The propagation of sound waves in a horizontally stratified environment, classic problem ocean acoustics, has traditionally been calculated using normal modes. Most programs based on the mode model are discretized finite difference method (FDM). In this paper, Legendre collocation (LCM) domain decomposition is proposed to solve problem. A set points cannot penetrate multiple layers media, thus ...

The Karhunen-Loève expansion of a Gaussian process, a common tool on finite element methods for differential equations with stochastic coefficients, is based on the spectral decomposition of its covariance function. The eigenpairs of the covariance are expressed as a Fredholm integral equation of second kind, which can be readily approximated with piecewise-constant finite elements. In this wor...

In this study, we consider a nonlinear system of three connected delay differential neoclassical growth models along with stochastic effect and additive white noise, which is influenced by perturbation. We derived the conditions for positive equilibria, stability solutions system. It observed that when constant reaches certain threshold steady state, asymptotic lost, Hopf bifurcation occurs. ca...

In this paper, interval Legendre wavelet method is investigated to approximated the solution of the interval Volterra-Fredholm-Hammerstein integral equation. The shifted interval Legendre polynomials are introduced and based on interval Legendre wavelet method is defined. The existence and uniqueness theorem for the interval Volterra-Fredholm-Hammerstein integral equations is proved. Some examp...

A systematic approach is presented for developing entropy stable (SS) formulations of any order for the Navier-Stokes equations. These SS formulations discretely conserve mass, momentum, energy and satisfy a mathematical entropy inequality. They are valid for smooth as well as discontinuous flows provided sufficient dissipation is added at shocks and discontinuities. Entropy stable formulations...

In this work we provide a convergence analysis for the quasi-optimal version of the Stochastic Sparse Grid Collocation method we had presented in our previous work “On the optimal polynomial approximation of Stochastic PDEs by Galerkin and Collocation methods” [6]. Here the construction of a sparse grid is recast into a knapsack problem: a profit is assigned to each hierarchical surplus and onl...

Identifying computational tasks suitable for (future) quantum computers is an active field of research. Here we explore utilizing the purpose solving differential equations. We consider two approaches: (i) basis encoding and fixed-point arithmetic on a digital computer, (ii) representing high-order Runge-Kutta methods as optimization problems annealers. As realizations applied to two-dimensiona...

Abstract This paper describes an experiment that attempts to compare a range of existing collocation extraction techniques as well as the implementation of a new technique based on tests for lexical substitutability. After a description of the experiment details, the techniques are discussed with particular emphasis on any adaptations that are required in order to evaluate it in the way propose...

Local projection methods which yield c'm_1) piecewise polynomials of order m + k as approximate solutions of a boundary value problem for an mth order ordinary differential equation are determined by the k linear functional at which the residual error in each partition interval is required to vanish on. We develop a condition on these k f unctionals which implies breakpoint superconvergence (of...