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.

Time trials on second-order and variable-learning-rate algorithms

The performance of seven minimization algorithms are compared on five neural network problems. These include a variable-step-size algorithm, conjugate gradient, and several methods with explicit analytic or numerical approximations to the Hessian.

Second-Order Risk Constraints

This paper discusses how numerically imprecise information can be modelled and how a risk evaluation process can be elaborated by integrating procedures for numerically imprecise probabilities and utilities. More recently, representations and methods for stating and analysing probabilities and values (utilities) with belief distributions over them (second order representations) have been suggested. In this paper, we are discussing some shortcomings in the use of the principle of maximising the expected utility and of utility theory in general, and offer remedies by the introduction of supplementary decision rules based on a concept of risk constraints taking advantage of second-order distributions.

Discontinuous Galerkin Methods for Advection-Diffusion-Reaction Problems on Anisotropically Refined Meshes

In this paper we consider the a posteriori and a priori error analysis of discontinuous Galerkin interior penalty methods for second-order partial differential equations with nonnegative characteristic form on anisotropically refined computational meshes. In particular, we discuss the question of error estimation for linear target functionals, such as the outflow flux and the local average of the solution. Based on our a posteriori error bound we design and implement the corresponding adaptive algorithm to ensure reliable and efficient control of the error in the prescribed functional to within a given tolerance. This involves exploiting both local isotropic and anisotropic mesh refinement. The theoretical results are illustrated by a series of numerical experiments.

Discontinuous Galerkin Methods on hp-Anisotropic Meshes I: A Priori Error Analysis

We consider the a priori error analysis of hp-version interior penalty discontinuous Galerkin methods for second-order partial differential equations with nonnegative characteristic form under weak assumptions on the mesh design and the local finite element spaces employed. In particular, we prove a priori hp-error bounds for linear target functionals of the solution, on (possibly) anisotropic computational meshes with anisotropic tensor-product polynomial basis functions. The theoretical results are illustrated by a numerical experiment.

Discontinuous Galerkin Methods on hp-Anisotropic Meshes II: A Posteriori Error Analysis and Adaptivity

We consider the a posteriori error analysis and hp-adaptation strategies for hp-version interior penalty discontinuous Galerkin methods for second-order partial differential equations with nonnegative characteristic form on anisotropically refined computational meshes with anisotropically enriched elemental polynomial degrees. In particular, we exploit duality based hp-error estimates for linear target functionals of the solution and design and implement the corresponding adaptive algorithms to ensure reliable and efficient control of the error in the prescribed functional to within a given tolerance. This involves exploiting both local isotropic and anisotropic mesh refinement and isotropic and anisotropic polynomial degree enrichment. The superiority of the proposed algorithm in comparison with standard hp-isotropic mesh refinement algorithms and an h-anisotropic/p-isotropic adaptive procedure is illustrated by a series of numerical experiments.

Existence of Homoclinic Solutions for Nonlinear Second-Order Problems

In this work, we consider the second-order discontinuous equation in the real line, u′′(t)−ku(t)=f(t,u(t),u′(t)),a.e.t∈R, with k>0 and f:R3→R an L1 -Carathéodory function. The existence of homoclinic solutions in presence of not necessarily ordered lower and upper solutions is proved, without periodicity assumptions or asymptotic conditions. Some applications to Duffing-like equations are presented in last section.

A second order control-volume finite-element least-squares strategy for simulating diffusion in strongly anisotropic media

An unstructured mesh �nite volume discretisation method for simulating di�usion in anisotropic media in two-dimensional space is discussed. This technique is considered as an extension of the fully implicit hybrid control-volume �nite-element method and it retains the local continuity of the ux at the control volume faces. A least squares function recon- struction technique together with a new ux decomposition strategy is used to obtain an accurate ux approximation at the control volume face, ensuring that the overall accuracy of the spatial discretisation maintains second order. This paper highlights that the new technique coincides with the traditional shape function technique when the correction term is neglected and that it signi�cantly increases the accuracy of the previous linear scheme on coarse meshes when applied to media that exhibit very strong to extreme anisotropy ratios. It is concluded that the method can be used on both regular and irregular meshes, and appears independent of the mesh quality.

