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2018
This chapter offers a peek at the vast literature on numerical methods for partial differential equations. The focus is on finite difference methods (FDM): approximating differential operators by functions of difference operators. Padé approximants (Fornberg) give a unifying principle for deriving the various stencils used by numericists.
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This chapter offers a peek at the vast literature on numerical methods for partial differential equations. The focus is on finite difference methods (FDM): approximating differential operators by functions of difference operators. Padé approximants (Fornberg) give a unifying principle for deriving the various stencils used by numericists.
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Exact finite difference and non-standard finite difference schemes for
Journal of Difference Equations and Applications, 2012Exact finite difference schemes and non-standard finite difference schemes are constructed for the first-order differential equation , for and . In particular, we show that the central finite difference scheme is an exact scheme for the differential equation .
Ronald E. Mickens, Lih-Ing W. Roeger
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Finite Differences and Orthogonal Polynomials
The Ramanujan Journal, 1999By combining finite differences with symmetric functions, we present an elementary demonstration for the limit relation from Laguerre to Hermite polynomials, proposed by Richard Askey. Another limit relation between these two polynomials is also established.
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Finite-difference and finite-element methods of approximation [PDF]
Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 1971 The finite-difference and finite-element methods for approximating the solution of elliptic boundary-value problems are discussed. The analysis of the order of accuracy is outlined, and the results compared, with some comment on special problems connected with singularities.
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1979
Publisher Summary Digital computer is not suited to perform computation by way of calculus. The automatic programming of the fundamental theorem of the integral calculus on the digital computer is a formidable, if not impossible, task of providing the computer with a symbolic list of functions and the corresponding antiderivatives, that is, a table of
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Publisher Summary Digital computer is not suited to perform computation by way of calculus. The automatic programming of the fundamental theorem of the integral calculus on the digital computer is a formidable, if not impossible, task of providing the computer with a symbolic list of functions and the corresponding antiderivatives, that is, a table of
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Finite differences on triangular grids
Numerical Methods for Partial Differential Equations, 1998A finite difference method is defined on a triangular grid for the Dirichlet problem of the Poisson equation. A relation between this finite difference scheme and the finite element method is given. From this new point of view the properties (maximum principle) of the solution and convergence are analyzed.
Brighi, B, Chipot, M, Gut, E
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Finite Element and Finite Difference Methods
2006Finite element methods (FEM) and finite difference methods (FDM) are numerical procedures for obtaining approximated solutions to boundary-value or initial-value problems. They can be applied to various areas of materials measurement and testing, especially for the characterization of mechanically or thermally loaded specimens or components ...
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A combined scheme of finite elements and finite differences
Journal of the Franklin Institute, 1989Two-dimensional boundary value problems for the Laplace equation in a ring-like region are considered. The differential equation is written first in polar coordinates and then reformulated in terms of a dimensionless radial variable. An approximate solution is assumed as the sum of products of functions of the radial and the angular variables, for ...
H. Yano, K. Nishioka, A. Kieda
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Fundamentals of finite elements and finite differences
2012In this introductory chapter we summarize some fundamental topics of approximation theory by finite elements and finite differences. The aim is to introduce the reader to notations and conventions used in the book and to gather in an organic way some basic results that are scattered in the vast literature concerning finite elements and finite ...
Alessandro Veneziani +2 more
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Finite Difference and Finite Element Methods
1999Applications of the finite difference and finite element techniques to vibroacoustic problems are presented. The basic ideas and the mathematical descriptions are outlined for both of the methods and examples are given to demonstrate the potential of such numerical techniques.
Tonni F. Johansen +2 more
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