q-Bernoulli polynomials and q-umbral calculus

In this paper, we investigate some properties of q-Bernoulli polynomials arising from q-umbral calculus. We find a formula for expressing any polynomial as a linear combination of q-Bernoulli polynomials with explicit coefficients. Also, we establish some connections between q-Bernoulli polynomials and higher-order q-Bernoulli polynomials.     

与广义$(p,q)$量子积分算子有关的调和单叶函数

利用广义的$(p,q)$量子积分算子, 本文引入了与Janowski函数相关的调和单叶函数的某些子类, 得到了这些函数类的充分系数条件, 极值点, 偏差估计及部分和等性质.     

On the Finite Increment Calculus method for stabilizing advection‐diffusion equations,analysis and computation of the stabilization parameter

In this work we are concerned with the finite increment calculus (FIC) method. The method has been developed for efficient approximation of advection‐diffusion equations with high Péclet numbers. Since the natural application of FIC is within the framework of the FEM, we consider the BVP in a weak sense on finite dimensional spaces. Here we provide a result on existence and uniqueness of the solution as well as an error analysis. Also we propose a choice of the stabilization parameter. We test the method on some troublesome 2D problems. Copyright © 2011 John Wiley & Sons, Ltd.     

A support learning programme for first-year mathematics*

ABSTRACT

This article is a follow-up to an earlier paper on the mathematics support learning tutorial programme (SLT), an intervention programme at The University of Queensland that targets students considered to be at risk of failing Calculus and Linear Algebra I, the first tertiary level mathematics subject at The University of Queensland. The first paper (Hillock, P., Jennings, M., Roberts, A., & Scharaschkin, V. (2013). Amathematics support programme for first-year engineering students. International Journal of Mathematical Education in Science and Technology, 44(7), 1030–1044) reported on the inaugural programme implemented in 2012. This article provides an update of the progress of the SLT since 2012. We provide statistics for the subsequent 12 semesters to Semester 2, 2018 and describe the evolution of the SLT since its implementation. Statistical analysis of the additional data and student feedback indicate that the SLT continues to have a positive impact on student learning, with weak students making significant gains from attending the programme.     

Paralinearization of the Dirichlet-Neumann operator and applications to progressive gravity waves

This note presents a paradifferential approach to the analysis of the water waves equations.     

An analysis of the sharpness of monotonicity results via homotopy for sequential fractional operators

We investigate the connection between the sign of $\left({\Delta }_{1-\mu +a}^{\nu }{\Delta }_a^{\mu }f\right)\left(t\right)$ and the monotone behavior of $t\mapsto f\left(t\right)$. In particular, given a function $f:{\mathbb{N}}_a\to \mathbb{R}$ we consider the homotopy $G_{\nu }:\mathbb{R}\times [0,1]\to \mathbb{R}$ defined by $G_{\nu }\left(f\left(a\right),\eta \right)≔\eta \cdot \frac{1}{2}(1-\nu )\left(\nu \right)f\left(a\right)$, and we analyze the effect of changing $\eta \in [0,1]$ in the condition $\left({\Delta }_{1-\mu +a}^{\nu }{\Delta }_a^{\mu }f\right)\left(t\right)\ge G_{\nu }\left(f\left(a\right),\eta \right)$. We show that as $\eta \to 1^{-}$ there is an increasingly strong connection between the sign of $\left({\Delta }_{1-\mu +a}^{\nu }{\Delta }_a^{\mu }f\right)\left(t\right)$ and the monotone behavior of $f$.     

Discussion on the Leibniz rule and Laplace transform of fractional derivatives using series representation

ABSTRACT

Taylor series is a useful mathematical tool when describing and constructing a function. With the series representation, some properties of fractional calculus can be revealed clearly. On this basis, the Lebiniz rule and Laplace transform of fractional calculus is investigated. It is analytically shown that the commonly used Leibniz rule cannot be applied for Caputo derivative. Similarly, the well-known Laplace transform of Riemann–Liouville derivative is doubtful for n-th continuously differentiable function. After pointing out such problems, the exact formula of Caputo Leibniz rule and the explanation of Riemann–Liouville Laplace transform are presented. Finally, three illustrative examples are revisited to confirm the obtained results.