基本解在半无界区域连续，但解的导数在奇异点发生间断，在奇异点处基本解取最大值。基本解的奇异点就是永久美式期权最佳实施边界点。 In this paper, the permanent American options determine optimal exercise boundary problem is boiling down to the Euler equation in a semi infinite domain boundary value problem of the basic solution to study. We obtained the expression of the basic solution and the singular point of the basic solution. It is proved that the basic solution is continuous in the semi infinite domain, but the derivative of the solution is discontinuous at the singular point. The maximum value is obtained in the singular point. The singular point of the basic solution is the best implementation point of the permanent American option.

Existence of two positive solutions for a class of second order impulsive singular integro-differential equations on the half line

In this paper, the author discusses the existence of two positive solutions for an infinite boundary value problem of second order impulsive singular integro-differential equations on the half line by means of the fixed point theorem of cone expansion and compression with norm type.

Multiple positive solutions for first-order impulsive singular integro-differential equations on the half line in a Banach space

In this paper, the author discusses the multiple positive solutions for an infinite three-point boundary value problem of first-order impulsive superlinear singular integro-differential equations on the half line in a Banach space by means of the fixed-point theorem of cone expansion and compression with norm type. MSC: 45J05; 34G20; 47H10

Multiple Positive Solutions for First Order Impulsive Singular Integro-Differential Equations on the Half Line

In this paper, the author discusses the multiple positive solutions for an infinite boundary value problem of first order impulsive singular integro-differential equations on the half line by means of the fixed point theorem of cone expansion and compression with norm type.

Positive solutions of an infinite boundary value problem for nth-order nonlinear impulsive singular integro-differential equations in Banach spaces

In this paper, the cone theory and Mönch fixed point theorem combined with a monotone iterative technique are used to investigate the positive solutions of a class of boundary problems for nth-order nonlinear impulsive singular integro-differential equations of mixed type on an infinite interval in Banach spaces. The conditions for the existence of a positive solution are established. In addition, an explicit iterative approximation of the solution for the boundary value problem is derived.

Fully computable robust a posteriori error bounds for singularly perturbed reaction–diffusion problems

A procedure for the construction of robust, upper bounds for the error in the finite element approximation of singularly perturbed reaction–diffusion problems was presented in Ainsworth and Babuska (SIAM J Numer Anal 36(2):331–353, 1999) which entailed the solution of an infinite dimensional local boundary value problem. It is not possible to solve this problem exactly and this fact was recognised in the above work where it was indicated that the limitation would be addressed in a subsequent article. We view the present work as fulfilling that promise and as completing the investigation begun in Ainsworth and Babuska (SIAM J Numer Anal 36(2):331–353, 1999) by removing the obligation to solve a local problem exactly. The resulting new estimator is indeed fully computable and the first to provide fully computable, robust upper bounds in the setting of singularly perturbed problems discretised by the finite element method.

Multiple positive solutions for first order impulsive superlinear integro-differential equations on the half line

In this paper, the author discusses the multiple positive solutions for an infinite boundary value problem of first order impulsive superlinear integro-differential equations on the half line by means of the fixed point theorem of cone expansion and compression with norm type.

Multiple Positive Solutions for First‐Order Impulsive Integrodifferential Equations on the Half Line in Banach Spaces

The author discusses the multiple positive solutions for an infinite boundary value problem of first‐order impulsive superlinear integrodifferential equations on the half line in a Banach space by means of the fixed point theorem of cone expansion and compression with norm type.

Optimal Control of Linear Impulsive Antiperiodic Boundary Value Problem on Infinite Dimensional Spaces

A class of optimal control problems for infinite dimensional impulsive antiperiodic boundary value problem is considered. Using exponential stabilizability and discussing the impulsive evolution operators, without compactness and exponential stability of the semigroup governed by original principle operator, we present the existence of optimal controls. At last, an example is given for demonstration.

Infinite Boundary Value Problems for Second‐Order Nonlinear Impulsive Differential Equations with Supremum

We investigate the infinite boundary value problems for second‐order impulsive differential equations with supremum by establishing a new comparison result and using the lower and upper solution method, and obtain the existence results for their maximal and minimal solutions.