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Title: Mathematical and Numerical Studies of Nonstandard Difference Equation Models of Differential Equations

Abstract

This research examined the following items/issues: the NSFD methodology, technical achievements and applications, dissemination efforts and research related professional activities. Also a list of unresolved issues were identified that could form the basis for future research in the area of constructing and analyzing NSFD schemes for both ODE's and PDE's.

Authors:
Publication Date:
Research Org.:
Clark Atlanta University
Sponsoring Org.:
USDOE
OSTI Identifier:
965764
Report Number(s):
DOE/ER/25515
TRN: US201002%%1171
DOE Contract Number:
FG02-02ER25515
Resource Type:
Technical Report
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; DIFFERENTIAL EQUATIONS; MATHEMATICS

Citation Formats

Mickens, Ronald E. Mathematical and Numerical Studies of Nonstandard Difference Equation Models of Differential Equations. United States: N. p., 2008. Web. doi:10.2172/965764.
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Mickens, Ronald E. Mathematical and Numerical Studies of Nonstandard Difference Equation Models of Differential Equations. United States. doi:10.2172/965764.
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Mickens, Ronald E. Mon . "Mathematical and Numerical Studies of Nonstandard Difference Equation Models of Differential Equations". United States. doi:10.2172/965764. https://www.osti.gov/servlets/purl/965764.
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@article{osti_965764,
title = {Mathematical and Numerical Studies of Nonstandard Difference Equation Models of Differential Equations},
author = {Mickens, Ronald E.},
abstractNote = {This research examined the following items/issues: the NSFD methodology, technical achievements and applications, dissemination efforts and research related professional activities. Also a list of unresolved issues were identified that could form the basis for future research in the area of constructing and analyzing NSFD schemes for both ODE's and PDE's.},
doi = {10.2172/965764},
journal = {},
number = ,
volume = ,
place = {United States},
year = {Mon Dec 22 00:00:00 EST 2008},
month = {Mon Dec 22 00:00:00 EST 2008}
}
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Technical Report:
View Technical Report (0.07 MB)
DOI:  10.2172/965764
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  • The major thrust of this proposal was to continue our investigations of so-called non-standard finite-difference schemes as formulated by other authors. These schemes do not follow the standard rules used to model continuous differential equations by discrete difference equations. The two major aspects of this procedure consist of generalizing the definition of the discrete derivative and using a nonlocal model (on the computational grid or lattice) for nonlinear terms that may occur in the differential equations. Our aim was to investigate the construction of nonstandard finite-difference schemes for several classes of ordinary and partial differential equations. These equations are simplemore » enough to be tractable, yet, have enough complexity to be both mathematically and scientifically interesting. It should be noted that all of these equations differential equations model some physical phenomena under an appropriate set of experimental conditions. The major goal of the project was to better understand the process of constructing finite-difference models for differential equations. In particular, it demonstrates the value of using nonstandard finite-difference procedures. A secondary goal was to construct and study a variety of analytical techniques that can be used to investigate the mathematical properties of the obtained difference equations. These mathematical procedures are of interest in their own right and should be a valuable contribution to the mathematics research literature in difference equations. All of the results obtained from the research done under this project have been published in the relevant research/technical journals or submitted for publication. Our expectation is that these results will lead to improved finite difference schemes for the numerical integration of both ordinary and partial differential equations. Section G of the Appendix gives a concise summary of the major results obtained under funding by the grant.« less

  • This report summarizes the complete research findings of the PI. Included are titles and places of publication of all journal, book, and conference papers, and abstracts. A listing of major conferences and meetings where these research results were reported is also provided.

  • The paper is divided into three parts. In the first part the one- dimensional heat equation with constant coefficients is discussed. The next section is devoted to the proof of the equivalence for the one-dimensional heat equation with variable coefficients. The technique used will carry over the general case which is treated briefly in the third section. (W.L.H.)

  • An investigation was made to determine whether a cyclically reduced system of equations or the original system may be solved by an iterative method in the most efficient manner. Sohroder and Varga used the special form of a 2- cyclic matrix to obtain improvements in the convergence rate for certain point iteration techniques. It is shown that improvements in the rate of convergence may also be obtained for various block iteration techniques and that the cyclic reduction process leads to an efficient method of solution for the two- dimensional self-adjoint second-order elliptic difference equations. A discussion of iterative techniques andmore » the concept of rate of convergence is given. For various block iteration methods, sufficient conditions were obtained which ensure a higher asymptotic rate of convergence for the cyclically reduced system than for the original system. The use of cyclic reduction in the numerical solution of two-dimensional second-order self-adjoint elliptic differential equations is illustrated. (M.C.G.)« less