• Title/Summary/Keyword: generalized Taylor's formula

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NEW BOUNDS FOR A PERTURBED GENERALIZED TAYLOR'S FORMULA

  • Cerone, P.;Dragomir, S.S.
    • East Asian mathematical journal
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    • v.17 no.2
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    • pp.197-215
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    • 2001
  • A generalised Taylor series with integral remainder involving a convex combination of the end points of the interval under consideration is investigated. Perturbed generalised Taylor series are bounded in terms of Lebesgue p-norms on $[a,b]^2$ for $f_{\Delta}:[a,b]^2{\rightarrow}\mathbb{R}$ with $f_{\Delta}(t,s)=f(t)-f(s)$.

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The Origin of Newton's Generalized Binomial Theorem (뉴턴의 일반화된 이항정리의 기원)

  • Koh, Youngmee;Ree, Sangwook
    • Journal for History of Mathematics
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    • v.27 no.2
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    • pp.127-138
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    • 2014
  • In this paper we investigate how Newton discovered the generalized binomial theorem. Newton's binomial theorem, or binomial series can be found in Calculus text books as a special case of Taylor series. It can also be understood as a formal power series which was first conceived by Euler if convergence does not matter much. Discovered before Taylor or Euler, Newton's binomial theorem must have a good explanation of its birth and validity. Newton learned the interpolation method from Wallis' famous book ${\ll}$Arithmetica Infinitorum${\gg}$ and employed it to get the theorem. The interpolation method, which Wallis devised to find the areas under a family of curves, was by nature arithmetrical but not geometrical. Newton himself used the method as a way of finding areas under curves. He noticed certain patterns hidden in the integer binomial sequence appeared in relation with curves and then applied them to rationals, finally obtained the generalized binomial sequence and the generalized binomial theorem.

SOLVING THE GENERALIZED FISHER'S EQUATION BY DIFFERENTIAL TRANSFORM METHOD

  • Matinfar, M.;Bahar, S.R.;Ghasemi, M.
    • Journal of applied mathematics & informatics
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    • v.30 no.3_4
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    • pp.555-560
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    • 2012
  • In this paper, differential transform method (DTM) is considered to obtain solution to the generalized Fisher's equation. This method is easy to apply and because of high level of accuracy can be used to solve other linear and nonlinear problems. Furthermore, is capable of reducing the size of computational work. In the present work, the generalization of the two-dimensional transform method that is based on generalized Taylor's formula is applied to solve the generalized Fisher equation and numerical example demonstrates the accuracy of the present method.

AN ALGORITHM FOR THE NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

  • Odibat, Zaid M.;Momani, Shaher
    • Journal of applied mathematics & informatics
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    • v.26 no.1_2
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    • pp.15-27
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    • 2008
  • We present and discuss an algorithm for the numerical solution of initial value problems of the form $D_*^\alpha$y(t) = f(t, y(t)), y(0) = y0, where $D_*^\alpha$y is the derivative of y of order $\alpha$ in the sense of Caputo and 0<${\alpha}{\leq}1$. The algorithm is based on the fractional Euler's method which can be seen as a generalization of the classical Euler's method. Numerical examples are given and the results show that the present algorithm is very effective and convenient.

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