First-order Boundary Value Problems Research Articles

Relation-Theoretic Nonlinear Almost Contractions with an Application to Boundary Value Problems

This article investigates certain fixed-point results enjoying nonlinear almost contraction conditions in the setup of relational metric space. Some examples are constructed in order to indicate the profitability of our results. As a practical use of our findings, we demonstrate the existence of a unique solution to a specific first-order boundary value problem.

Nonlinear contractions on directed graphs with applications to boundary value problems

<abstract><p>This article contains some outcomes on fixed points for a graph preserving nonlinear contraction in a metric space endued with a transitive directed graph. Our results improved, enriched, and subsumed various known fixed point theorems. To argue for reliability of our results, we presented two examples. We concluded the manuscript to investigate a unique solution of a certain first-order boundary value problem by means of our results.</p></abstract>

Boyd-Wong type functional contractions under locally transitive binary relation with applications to boundary value problems

<abstract><p>The area of metric fixed point theory applied to relational metric spaces has received significant attention since the appearance of the relation-theoretic contraction principle. In recent times, a number of fixed point theorems addressing the various contractivity conditions in the relational metric space has been investigated. Such results are extremely advantageous in solving a variety of boundary value problems, matrix equations, and integral equations. This article offerred some fixed point results for a functional contractive mapping depending on a control function due to Boyd and Wong in a metric space endued with a local class of transitive relations. Our findings improved, developed, enhanced, combined and strengthened several fixed point theorems found in the literature. Several illustrative examples were delivered to argue for the reliability of our findings. To verify the relevance of our findings, we conveyed an existence and uniqueness theorem regarding the solution of a first-order boundary value problem.</p></abstract>

Traveling waves for a Fisher-type reaction–diffusion equation with a flux in divergence form

Analysis of the speed of propagation in parabolic operators is frequently carried out considering the minimal speed at which its traveling waves (TWs) move. This value depends on the solution concept being considered. We analyze an extensive class of Fisher-type reaction–diffusion equations with flows in divergence form. We work with regular flows, which may not meet the standard elliptical conditions, but without other types of singularities. We show that the range of speeds at which classic TWs move is an interval unbounded to the right. Contrary to classic examples, the infimum may not be reached. When the flow is elliptic or over-elliptic, the minimum speed of propagation is achieved. The classic TW speed threshold is complemented by another value by analyzing an extension of the first-order boundary value problem to which the classic case is reduced. This singular minimum speed can be justified as a viscous limit of classic minimal speeds in elliptic or over-elliptic flows. We construct a singular profile for each speed between the minimum singular speed and the speeds at which classic TWs move. Under additional assumptions, the constructed profile can be justified as that of a TW of the starting equation in the framework of bounded variation functions. We also show that saturated fronts verifying the Rankine–Hugoniot condition can appear for strictly lower speeds even in the framework of bounded variation functions.

A Fast Eulerian Approach for Computation of Global Isochrons in High Dimensions

We present a novel Eulerian numerical method to compute global isochrons of a stable periodic orbit in high dimensions. Our approach is to formulate the asymptotic phase as a solution to a first-order boundary value problem and solve the resulting Hamilton--Jacobi equation with the parallel fast sweeping method. All isochrons are then given as isocontours of the phase. We apply this method to the Hodgkin--Huxley equations and a model of a dopaminergic neuron which exhibits mixed mode oscillations. Our results show that this Eulerian scheme is an efficient, accurate method for computing the asymptotic phase of a periodic dynamical system. Furthermore, by computing the phase on a Cartesian grid, it is simple to compute the gradient of phase, and thus compute an “almost phaseless” target set for the purposes of desynchronization of a system of oscillators.

Boundary Value Problems for First-Order Impulsive Functionalq-Integrodifference Equations

We discuss the existence and uniqueness of solutions for a first-order boundary value problem for impulsive functionalqk-integrodifference equations. The main results are obtained with the aid of some classical fixed point theorems. Illustrative examples are also presented.

