General quadrature formulae based on the weighted Montgomery identity and related inequalities

In this paper two families of general two-point and closed four-point weighted quadrature formulae are established. Obtained formulae are used to present several Hadamard type and Ostrowski type inequalities for alpha-L-Holder functions. These results are applied to establish error estimates for the Gauss-Chebyshev quadratures

NEW DOCTORAL DEGREE Montgomery identity, quadrature formulae and derived inequalities

The aim of this PhD dissertation is to give generalizations of classical quadrature formulae with two, three and four nodes using some generalizations of the weighted Montgomery identity. Thereby families of weighted and non-weighted quadrature formulae are considered, some error estimates are derived, and sharp and the best possible inequalities as well as Ostrowski type inequalities are proved. Classes of weighted and non-weighted two-point quadrature formulae are studied and corresponding error estimates are calculated. Two-point Gauss-Chebyshev formulae of the first and of the second kind as well as genera-lizations of the trapezoidal formula, Newton-Cotes two-point formula, Maclaurin two-point formula and midpoint formula are obtained as special cases of these formulae. The dissertation deals with three-point quadrature formulae, generalizations of Simpson\u27s, dual Simpson\u27s and Maclaurin\u27s formula, three-point Gauss-Chebyshev formulae of the first kind and of the second kind that follow from a general formula, as well as corresponding error estimates. It is also dedicated to closed four-point quadrature formulae from which a we-ight-ed and non-weighted generalization of Bullen type inequalities for $(2n)-$ convex functions is obtained. As a special case, Simpson\u27s $3/8$ formula and Lobatto four-point formula with related inequalities are considered. Weighted Euler type identities, which represent weighted integral one-point formulae, are worked out in the dissertation as well. By means of these identities, generalized weighted quadrature formulae are derived in which the integral is estimated by function values in $n$ nodes and generalizations of Gauss-Chebyshev formulae of the first and of the second kind are given. Error estimates are derived and some sharp and best possible inequalities are proved for all given formulae

Asymptotic expansions and fast computation of oscillatory Hilbert transforms

In this paper, we study the asymptotics and fast computation of the one-sided oscillatory Hilbert transforms of the form $$H^{+}(f(t)e^{i\omega t})(x)=-int_{0}^{\infty}e^{i\omega t}\frac{f(t)}{t-x}dt,\qquad \omega>0,\qquad x\geq 0,$$ where the bar indicates the Cauchy principal value and $f$ is a real-valued function with analytic continuation in the first quadrant, except possibly a branch point of algebraic type at the origin. When $x=0$, the integral is interpreted as a Hadamard finite-part integral, provided it is divergent. Asymptotic expansions in inverse powers of $\omega$ are derived for each fixed $x\geq 0$, which clarify the large $\omega$ behavior of this transform. We then present efficient and affordable approaches for numerical evaluation of such oscillatory transforms. Depending on the position of $x$, we classify our discussion into three regimes, namely, $x=\mathcal{O}(1)$ or $x\gg1$, $0<x\ll 1$ and $x=0$. Numerical experiments show that the convergence of the proposed methods greatly improve when the frequency $\omega$ increases. Some extensions to oscillatory Hilbert transforms with Bessel oscillators are briefly discussed as well.Comment: 32 pages, 6 figures, 4 table

Spectral method for the unsteady incompressible Navier-Stokes equations in gauge formulation

A spectral method which uses a gauge method, as opposed to a projection method, to decouple the computation of velocity and pressure in the unsteady incompressible Navier-Stokes equations, is presented. Gauge methods decompose velocity into the sum of an auxilary field and the gradient of a gauge variable, which may, in principle, be assigned arbitrary boundary conditions, thus overcoming the issue of artificial pressure boundary conditions in projection methods. A lid-driven cavity flow is used as a test problem. A subtraction method is used to reduce the pollution effect of singularities at the top corners of the cavity. A Chebyshev spectral collocation method is used to discretize spatially. An exponential time differencing method is used to discretize temporally. Matrix diagonalization procedures are used to compute solutions directly and efficiently. Numerical results for the flow at Reynolds number Re = 1000 are presented, and compared to benchmark results. It is shown that the method, called the spectral gauge method, is straightforward to implement, and yields accurate solutions if Neumann boundary conditions are imposed on the gauge variable, but suffers from reduced convergence rates if Dirichlet boundary conditions are imposed on the gauge variable