We introduce a quadrature scheme—QBKIX—for the ubiquitous highorder accurate evaluation of singular layer potentials associated with general elliptic PDEs, i.e., a scheme that yields high accuracy at all distances to the domain boundary as well as on the boundary itself. Relying solely on point evaluations of the underlying kernel, our scheme is essentially PDE-independent; in particular, no an...

We give a Taylor series expansion for the joint Laplace transform of stationary waiting times in open (max,+)-linear stochastic systems with Poisson input. Probabilistic expressions are derived for coefficients of all orders. Even though the computation of these coefficients can be hard for certain systems, it is sufficient to compute only a few coefficients to obtain good approximations (espec...

The mission time analysis of large dependable systems that can be described by Markov Reward Models (MRM) with phase type distributed impulse and constant rate rewards is considered in the paper. A single Laplace transform domain description of the distribution of completion time is provided through a new analysis approach. Based on this description an effective numerical method is introduced w...

It is known that the Laplace transform is frequently used to solve fractional-order differential equations. Unlike integer-order differential equations, fractional-order differential equations always lead to difficulties in calculating inversion of Laplace transforms. Motivated by finding an easy way to numerically solve the fractional-order differential equations, we investigated the validity ...

We give a unified proof of Hölder regularity weak solutions for mixed local and nonlocal \begin{document}$ p $\end{document}-Laplace type parabolic equations with the full range exponents id="M3">\begin{document}$ 1<p<\infty $\end{document}. Our is based on expansion posi...

Abstract We present a family of high-order trapezoidal rule-based quadratures for class singular integrals, where the integrand has point singularity. The part is expanded in Taylor series involving terms increasing smoothness. are based on rule, with quadrature weights Cartesian nodes close to singularity judiciously corrected expansion. High-order accuracy can be achieved by utilizing suffici...

A modification of Watson's lemma for Laplace transforms ∫0∞f(t)e−ztdt was introduced in [Nielsen, 1906], deriving a new asymptotic expansion large |z| with the extra property being convergent as well. Inspired that idea, this paper we derive expansions two-dimensional F(x,y):=∫0∞∫0∞f(t,s)e−xt−ysdtds |x| and |y| are also convergent. The F(x,y) accompanied by error bounds. Asymptotic some special...