A second-order accurate numerical approximation for the Riesz space fractional advection-dispersion equation

In this paper, we consider a space Riesz fractional advection-dispersion equation. The equation is obtained from the standard advection-diffusion equation by replacing the ¯rst-order and second-order space derivatives by the Riesz fractional derivatives of order β 1 Є (0; 1) and β2 Є(1; 2], respectively. Riesz fractional advection and dispersion terms are approximated by using two fractional centered difference schemes, respectively. A new weighted Riesz fractional ¯nite difference approximation scheme is proposed. When the weighting factor Ѳ = 1/2, a second- order accurate numerical approximation scheme for the Riesz fractional advection-dispersion equation is obtained. Stability, consistency and convergence of the numerical approximation scheme are discussed. A numerical example is given to show that the numerical results are in good agreement with our theoretical analysis.

Solution methods for advection-diffusion-reaction equations on growing domains and subdomains, with application to modelling skin substitutes

Problems involving the solution of advection-diffusion-reaction equations on domains and subdomains whose growth affects and is affected by these equations, commonly arise in developmental biology. Here, a mathematical framework for these situations, together with methods for obtaining spatio-temporal solutions and steady states of models built from this framework, is presented. The framework and methods are applied to a recently published model of epidermal skin substitutes. Despite the use of Eulerian schemes, excellent agreement is obtained between the numerical spatio-temporal, numerical steady state, and analytical solutions of the model.

Testing second order cyclostationarity in the squared envelope spectrum of non-white vibration signals

Cyclostationary models for the diagnostic signals measured on faulty rotating machineries have proved to be successful in many laboratory tests and industrial applications. The squared envelope spectrum has been pointed out as the most efficient indicator for the assessment of second order cyclostationary symptoms of damages, which are typical, for instance, of rolling element bearing faults. In an attempt to foster the spread of rotating machinery diagnostics, the current trend in the field is to reach higher levels of automation of the condition monitoring systems. For this purpose, statistical tests for the presence of cyclostationarity have been proposed during the last years. The statistical thresholds proposed in the past for the identification of cyclostationary components have been obtained under the hypothesis of having a white noise signal when the component is healthy. This need, coupled with the non-white nature of the real signals implies the necessity of pre-whitening or filtering the signal in optimal narrow-bands, increasing the complexity of the algorithm and the risk of losing diagnostic information or introducing biases on the result. In this paper, the authors introduce an original analytical derivation of the statistical tests for cyclostationarity in the squared envelope spectrum, dropping the hypothesis of white noise from the beginning. The effect of first order and second order cyclostationary components on the distribution of the squared envelope spectrum will be quantified and the effectiveness of the newly proposed threshold verified, providing a sound theoretical basis and a practical starting point for efficient automated diagnostics of machine components such as rolling element bearings. The analytical results will be verified by means of numerical simulations and by using experimental vibration data of rolling element bearings.

Second-order inelastic analysis of composite framed structures based on the refined plastic hinge method

Composite steel-concrete structures experience non-linear effects which arise from both instability-related geometric non-linearity and from material non-linearity in all of their component members. Because of this, conventional design procedures cannot capture the true behaviour of a composite frame throughout its full loading range, and so a procedure to account for those non-linearities is much needed. This paper therefore presents a numerical procedure capable of addressing geometric and material non-linearities at the strength limit state based on the refined plastic hinge method. Different material non-linearity for different composite structural components such as T-beams, concrete-filled tubular (CFT) and steel-encased reinforced concrete (SRC) sections can be treated using a routine numerical procedure for their section properties in this plastic hinge approach. Simple and conservative initial and full yield surfaces for general composite sections are proposed in this paper. The refined plastic hinge approach models springs at the ends of the element which are activated when the surface defining the interaction of bending and axial force at first yield is reached; a transition from the first yield interaction surface to the fully plastic interaction surface is postulated based on a proposed refined spring stiffness, which formulates the load-displacement relation for material non-linearity under the interaction of bending and axial actions. This produces a benign method for a beam-column composite element under general loading cases. Another main feature of this paper is that, for members containing a point of contraflexure, its location is determined with a simple application of the method herein and a node is then located at this position to reproduce the real flexural behaviour and associated material non-linearity of the member. Recourse is made to an updated Lagrangian formulation to consider geometric non-linear behaviour and to develop a non-linear solution strategy. The formulation with the refined plastic hinge approach is efficacious and robust, and so a full frame analysis incorporating geometric and material non-linearity is tractable. By way of contrast, the plastic zone approach possesses the drawback of strain-based procedures which rely on determining plastic zones within a cross-section and which require lengthwise integration. Following development of the theory, its application is illustrated with a number of varied examples.