The authors give convergence theorems for interpolatory product rules for evaluating Cauchy singular integrals and obtain asymptotic estimates of the remainder. Some results, previously established by other authors, are generalized and improved.

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In this paper, we are interested in the Cauchy problem for viscoelastic damped wave equation with memory of type I. By applying WKB analysis and Fourier analysis, explain memory’s influence on dissipative structures asymptotic profiles solutions to model additionally weighted L 1 initial data. Furthermore, concerning standard energy solution itself, establish singular limit between Moore–Gibson...

Abstract This paper studies the problem of construction of the optimal quadrature formula in the sense of Sard in (2) 2 ( 1,1) L − S.L.Sobolev space for approximate calculation of the Cauchy type singular integral. Using the discrete analogue of the operator 4 4 / d dx we obtain new optimal quadrature formulas. Furthermore, explicit formulas of the optimal coefficients are obtained. Finally, in...

In this paper, we use the no-response test idea, introduced in ([L-P], [P1]) for the inverse obstacle problem, to identify the interface of the discontinuity of the coefficient γ of the equation ∇ · γ(x)∇+ c(x) with piecewise regular γ and bounded function c(x). We use infinitely many Cauchy data as measurement and give a reconstructive method to localize the interface. We will base this multiw...

Every infinitely divisible law defines a convolution semigroup that solves an abstract Cauchy problem. In the fractional Cauchy problem, we replace the first order time derivative by a fractional derivative. Solutions to fractional Cauchy problems are obtained by subordinating the solution to the original Cauchy problem. Fractional Cauchy problems are useful in physics to model anomalous diffus...

We investigate the interior of regular axisymmetric and stationary black holes surrounded by matter and find that for non-vanishing angular momentum of the black hole the spacetime can always be extended regularly up to and including an inner Cauchy horizon. We provide an explicit relation for the regular metric at the inner Cauchy horizon in terms of that at the event horizon. As a consequence...

We use integrable systems techniques to study the singularities of timelike non-minimal constant mean curvature (CMC) surfaces in the LorentzMinkowski 3-space. The singularities arise at the boundary of the Birkhoff big cell of the loop group involved. We examine the behaviour of the surfaces at the big cell boundary, generalize the definition of CMC surfaces to include those with finite, gener...

We present new O(n) algorithms to compute very accurate SVDs of Cauchy matrices, Vandermonde matrices, and related \unit-displacement-rank" matrices. These algorithms compute all the singular values with guaranteed relative accuracy, independent of their dynamic range. In contrast, previous O(n) algorithms can potentially lose all relative accuracy in the tiniest singular values. LAPACK Working...