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Mathematical and Computer Two algorithms are presented to find conserved densities and fluxes of ... more Mathematical and Computer Two algorithms are presented to find conserved densities and fluxes of nonlinear systems of differential-difference equations. Both algorithms utilize the scaling properties of lattice equations to reduce the problem to a calculus and linear algebra problem. The two algorithms are illustrated for the Kac-van Moerbeke, Toda, and Ablowitz-Ladik lattices. The first method leads to a three step algorithm which utilizes the dilation invariance of the conservation laws to construct the form of the density. For this method, the discrete Euler operator or discrete variational derivative is an advantageous tool. The algorithm is implemented in Mathematica. The package is called DDEDensityFlux.m. The key applications are to analyze the discretizations of the Korteweg-de Vries (KdV), and modified Korteweg-de Vries (mKdV) lattices. A combination of the KdV and mKdV lattices is also considered. The second method leads to a five step algorithm that is primarily useful in...
Advances in Nonlinear …, 2009
PRIMUS
Abstract The purpose of this paper is to explain a newly implemented interactive delivery model f... more Abstract The purpose of this paper is to explain a newly implemented interactive delivery model for students at the Colorado School of Mines (Mines) repeating Calculus for Scientists & Engineers I during the spring of 2017. Since calculus is a necessary component for so many STEM degrees, it will continue to be a gateway to obtaining such degrees and entering the highly sought after technological professions. Educators have a responsibility to further investigate how to support students who are not immediately successful. This interactive delivery model shows promise in improving student performance on final exams, increasing student learning, and encouraging struggling students to persist in STEM.
We present direct methods, algorithms, and symbolic software for the computation of conservation ... more We present direct methods, algorithms, and symbolic software for the computation of conservation laws of nonlinear partial differential equations (PDEs) and differential-difference equations (DDEs).
Mathematical and Computer Two algorithms are presented to find conserved densities and fluxes of ... more Mathematical and Computer Two algorithms are presented to find conserved densities and fluxes of nonlinear systems of differential-difference equations. Both algorithms utilize the scaling properties of lattice equations to reduce the problem to a calculus and linear algebra problem. The two algorithms are illustrated for the Kac-van Moerbeke, Toda, and Ablowitz-Ladik lattices. The first method leads to a three step algorithm which utilizes the dilation invariance of the conservation laws to construct the form of the density. For this method, the discrete Euler operator or discrete variational derivative is an advantageous tool. The algorithm is implemented in Mathematica. The package is called DDEDensityFlux.m. The key applications are to analyze the discretizations of the Korteweg-de Vries (KdV), and modified Korteweg-de Vries (mKdV) lattices. A combination of the KdV and mKdV lattices is also considered. The second method leads to a five step algorithm that is primarily useful in...
Advances in Nonlinear …, 2009
PRIMUS
Abstract The purpose of this paper is to explain a newly implemented interactive delivery model f... more Abstract The purpose of this paper is to explain a newly implemented interactive delivery model for students at the Colorado School of Mines (Mines) repeating Calculus for Scientists & Engineers I during the spring of 2017. Since calculus is a necessary component for so many STEM degrees, it will continue to be a gateway to obtaining such degrees and entering the highly sought after technological professions. Educators have a responsibility to further investigate how to support students who are not immediately successful. This interactive delivery model shows promise in improving student performance on final exams, increasing student learning, and encouraging struggling students to persist in STEM.
We present direct methods, algorithms, and symbolic software for the computation of conservation ... more We present direct methods, algorithms, and symbolic software for the computation of conservation laws of nonlinear partial differential equations (PDEs) and differential-difference equations (DDEs).