scholarly journals A numerical method for two-dimensional Hammerstein integral equations

2021 ◽  
Vol 66 (2) ◽  
pp. 267-277
Author(s):  
Sanda Micula
Keyword(s):  
Integral Equations ◽  
Collocation Method ◽  
Piecewise Linear ◽  
Computational Cost ◽  
Linear Interpolation ◽  
Two Dimensions ◽  
Future Research ◽  
Integration Scheme ◽  
Hammerstein Integral Equations ◽  
Piecewise Linear Interpolation

"In this paper we investigate a collocation method for the approximate solution of Hammerstein integral equations in two dimensions. As in [8], col- location is applied to a reformulation of the equation in a new unknown, thus reducing the computational cost and simplifying the implementation. We start with a special type of piecewise linear interpolation over triangles for a refor- mulation of the equation. This leads to a numerical integration scheme that can then be extended to any bounded domain in R2, which is used in collocation. We analyze and prove the convergence of the method and give error estimates. As the quadrature formula has a higher degree of precision than expected with linear interpolation, the resulting collocation method is superconvergent, thus requiring fewer iterations for a desired accuracy. We show the applicability of the proposed scheme on numerical examples and discuss future research ideas in this area."

scholarly journals Crank-Nicolson scheme for two-dimensional in space fractional diffusion equations with functional delay

2021 ◽  
Vol 57 ◽  
pp. 128-141
Author(s):  
M. Ibrahim ◽  
V.G. Pimenov
Keyword(s):  
Piecewise Linear ◽  
Linear Interpolation ◽  
Fractional Diffusion ◽  
Dimensional System ◽  
Two Dimensional ◽  
Piecewise Linear Interpolation ◽  
Functional Delay ◽  
General Difference ◽  
Nicolson Scheme ◽  
Crank Nicolson

A two-dimensional in space fractional diffusion equation with functional delay of a general form is considered. For this problem, the Crank-Nicolson method is constructed, based on shifted Grunwald-Letnikov formulas for approximating fractional derivatives with respect to each spatial variable and using piecewise linear interpolation of discrete history with continuation extrapolation to take into account the delay effect. The Douglas scheme is used to reduce the emerging high-dimensional system to tridiagonal systems. The residual of the method is investigated. To obtain the order of the method, we reduce the systems to constructions of the general difference scheme with heredity. A theorem on the second order of convergence of the method in time and space steps is proved. The results of numerical experiments are presented.

Median based algorithm for sub-pixel estimation of extrema positions of diffuse light reflection signal

2021 ◽  
Vol 2021 (49) ◽  
pp. 37-44
Author(s):  
I. B. Ivasiv ◽  
Keyword(s):  
Linear Array ◽  
Piecewise Linear ◽  
Linear Interpolation ◽  
Diffuse Light ◽  
Random Errors ◽  
Centroid Algorithm ◽  
Piecewise Linear Interpolation ◽  
Light Reflectance ◽  
Cumulative Function

It has been proposed to utilize the median algorithm for determination of the extrema positions of diffuse light reflectance intensity distribution by a discrete signal of a photodiode linear array. The algorithm formula has been deduced on the base of piecewise-linear interpolation for signal representation by cumulative function. It has been shown that this formula is much simpler for implementation than known centroid algorithm and the noise immune Blais and Rioux detector algorithm. Also, the methodical systematic errors for zero noise as well as the random errors for full common mode additive noises and uncorrelated noises have been estimated and compared for mentioned algorithms. In these terms, the proposed median algorithm is proportionate to Blais and Rioux algorithm and considerably better then centroid algorithm.

A Normalized Numerical Scaling Method for the Unbalanced Multi-Granular Linguistic Sets

2015 ◽  
Vol 23 (02) ◽  
pp. 221-243 ◽  
Author(s):  
Baoli Wang ◽  
Jiye Liang ◽  
Yuhua Qian ◽  
Chuangyin Dang
Keyword(s):  
Piecewise Linear ◽  
Linear Interpolation ◽  
Decision Makers ◽  
Decision Problems ◽  
Scaling Functions ◽  
Linguistic Terms ◽  
Scaling Method ◽  
Piecewise Linear Interpolation ◽  
Transformation Functions

Decision makers often express their evaluations on decision problems with multi-granular linguistic terms. This fact leads to the unification of the multi-granular linguistic terms into a single linguistic set in the literature. However, this unification process increases the complexity of computation and the subjectivity in the determination of transformation functions. To overcome this deficiency, this paper aims to develop a normalized numerical scaling method for determining the semantics of multi-granular linguistic terms in the same domain. We first introduce a class of numerical scaling functions to generate several balanced or unbalanced linguistic sets. Since these scaled linguistic sets have different domains, we then develop a normalized numerical scaling method to form them into the unique interval [0,1]. As a result of this development, two classes of normalized scaling functions are derived from the priori scale information and applications of piecewise linear interpolation and piecewise arc interpolation. Finally, an example is given to illustrate how the method works.

Rearch of Data Interpolation in Airlevel Posture Sensor

2011 ◽  
Vol 271-273 ◽  
pp. 225-228
Author(s):  
Bai Hua Li ◽  
Lin Hua Piao ◽  
Ming Ming Ji
Keyword(s):  
Lagrange Interpolation ◽  
Polynomial Interpolation ◽  
Piecewise Linear ◽  
Linear Interpolation ◽  
Piecewise Linear Interpolation ◽  
Runge Phenomenon ◽  
Software Compensation ◽  
The Given ◽  
The Relationship ◽  
Zero Voltage

The method of interpolation was used in the process of software compensation of temperature in order to improve the environmental performance of air level posture sensor. Two algorithms including Lagrange interpolation and piecewise linear interpolation were calculated and compared in Matlab software, and then the optimization scheme could be achieved. The result showed that although the curse of Lagrange interpolation included all the given data positions, the Runge phenomenon in polynomial interpolation made the accuracy of interpolation lower. Piecewise linear interpolation reflected the relationship between environment and zero voltage more accurately. Piecewise linear interpolation not only can be used to improve the accuracy of software compensation of air level posture sensor.

Sign in / Sign up

Export Citation Format

Share Document