This paper extends the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou in a critical new way. We present, in particular, an algorithm, to obtain the support of the Riemann-Hilbert problem for leading asymptotics. Applying this extended method to small dispersion KdV (Korteweg-de Vries) equation, we (i) recover the variational formulation of P. D. Lax and C. D. ...

We construct a lattice kinetic scheme to study electronic flow in graphene. For this purpose, we first derive a basis of orthogonal polynomials, using as the weight function the ultrarelativistic Fermi-Dirac distribution at rest. Later, we use these polynomials to expand the respective distribution in a moving frame, for both cases, undoped and doped graphene. In order to discretize the Boltzma...

This paper considers the Riemann problem and an associated Godunov method for a model of compressible two-phase flow. The model is a reduced form of the wellknown Baer-Nunziato model which describes the behavior of granular explosives. In the analysis presented here, we focus on the effect of non-conservative nozzling terms in the two-phase model and omit source terms representing the exchange ...

In this paper, we study the existence and uniqueness of solutions for a coupled implicit system involving ψ-Riemann–Liouville fractional derivative with nonlocal conditions. We first transformed problem into an integral then analyzed by means Banach fixed-point theorem Krasnoselskiis theorem. Some known results in literature are extended. Finally, example is given to illustrate our theoretical ...

We present a finite volume method that is applicable to general hyperbolic PDEs, including non-conservative and spatially varying systems. The method can be extended to arbitrarily high order of accuracy and allows a well-balanced implementation for capturing solutions of balance laws near steady state. This well-balancing is achieved through the f -wave Riemann solver and a novel wave-slope WE...

We present a finite volume method that is applicable to hyperbolic PDEs including spatially varying and semilinear nonconservative systems. The spatial discretization, like that of the well-known Clawpack software, is based on solving Riemann problems and calculating fluctuations (not fluxes). The implementation employs weighted essentially nonoscillatory reconstruction in space and strong stab...

Let a ∈ (0, 1) and let Fs(a) be the periodized zeta function that is defined as Fs(a) = ∑ n−s exp(2πina) for 1, and extended to the complex plane via analytic continuation. Let sn = σn + itn, tn > 0, denote the sequence of nontrivial zeros of the Riemann zeta function in the upper halfplane ordered according to nondecreasing ordinates. We demonstrate that, assuming the Riemann Hypothesis, ...

This is the second in a series of papers on the numerical treatment of hyperelliptic theta-functions with spectral methods. A code for the numerical evaluation of solutions to the Ernst equation on hyperelliptic surfaces of genus 2 is extended to arbitrary genus and general position of the branch points. The use of spectral approximations allows for an efficient calculation of all characteristi...

The well-known root-locus method is developed for special subset of linear time-invariant systems known as fractional-order systems. Transfer functions of these systems are rational functions with polynomials of rational powers of the Laplace variable s. Such systems are defined on a Riemann surface because of their multivalued nature. A set of rules for plotting the root loci on the first Riem...

In this paper, based on the fractional Riccati equation, we propose an extended fractional Riccati sub-equation method for solving fractional partial differential equations. The fractional derivative is defined in the sense of the modified Riemann-Liouville derivative. By a proposed variable transformation, certain fractional partial differential equations are turned into fractional ordinary di...