A central WENO-TVD scheme for hyperbolic conservation laws.
A comparison of the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method for solutions of partial differential equations
We compare numerical experiments from the String Gradient Weighted Moving Finite Element method and a Parabolic Moving Mesh Partial Differential Equation method, applied to three benchmark problems based on two different partial differential equations. Both methods are described in detail and we highlight some strengths and weaknesses of each method via the numerical comparisons. The two equations used in the benchmark problems are the viscous Burgers’ equation and the porous medium equation, both...
A Computer Algebra Application to Determination of Lie Symmetries of Partial Differential Equations
The paper has been presented at the 12th International Conference on Applications of Computer Algebra, Varna, Bulgaria, June, 2006A MATHEMATICA package for finding Lie symmetries of partial differential equations is presented. The package is designed to create and solve the associated determining system of equations, the full set of solutions of which generates the widest permissible local Lie group of point symmetry transformations. Examples illustrating the functionality of the package's tools...
A further improved tanh function method exactly solving the -dimensional dispersive long wave equations.
A generalization for Peakon's solitary wave and Camassa-Holm equation.
A gradient weighted moving finite-element method with polynomial approximation of any degree.
A kinetic formulation for multi-branch entropy solutions of scalar conservation laws
A nonlinear elliptic equation with singular potential and applications to nonlinear field equations
We prove the existence of cylindrical solutions to the semilinear elliptic problem , , , where , and has a double-power behaviour, subcritical at infinity and supercritical near the origin. This result also implies the existence of solitary waves with nonvanishing angular momentum for nonlinear Schr¨odinger and Klein–Gordon equations.
A note on local smoothing effects for the unitary group associated with the KdV equation.
A numerical scheme using multi-shockpeakons to compute solutions of the Degasperis-Procesi equation.
A spatially periodic Kuramoto-Sivashinsky equation as a model problem for inclined film flow over wavy bottom.
A supraconvergent scheme for the Korteweg-de Vries equation.
A wave equation with a Dirac distribution.
A wavelet based space-time adaptive numerical method for partial differential equations
About the existence and uniqueness of solution to fractional Bürger's equation.
Active optimal control of the KdV equation using the variational iteration method.
Adaptive finite element relaxation schemes for hyperbolic conservation laws
We propose and study semidiscrete and fully discrete finite element schemes based on appropriate relaxation models for systems of Hyperbolic Conservation Laws. These schemes are using piecewise polynomials of arbitrary degree and their consistency error is of high order. The methods are combined with an adaptive strategy that yields fine mesh in shock regions and coarser mesh in the smooth parts of the solution. The computational performance of these methods is demonstrated by considering scalar...
Adaptive Finite Element Relaxation Schemes for Hyperbolic Conservation Laws
We propose and study semidiscrete and fully discrete finite element schemes based on appropriate relaxation models for systems of Hyperbolic Conservation Laws. These schemes are using piecewise polynomials of arbitrary degree and their consistency error is of high order. The methods are combined with an adaptive strategy that yields fine mesh in shock regions and coarser mesh in the smooth parts of the solution. The computational performance of these methods is demonstrated by considering scalar...
Addendum to a paper of Craig and Goodman.
Adiabatic Evolution of Coupled Waves for a Schrödinger-Korteweg-de Vries System
The effective dynamics of interacting waves for coupled Schrödinger-Korteweg-de Vries equations over a slowly varying random bottom is rigorously studied. One motivation for studying such a system is better understanding the unidirectional motion of interacting surface and internal waves for a fluid system that is formed of two immiscible layers. It was shown recently by Craig-Guyenne-Sulem [1] that in the regime where the internal wave has a large...