scholarly journals Numerical Approximations of the Relative Rearrangement: The piecewise linear case. Application to some Nonlocal Problems

2000 ◽  
Vol 34 (2) ◽  
pp. 477-499 ◽  
Author(s):  
Jean-Michel Rakotoson ◽  
Maria Luisa Seoane
Keyword(s):  
Piecewise Linear ◽  
Linear Case ◽  
Nonlocal Problems ◽  
Numerical Approximations ◽  
Relative Rearrangement ◽  
Case Application

scholarly journals On variation in birth processes

1990 ◽  
Vol 22 (02) ◽  
pp. 480-483 ◽  
Author(s):  
M. J. Faddy
Keyword(s):  
Piecewise Linear ◽  
Numerical Results ◽  
Linear Case ◽  
Birth Rates ◽  
Birth Processes

Birth processes with piecewise linear birth rates are analysed, and numerical results suggest that, relative to the linear case, convex birth rates increase variability and concave birth rates decrease variability.

scholarly journals A sharp error estimate of piecewise polynomial collocation for nonlocal problems with weakly singular kernels

2020 ◽  
Author(s):  
Minghua Chen ◽  
Wenya Qi ◽  
Jiankang Shi ◽  
Jiming Wu
Keyword(s):  
Global Convergence ◽  
Integral Equations ◽  
Singular Integrals ◽  
Convergence Rates ◽  
Piecewise Linear ◽  
Singular Integral Equations ◽  
Fredholm Integral Equations ◽  
Nonlocal Problems ◽  
Weakly Singular ◽  
Polynomial Collocation

Abstract As is well known, piecewise linear polynomial collocation (PLC) and piecewise quadratic polynomial collocation (PQC) are used to approximate the weakly singular integrals $$\begin{equation*}I(a,b,x) =\int^b_a \frac{u(y)}{|x-y|^\gamma}\textrm{d}y, \quad x \in (a,b),\quad 0< \gamma <1,\end{equation*}$$which have local truncation errors $\mathcal{O} (h^2 )$ and $\mathcal{O} (h^{4-\gamma } )$, respectively. Moreover, for Fredholm weakly singular integral equations of the second kind, i.e., $\lambda u(x)- I(a,b,x) =f(x)$, $\lambda \neq 0$, the global convergence rates are also $\mathcal{O} (h^2 )$ and $\mathcal{O} (h^{4-\gamma } )$ by PLC and PQC in Atkinson (2009, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press). In this work we study the following nonlocal problems, which are similar to the above Fredholm integral equations: $$\begin{equation*}\int^b_a \frac{u(x)-u(y)}{|x-y|^\gamma}\textrm{d}y =f(x), \quad x \in (a,b),\quad 0< \gamma <1. \end{equation*}$$In the first part of this paper we prove that the weakly singular integrals $I(a,b,x)$ have optimal local truncation error $\mathcal{O}(h^4\eta _i^{-\gamma } )$ by PQC, where $\eta _i=\min \left \{x_i-a,b-x_i\right \}$ and $x_i$ coincides with an element junction point. Then the sharp global convergence orders $\mathcal{O}\left (h\right )$ and $\mathcal{O} (h^3)$ by PLC and PQC, respectively, are established for nonlocal problems. Finally, numerical experiments are shown to illustrate the effectiveness of the presented methods.

scholarly journals On variation in birth processes

1990 ◽  
Vol 22 (2) ◽  
pp. 480-483 ◽  
Author(s):  
M. J. Faddy
Keyword(s):  
Piecewise Linear ◽  
Numerical Results ◽  
Linear Case ◽  
Birth Rates ◽  
Birth Processes

Birth processes with piecewise linear birth rates are analysed, and numerical results suggest that, relative to the linear case, convex birth rates increase variability and concave birth rates decrease variability.

Least-square refinement of structure parameters from CBED integrated intensities: application to determination of temperature factors in Y BA2CU3O7

1992 ◽  
Vol 50 (2) ◽  
pp. 1456-1457
Author(s):  
Kjersti Gjønnes ◽  
Jon Gjønnes
Keyword(s):  
Least Square ◽  
Measured Intensity ◽  
Structure Information ◽  
Accurate Temperature ◽  
Least Square Fit ◽  
Case Application ◽  
Integrated Intensities ◽  
Temperature Factors ◽  
Narrow Bands

Electron diffraction intensities can be obtained at large scattering angles (sinθ/λ ≥ 2.0), and thus structure information can be collected in regions of reciprocal space that are not accessable with other diffraction methods. LACBED intensities in this range can be utilized for determination of accurate temperature factors or for refinement of coordinates. Such high index reflections can usually be treated kinematically or as a pertubed two-beam case. Application to Y Ba2Cu3O7 shows that a least square refinememt based on integrated intensities can determine temperature factors or coordinates.LACBED patterns taken in the (00l) systematic row show an easily recognisable pattern of narrow bands from reflections in the range 15 < l < 40 (figure 1). Integrated intensities obtained from measured intensity profiles after subtraction of inelastic background (figure 2) were used in the least square fit for determination of temperature factors and refinement of z-coordinates for the Ba- and Cu-atoms.

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