on the basis of a reproducing kernel space, an iterative algorithm for solving the one-dimensional linear and nonlinear schrödinger equations is presented. the analytical solution is shown in a series form in the reproducing kernel space and the approximate solution is constructed by truncating the series. the convergence of the approximate solution to the analytical solution is also proved. th...

in this paper, we studied the numerical solution of nonlinear weakly singular volterra-fredholm integral equations by using the product integration method. also, we shall study the convergence behavior of a fully discrete version of a product integration method for numerical solution of the nonlinear volterra-fredholm integral equations. the reliability and efficiency of the proposed scheme are...

In this study, a novel stochastic computational frameworks based on fractional Meyer wavelet artificial neural network (FMW-ANN) is designed for nonlinear-singular Lane-Emden (NS-FLE) differential equation. The modeling strength of FMW-ANN used to transformed the NS-FLE system difference equations and approximate theory implemented in mean squared error sense develop merit function equations. M...

Weak gravitational lensing is one of the few direct methods to map dark-matter distribution on large scales in Universe, and estimate cosmological parameters. We study a Bayesian inference problem where data covariance C, estimated from number ns numerical simulations, singular. In context large-scale structure observations, creation such N-body simulations often prohibitively expensive. Infere...

We summarize a recently developed integral equation approach to tackling the long-time existence problem for smooth solution v(x, t) to the 3-D Navier-Stokes equation in the context of a periodic box problem with smooth time-independent forcing and initial condition v0. Using an inverse Laplace transform of v̂(k, t) − v̂0 in 1/t, we arrive at an integral equation for Û(k, p), where p is inverse-L...

We describe how the Borel summability of a divergent asymptotic expansion can be expanded and applied to nonlinear partial differential equations (PDEs). While Borel summation does not apply for non-analytic initial data, the present approach generates an integral equation (IE) applicable to much more general data. We apply these concepts to the three-dimensional Navier-Stokes (NS) system and s...

Up to an arbitrary order of the Chapman–Enskog expansion of the kinetic Bhatnagar– Gross–Krook (BGK) equation, a corresponding analytic solution can be obtained. Based on such a compact exact solution, a new gas-kinetic scheme is constructed for the compressible Navier–Stokes equations. Instead of using a discontinuous initial condition in the gas-kinetic BGK–NS method Xu [K. Xu, A gas-kinetic ...

In this paper we are concerned with the problem of solving numerically isospectral ows. These ows are characterized by the diierential equation L 0 = B(L);L]; L(0) = L 0 ; where L 0 is a d d symmetric matrix, B(L) is a skew-symmetric matrix function of L and B;L] is the Lie bracket operator. We show that standard Runge{Kutta schemes fail in recovering the main qualitative feature of these ows, ...

One of the outstanding problems in the numerical simulation of mechanical systems is the development of eecient methods for dealing with highly oscilla-tory systems. These types of systems arise for example in vehicle simulation in modeling the suspension system or tires, in models for contact and impact, in exible body simulation from vibrations in the structural model, and in molecular dynami...

to control the nonlinear numerical instability, throughout the time evolution of the eulerian form of the nonlinear rotating shallow water equations, it is necessary to add numerical diffusion to the solution. it is clear that, this extra numerical diffusion degrades the accuracy of the numerical solution and should be kept as small as possible. in a conventional approach a hyper-diffusion is u...