Van der Pol's equation for a relaxation oscillator is generalized by the addition of terms to produce a pair of non-linear differential equations with either a stable singular point or a limit cycle. The resulting "BVP model" has two variables of state, representing excitability and refractoriness, and qualitatively resembles Bonhoeffer's theoretical model for the iron wire model of nerve. This...

ABSTRACT In this article, we propose the application of a modified Taylor series method (MTSM) for the approximation of nonlinear problems described on finite intervals. The issue of Taylor series method with mixed boundary conditions is circumvented using shooting constants and extra derivatives of the problem. In order to show the benefits of this proposal, three different kinds of problems a...

This article studies the existence of nonnegative solutions for a boundary value problem (BVP) of nonlinear fractional differential equations. Some new existence results are obtained by applying Kranoselskii‘s fixed–point theorem in a cone. First we prove the existence of solutions of an auxiliary BVP formulated by truncating the response function. Then the Arzela–Ascoli theorem is used to take...

The main aim of this paper is to construct quantum extension of the discrete Sturm–Liouville equation consisting of second-order difference equation and boundary conditions that depend on a quadratic eigenvalue parameter. We consider a boundary value problem (BVP) consisting of a second-order quantum difference equation and boundary conditions that depend on the quadratic eigenvalue parameter. ...

For a second order nonlinear ordinary differential equation (ODE), a singular boundary value problem (BVP) is investigated which arises in hydromechanics and nonlinear field theory when static centrally symmetric bubble type (droplet type) solutions are sought. Being defined on a semi infinite interval 0 < r <∞, this ODE, with a polynomial nonlinearity of the third order with respect to a desir...

This paper is concerned with the numerical solution of a system of ordinary di erential equations ODEs y Sy t f t y p on an inter val b subject to boundary conditions g y y b p The ODEs have a coe cient that is singular at t but it is assumed that the boundary value problem BVP has a smooth solution Some popular methods for BVPs evaluate the ODEs at t This paper deals with the practical issues ...

This study focuses on nonlocal boundary value problems (BVP) for linear and nonlinear elliptic differential-operator equations (DOE) that are defined in Banachvalued function spaces. The considered domain is a region with varying bound and depends on a certain parameter. Some conditions that guarantee the maximal Lpregularity and Fredholmness of linear BVP, uniformly with respect to this parame...

Nonlinear two-point boundary value problems (BVPs) may have none or more than one solution. For the singularly perturbed two-point BVP εu′′+2u′+f(u) = 0, 0 < x < 1, u(0) = 0, u(1) = 0, a condition is given to have one and only one solution; also cases of more solutions have been analyzed. After attention to the form and validity of the corresponding asymptotic expansions, partially based on slo...

This paper investigates the existence of positive solutions for a class of boundary value problems (BVP) of fractional impulsive differential equations and presents a number of new results. First, by constructing a novel transformation, the considered impulsive system is convert into a continuous system. Second, using a specially constructed cone, the Krein-Rutman theorem, topological degree th...