Number Theory

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By Ronald E Mickens

This quantity may be divided into elements: a in simple terms mathematical half with contributions on finance arithmetic, interactions among geometry and physics and diversified components of arithmetic; one other half at the popularization of arithmetic and the location of girls in arithmetic Nonstandard finite distinction schemes / Ronald E. Mickens -- Nonstandard equipment for advection-diffusion response equations / Hristo V. Kojouharov and Benito M. Chen -- software of nonstandard finite transformations to resolve the wave equation and Maxwell's equations / James B. Cole -- Non-standard discretization tools for a few organic versions / H. Al-Kahby, F. Dannan, and S. Elaydi -- An creation to numerical integrators holding actual homes / Martin J. Gander and Rita Meyer-Spasche

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The follow­ ing conclusions were reached: (i) The Mickens-Ramadhani scheme is (formally) of 0(h2), while the Numerov scheme is 0(h*). However, the Mickens-Ramadhani scheme performs much better than the Numerov scheme for large values of h. (ii) The Mickens-Ramadhani scheme is an exact finite difference model for f{x) = constant. This is not the situation for the Numerov method. 37 Applications (iii) The CNMFDS is of 0(/i 4 ), just like the Numerov method, and is an exact finite difference method for f(x) = constant.

68) by Mickens and Ramadhani [42]. 70) + Aum = 0, where h = Ax and um = u(xm). This result is correct whether A is positive or negative if use is made of the relation sin(i0) = isinh(0), i = yf^. 71) The Mickens-Ramadhani scheme for Eq. 68) is gotten by replacing A in Eq. 70) by fm - f(xm). 72) + fmUm = 0. 73) allows Eq. 74) Chen et al. 75) They carried out numerical studies and compared the Numerov [44], MickensRamadhani [42], and CNMFDS representations for Eq. 68). The follow­ ing conclusions were reached: (i) The Mickens-Ramadhani scheme is (formally) of 0(h2), while the Numerov scheme is 0(h*).

Note that c is exactly that given by the ODE. 3], the following ex­ pression is obtained for the solution to Eq. 95)

is an arbitrary constant. 97) 40 Nonstandard Finite Difference Schemes force (/i) to satisfy the constraint 4>(h) <—, h > 0. 98) u2 A particular explicit functional form for cj>(h) that satisfies Eqs. 98) is 1_ p-h"2 (h) = . 99) U2 The constant D can be calculated by selecting an arbitrary value for y0. 101) Since u2 > «i, it follows from the condition of Eq. 102) and this implies 0

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