Galerkin-method approach to nonlinear soliton classical stability

Stability Analysis of the Nonlinear Galerkin Method

Our object in this article is to describe some numerical schemes for the approximation of nonlinear evolution equations, and to study the stability of the schemes. Spatial discretization can be performed by either spectral or pseudospectral methods, finite elements or finite differences; time discretization is done by two-level schemes, partly or fully explicit. The algorithms that we present s...

Nonlinear Cable equation, Fractional differential equation, Radial point interpolation method, Meshless local Petrov – Galerkin, Stability analysis

The cable equation is one the most fundamental mathematical models in the neuroscience, which describes the electro-diffusion of ions in denderits. New findings indicate that the standard cable equation is inadequate for describing the process of electro-diffusion of ions. So, recently, the cable model has been modified based on the theory of fractional calculus. In this paper, the two dimensio...

The Post-processed Galerkin Method Applied to Nonlinear Shell Vibrations

A Discontinuous Galerkin Method Applied to Nonlinear Parabolic Equations

Semi-discrete and a family of discrete time locally conservative Dis-continuous Galerkin procedures are formulated for approximations to nonlinear parabolic equations. For the continuous time approximations a priori L 1 (L 2) and L 2 (H 1) estimates are derived and similarly, l 1 (L 2) and l 2 (H 1) for the discrete time schemes. Spatial rates in H 1 and time truncation errors in L 2 are optimal.

A numerical method for solving nonlinear partial differential equations based on Sinc-Galerkin method

In this paper, we consider two dimensional nonlinear elliptic equations of the form $ -{rm div}(a(u,nabla u)) = f $. Then, in order to solve these equations on rectangular domains, we propose a numerical method based on Sinc-Galerkin method. Finally, the presented method is tested on some examples. Numerical results show the accuracy and reliability of the proposed method.