Displaying similar documents to “On the estimate of the error of quadrature formulae”

Functional equations stemming from numerical analysis

Tomasz Szostok

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Always when a numerical method gives exact results an interesting functional equation arises. And, since no regularity is assumed, some unexpected solutions may appear. Here we deal with equations constructed in this spirit. The vast majority of this paper is devoted to the equation ∑ i = 0 l ( y - x ) i [ f 1 , i ( α 1 , i x + β 1 , i y ) + ⋯ + f k i , i ( α k i , i x + β k i , i y ) ] = 0 ( 1 ) and its particular cases. We use Sablik’s lemma to prove that all solutions of (1) are polynomial functions. Since a continuous polynomial function is an ordinary polynomial, the crucial problem throughout...

The finite element solution of parabolic equations

Josef Nedoma (1978)

Aplikace matematiky

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In contradistinction to former results, the error bounds introduced in this paper are given for fully discretized approximate soltuions of parabolic equations and for arbitrary curved domains. Simplicial isoparametric elements in n -dimensional space are applied. Degrees of accuracy of quadrature formulas are determined so that numerical integration does not worsen the optimal order of convergence in L 2 -norm of the method.

Fermat’s method of quadrature

Jaume Paradís, Josep Pla, Pelegrí Viader (2008)

Revue d'histoire des mathématiques

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The of Fermat (c. 1659), besides containing the first known proof of the computation of the area under a higher parabola, ∫ x + m / n d x , or under a higher hyperbola, ∫ x - m / n d x —with the appropriate limits of integration in each case—has a second part which was mostly unnoticed by Fermat’s contemporaries. This second part of the is obscure and difficult to read. In it Fermat reduced the quadrature of a great number of algebraic curves in implicit form to the quadrature of known curves: the higher parabolas...