摘要:

AbstractThe purpose of this article is to investigate graphically and numerically the topic of chaos in reaction‐diffusion equations. This article is based on the article by Mitchell and Bruch [1]. One‐ and two‐dimensional forms of the reaction‐diffusion equation are discretized using the explicit Euler finite difference scheme. Plots are presented to show the effect of bifurcation parameters on the difference equations. Varying these parameters produce single point, periodic, chaotic, intermittent, and divergent so

ISSN:0749-159X    

DOI:10.1002/num.1690030302

出版商:John Wiley&Sons, Inc.    

年代:1987

数据来源: WILEY

摘要:

AbstractA fully Galerkin method in both space and time is developed for the second‐order, linear hyperbolic problem. Sinc basis functions are used and error bounds are given which show the exponential convergence rate of the method. The matrices necessary for the formulation of the discrete system are easily assembled. They require no numerical integrations (merely point evaluations) to be filled. The discrete problem is formulated in two different ways and solution techniques for each are described. Consideration of the two formulations is motivated by the computational architecture available. Each has advantages for the appropriate hardware. Numerical results reported show that if 2N+ 1 basis functions are used then the exponential convergence rate\documentclass{article}\pagestyle{empty}\begin{document}$ 0\left[{\exp \left({- \kappa \sqrt N} \right)} \right] $\end{document}, κ>0, is attained for both analytic and singular proble

ISSN:0749-159X    

DOI:10.1002/num.1690030303

出版商:John Wiley&Sons, Inc.    

年代:1987

数据来源: WILEY

摘要:

AbstractA new boundary element method using discretization in time is proposed to solve a class of parabolic differential equations. The method treats the term containing the time derivative as a forcing term. This necessitates the introduction of additional unknowns in the interior of the domain. At the same time, however, values for the dependent variable are determined directly in the interior. The boundary element formulation is reduced to essentially solving a Poisson equation. The accuracy and efficiency of the method are demonstrated with several examples.

ISSN:0749-159X    

DOI:10.1002/num.1690030304

出版商:John Wiley&Sons, Inc.    

年代:1987

数据来源: WILEY

摘要:

AbstractThis article reports further developments of Herrera's algebraic theory approach to the numerical treatment of differential equations. A new solution procedure for ordinary differential equations is presented. Finite difference algorithms of 0(hr), for arbitrary “r” are developed. The method consists in constructing local approximate solutions and using them to extract information about the sought solution. Only nodal information is derived. The local approximate solutions are constructed by collocation, using polynomials of degreeG. When “n” collocation points are used at each subinterval,G=n+ 1and the order of accuracy is 0(h2n−1). The procedure here presented is very easy to implement. A program in whichncan be chosen arbitrarily, was constructed and applied to selected

ISSN:0749-159X    

DOI:10.1002/num.1690030305

出版商:John Wiley&Sons, Inc.    

年代:1987

数据来源: WILEY

摘要:

AbstractIn this article, we give a simple method for developing finite difference schemes on a uniform square gird. We consider a general, two‐dimensional, second‐order, partial differential equation with variable coefficients. In the case of a nine‐point scheme, we obtain the known results of Young and Dauwalder in a fairly elegant fashion. We show how this can be extended to obtain fourth‐order schemes on thirteen points. We derive two such schemes which are attractive because they can be adapted quite easily bnto obtain formulas for gird points near the boundary. In addition to this, these formulas only require nine evaluations for the typical forcing function. Numerical examples are given to demonstrate the performance of one of the fourth‐orde

ISSN:0749-159X    

DOI:10.1002/num.1690030306

出版商:John Wiley&Sons, Inc.    

年代:1987

数据来源: WILEY

摘要:

AbstractIn this article, we give a simple method for deriving finite difference schemes on a uniform cubic grid. We consider a general, three‐dimensional, second‐order, linear, elliptic partial differential equation with variable coefficients. We derive two simple fourth‐order schemes. When the coefficients of the second‐order mixed derivatives are equal to zero, the fourth‐order scheme requires only 19 grid points. When the coefficients of the mixed derivatives are not equal to zero and the coefficients ofUxx,Uyy, andUzzare equal, we require the 27 points of the cubic grid. Numerical examples are given to demonstrate the performance of the two schemes derived. There does not exist a fourth‐order scheme involving 27 grid points for the g

ISSN:0749-159X    

DOI:10.1002/num.1690030307

出版商:John Wiley&Sons, Inc.    

年代:1987

数据来源: WILEY

ISSN:0749-159X    

DOI:10.1002/num.1690030301

出版商:John Wiley&Sons, Inc.    

年代:1987

数据来源: WILEY