Hydrodynamics of a Free Floating Vertical Axisymmetric Oscillating Water Column Device

This paper aims at presenting a general formulation of the hydrodynamic problem of a floating or restrained oscillating water column device. Three types of first‐order boundary value problems are investigated in order to calculate the velocity potential of the flow field around the device. The horizontal and vertical exciting wave forces, the rolling moment, the hydrodynamic parameters, the volume flows, and the drift forces are obtained in order to find the loads on the structure. The efficiency rate of the device is calculated in connection with the absorbed power and the capture length of energy absorption. Finally, the resulting wave motion inside and outside the device and the inner air pressure are examined.

Almost automorphic mild solutions to some partial neutral functional-differential equations and applications

The paper considers the existence and uniqueness of almost automorphic mild solutions to some classes of first-order partial neutral functional-differential equations. Sufficient conditions for the existence and uniqueness of almost automorphic mild solutions to the above-mentioned equations are obtained. As an application, a first-order boundary value problem arising in control systems is considered.

Twin positive solutions of nonlinear first-order boundary value problems on time scales

In this paper we consider the following nonlinear first-order boundary value problem on time scale T : { x Δ ( t ) = f ( x ( σ ( t ) ) ) , t ∈ [ 0 , T ] T , x ( 0 ) = β x ( σ ( T ) ) , where 0 < β < 1 . Some existence criteria for at least two positive solutions are established by using the well-known twin-fixed-point theorem due to Avery and Henderson.

Some observations and remarks on differential operators generated by first-order boundary value problems

This paper deals with the study of the set of all self-adjoint differential operators which are generated from first-order, linear, ordinary boundary value problems. These operators are defined on a weighted Hilbert function space and are examined as an application of the result obtained by Everitt and Markus in their paper in 1997. An investigation is given so that first-order self-adjoint boundary value problems are transformed to a study of the nature of the spectrum of associated self-adjoint operators. However, the analysis of this paper is restricted to consideration of conditions under which the spectral properties of these operators yield a discrete spectrum, and consequently to the determination of conditions under which the construction of Kramer analytic kernels, from the above boundary value problems, can be accomplished.

Kramer analytic kernels and first-order boundary value problems

This paper is concerned with the generation of Kramer analytic kernels from first-order, linear, ordinary boundary-value problems. These kernels are obtained from boundary-value problems that are represented by self-adjoint differential operators. Necessary and sufficient conditions are given to ensure that these differential operators have a discrete spectrum which then allows of the introduction of the associated Kramer analytic kernel. An example is considered which leads to the important Shannon–Whittaker interpolation expansion theorem.

A first-order boundary value problem with boundary condition on a countable set of points

LetE={E n } be the family of subspaces spanning the eigenfunctions and adjoint functions of the boundary-value problem $$ - i\frac{{dy}}{{dx}} = \lambda y, - \alpha \leqslant x \leqslant \alpha , U(y) \equiv \int_{ - a}^a {y(t)} d\sigma (t) = 0$$ that correspond to “close” eigenvalues (in the sense of the distance defined as the maximal of the Euclidean and the hyperbolic metrics). For a purely discrete measuredσ, it is shown that the systemE does not form an unconditional basis of subspaces inL2(−a, a) if at least one of the end points ±a is mass-free.

Convergence of Upwind Schemes for a Stationary Shock

A nonlinear first-order boundary value problem with discontinuous solutions is considered. It arises in the study of gasflow through a duct and allows, in general, for multiple solutions. New convergence results for three difference schemes are presented and the sharpness of numerical layers is established. For the EO-scheme, stability of a physically correct solution with respect to time evolution is shown.

Convergence of upwind schemes for a stationary shock

A nonlinear first-order boundary value problem with discontinuous solutions is considered. It arises in the study of gasflow through a duct and allows, in general, for multiple solutions. New convergence results for three difference schemes are presented and the sharpness of numerical layers is established. For the EO-scheme, stability of a physically correct solution with respect to time evolution is shown.

On a first-order boundary value problem from laminar flame theory

On a first-order boundary value problem from laminar flame theory