Existence of Multiple Solutions for Second Order Boundary Value Problems

We give conditions on f involving pairs of lower and upper solutions which lead to the existence of at least three solutions of the two point boundary value problem y" + f(x, y, y') = 0, x epsilon [0, 1], y(0) = 0 = y(1). In the special case f(x, y, y') = f(y) greater than or equal to 0 we give growth conditions on f and apply our general result to show the existence of three positive solutions. We give an example showing this latter result is sharp. Our results extend those of Avery and of Lakshmikantham et al.

Three-point boundary value problems for second-order, ordinary, differential equations

We establish existence results for solutions to three-point boundary value problems for nonlinear, second-order, ordinary differential equations with nonlinear boundary conditions. (C) 2001 Elsevier Science Ltd. All rights reserved.

Topological methods for some boundary value problems

We establish existence of solutions for a finite difference approximation to y = f(x, y, y ') on [0, 1], subject to nonlinear two-point Sturm-Liouville boundary conditions of the form g(i)(y(i),y ' (i)) = 0, i = 0, 1, assuming S satisfies one-sided growth bounds with respect to y '. (C) 2001 Elsevier Science Ltd. All rights reserved.

Solution techniques for transport problems involving steep concentration gradients: application to noncatalytic fluid solid reactions

Some efficient solution techniques for solving models of noncatalytic gas-solid and fluid-solid reactions are presented. These models include those with non-constant diffusivities for which the formulation reduces to that of a convection-diffusion problem. A singular perturbation problem results for such models in the presence of a large Thiele modulus, for which the classical numerical methods can present difficulties. For the convection-diffusion like case, the time-dependent partial differential equations are transformed by a semi-discrete Petrov-Galerkin finite element method into a system of ordinary differential equations of the initial-value type that can be readily solved. In the presence of a constant diffusivity, in slab geometry the convection-like terms are absent, and the combination of a fitted mesh finite difference method with a predictor-corrector method is used to solve the problem. Both the methods are found to converge, and general reaction rate forms can be treated. These methods are simple and highly efficient for arbitrary particle geometry and parameters, including a large Thiele modulus. (C) 2001 Elsevier Science Ltd. All rights reserved.

Boundary value problems for systems of difference equations associated with systems of second-order ordinary differential equations

We study difference equations which arise as discrete approximations to two-point boundary value problems for systems of second-order ordinary differential equations. We formulate conditions which guarantee a priori bounds on first differences of solutions to the discretized problem. We establish existence results for solutions to the discretized boundary value problems subject to nonlinear boundary conditions. We apply our results to show that solutions to the discrete problem converge to solutions of the continuous problem in an aggregate sense. (C) 2002 Elsevier Science Ltd. All rights reserved.

Existence of multiple solutions for second-order discrete boundary value problems

We give conditions on f involving pairs of discrete lower and discrete upper solutions which lead to the existence of at least three solutions of the discrete two-point boundary value problem yk+1 - 2yk + yk-1 + f (k, yk, vk) = 0, for k = 1,..., n - 1, y0 = 0 = yn,, where f is continuous and vk = yk - yk-1, for k = 1,..., n. In the special case f (k, t, p) = f (t) greater than or equal to 0, we give growth conditions on f and apply our general result to show the existence of three positive solutions. We give an example showing this latter result is sharp. Our results extend those of Avery and Peterson and are in the spirit of our results for the continuous analogue. (C) 2002 Elsevier Science Ltd. All rights reserved.

On the existence of solutions to boundary value problems on time scales

This work formulates existence theorems for solutions to two-point boundary value problems on time scales. The methods used include maximum principles, a priori bounds and topological degree theory.

Solvability of singular second order m-point boundary value problems of Dirichlet type

Let f : [0, 1] x R2 -> R be a function satisfying the Caxatheodory conditions and t(1 - t)e(t) epsilon L-1 (0, 1). Let a(i) epsilon R and xi(i) (0, 1) for i = 1,..., m - 2 where 0 < xi(1) < xi(2) < (...) < xi(m-2) < 1 - In this paper we study the existence of C[0, 1] solutions for the m-point boundary value problem [GRAPHICS] The proof of our main result is based on the Leray-Schauder continuation theorem.

Parameter setting and statistical learning

Three main models of parameter setting have been proposed: the Variational model proposed by Yang (2002; 2004), the Structured Acquisition model endorsed by Baker (2001; 2005), and the Very Early Parameter Setting (VEPS) model advanced by Wexler (1998). The VEPS model contends that parameters are set early. The Variational model supposes that children employ statistical learning mechanisms to decide among competing parameter values, so this model anticipates delays in parameter setting when critical input is sparse, and gradual setting of parameters. On the Structured Acquisition model, delays occur because parameters form a hierarchy, with higher-level parameters set before lower-level parameters. Assuming that children freely choose the initial value, children sometimes will miss-set parameters. However when that happens, the input is expected to trigger a precipitous rise in one parameter value and a corresponding decline in the other value. We will point to the kind of child language data that is needed in order to adjudicate among these competing models.

Repeatability of surface EMG variables in the sternocleidomastoid and anterior scalene muscles

In this study we examined the repeatability and reliability of the surface electromyographic (sEMG) signal mean frequency (MNF), average rectified value (ARV) and conduction velocity (CV) measured for the sternocleidomastoid (SCM) and the anterior scalene (AS) muscles in nine healthy volunteers during 15-s isometric cervical flexion contractions at 50% of the maximal voluntary contraction level over 3 non-consecutive days. Repeatability and reliability estimates were obtained for the initial values and rates of change of each sEMG variable by using both the Intraclass Correlation Coefficient (ICC) and the normalised standard error of the mean (nSEM). Results from SCM indicated good levels of repeatability for the initial value and slope of ARV (ICC > 65%). For the AS, high levels of repeatability were identified for the initial value of MNF (ICC > 70%) and the slope of ARV (ICC > 75%). Values of nSEM in the range 2.8-7.2% were obtained for the initial values of MNF and CV for both SCM and AS, indicating clinically acceptable measurement precision. The low value obtained for the nSEM of the initial value of MNF for the AS, in combination with the high ICC, indicates that of all of the variables examined, this variable could offer the best normative index to distinguish between subjects with and without neck pain, and represents the sEMG variable of choice for future evaluation purposes.

Predictor-corrector methods of Runge-Kutta type for stochastic differential equations

In this paper we construct predictor-corrector (PC) methods based on the trivial predictor and stochastic implicit Runge-Kutta (RK) correctors for solving stochastic differential equations. Using the colored rooted tree theory and stochastic B-series, the order condition theorem is derived for constructing stochastic RK methods based on PC implementations. We also present detailed order conditions of the PC methods using stochastic implicit RK correctors with strong global order 1.0 and 1.5. A two-stage implicit RK method with strong global order 1.0 and a four-stage implicit RK method with strong global order 1.5 used as the correctors are constructed in this paper. The mean-square stability properties and numerical results of the PC methods based on these two implicit RK correctors are reported.

Difference equations in Banach spaces

Difference equations which discretely approximate boundary value problems for second-order ordinary differential equations are analysed. It is well known that the existence of solutions to the continuous problem does not necessarily imply existence of solutions to the discrete problem and, even if solutions to the discrete problem are guaranteed, they may be unrelated and inapplicable to the continuous problem. Analogues to theorems for the continuous problem regarding a priori bounds and existence of solutions are formulated for the discrete problem. Solutions to the discrete problem are shown to converge to solutions of the continuous problem in an aggregate sense. An example which arises in the study of the finite deflections of an elastic string under a transverse load is investigated. The earlier results are applied to show the existence of a solution; the sufficient estimates on the step size are presented. (C) 2003 Elsevier Science Ltd. All rights reserved.

Myoelectric manifestations of sternocleidomastoid and anterior scalene muscle fatigue in chronic neck pain patients

Objective: This study compares myoelectric manifestations of fatigue of the sternocleidomastoid (SCM) and anterior scalene (AS) muscles between 10 chronic neck pain subjects and 10 normal matched controls. Methods: Surface electromyography (sEMG) signals were recorded from the sternal bead of SCM and AS muscles bilaterally during submaximal isometric cervical flexion contractions at 25 and 50% of the maximum voluntary contraction (MVC). The mean frequency, average rectified value and conduction velocity of the sEMG signal were calculated to quantify myoelectric manifestations of muscle fatigue. Results: For both the SCM and AS muscles, the Mann-Whitney U test indicated that the initial value and slope of the mean frequency in neck pain patients were greater than in healthy subjects (P < 0.05). This was significant both at 25 and 50% of MVC. Conclusions: These results suggest: (a) a predominance of type-II fibres in the neck pain patients and/or (b) greater fatigability of the superficial cervical flexors in neck pain patients. These results are in agreement with previous muscle biopsy studies in subjects with neck pain, which identified transformation of slow-twitch type-I fibres to fast-twitch type-IIB fibres, as well as the clinical observation of reduced endurance in the cervical flexors in neck pain patients. (C) 2003 Elsevier Science Ireland Ltd. All rights reserved.

A new analytical solution for water table fluctuations in coastal aquifers with sloping beaches

The Boussinesq equation appears as the zeroth-order term in the shallow water flow expansion of the non-linear equation describing the flow of fluid in an unconfined aquifer. One-dimensional models based on the Boussinesq equation have been used to analyse tide-induced water table fluctuations in coastal aquifers. Previous analytical solutions for a sloping beach are based on the perturbation parameter, epsilon(N) = alphaepsilon cot beta (in which beta is the beach slope, alpha is the amplitude parameter and epsilon is the shallow water parameter) and are limited to tan(-1) (alphaepsilon) much less than beta less than or equal to pi/2. In this paper, a new higher-order solution to the non-linear boundary value problem is derived. The results demonstrate the significant influence of the higher-order components and beach slope on the water table fluctuations. The relative difference between the linear solution and the present solution increases as 6 and a increase, and reaches 7% of the linear solution. (C) 2003 Elsevier Ltd. All rights reserved.

Neck flexor muscle fatigue is side specific in patients with unilateral neck pain

Despite the evidence of greater fatigability of the cervical flexor muscles in neck pain patients, the effect of unilaterality of neck pain on muscle fatigue has not been investigated. This study compared myoelectric manifestations of sternocleidomastoid (SCM) and anterior scalene (AS) muscle fatigue between the painful and non-painful sides in patients with chronic unilateral neck pain. Myoelectric signals were recorded from the sternal head of SCM and the AS muscles bilaterally during sub-maximal isometric cervical flexion contractions at 25% and 50% of the maximum voluntary contraction (MVC). The time course of the mean power frequency, average rectified value and conduction velocity of the electromyographic signals were calculated to quantify myoelectric manifestations of muscle fatigue. Results revealed greater estimates of the initial value and slope of the mean frequency for both the SCM and AS muscles on the side of the patient's neck pain at 25% and 50% of MVC. These results indicate greater myoelectric manifestations of muscle fatigue of the superficial cervical flexor muscles ipsilateral to the side of pain. This suggests a specificity of the effect of pain on muscle function and hence the need for specificity of therapeutic exercise in the management of neck pain patients. (C) 2003 European Federation of Chapters of the International Association for the Study of Pain. Published by Elsevier Ltd. All rights reserved.