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21 pages, 5633 KiB

Open AccessArticle

Polynomial Approximation over Arbitrary Shape Domains

by Mohammad J. Mahtabi, Arash Ghasemi, Amirehsan Ghasemi and James C. Newman

Math. Comput. Appl. 2024, 29(6), 110; https://doi.org/10.3390/mca29060110 - 25 Nov 2024

Viewed by 966

In spectral/finite element methods, a robust and stable high-order polynomial approximation method for the solution can significantly reduce the required number of degrees of freedom (DOFs) to achieve a certain level of accuracy. In this work, a closed-form relation is proposed to approximate the Fekete points (AFPs) on arbitrary shape domains based on the singular value decomposition (SVD) of the Vandermonde matrix. In addition, a novel method is derived to compute the moments on highly complex domains, which may include discontinuities. Then, AFPs are used to generate compatible basis functions using SVD. Equations are derived and presented to determine orthogonal/orthonormal modal basis functions, as well as the Lagrange basis. Furthermore, theorems are proved to show the convergence and accuracy of the proposed method, together with an explicit form of the Weierstrass theorem for polynomial approximation. The method was implemented and some classical cases were analyzed. The results show the superior performance of the proposed method in terms of convergence and accuracy using many fewer DOFs and, thus, a much lower computational cost. It was shown that the orthogonal modal basis is the best choice to decrease the DOFs while maintaining a small Lebesgue constant when very high degree of polynomial is employed. Full article

(This article belongs to the Special Issue Mathematical and Computational Approaches in Applied Mechanics: A Themed Issue Dedicated to Professor J.N. Reddy)

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18 pages, 1507 KiB

Open AccessArticle

Predicting the Moisture Ratio of a Hami Melon Drying Process Using Image Processing Technology

by Guanyu Zhu, G.S.V. Raghavan and Zhenfeng Li

Foods 2023, 12(3), 672; https://doi.org/10.3390/foods12030672 - 3 Feb 2023

Cited by 5 | Viewed by 2281

Abstract

For food drying, moisture content and shrinkage are vital in the drying process. This paper is concerned with the moisture ratio modeling and prediction issues of the Hami melon drying process. First, an experimental system was developed; it included an adjustable-power microwave drying unit and an image-processing unit. The moisture contents and the areas of Hami melon slices at different times were sampled in real time. Then, the expression of the moisture ratio with regard to shrinkage was derived by using the Weierstrass approximation theorem. A maximum likelihood fitness function-based population evolution (MLFF-PE) algorithm was then put forward to fit the moisture ratio model and predict the moisture ratio. The results showed that the proposed MLFF-PE algorithm was effective at fitting and predicting the moisture ratio model of the drying process of Hami melon slices. Full article

(This article belongs to the Section Food Engineering and Technology)

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11 pages, 1027 KiB

Open AccessArticle

Some Probabilistic Generalizations of the Cheney–Sharma and Bernstein Approximation Operators

by Seng Huat Ong, Choung Min Ng, Hong Keat Yap and Hari Mohan Srivastava

Axioms 2022, 11(10), 537; https://doi.org/10.3390/axioms11100537 - 8 Oct 2022

Cited by 6 | Viewed by 1835

Abstract

The objective of this paper is to give some probabilistic derivations of the Cheney, Sharma, and Bernstein approximation operators. Motivated by these probabilistic derivations, generalizations of the Cheney, Sharma, and Bernstein operators are defined. The convergence property of the Bernstein generalization is established. [...] Read more.

(This article belongs to the Special Issue Mathematical Tools and Techniques Applicable to Probability Theory and Statistics)

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21 pages, 617 KiB

Open AccessArticle

Rate of Weighted Statistical Convergence for Generalized Blending-Type Bernstein-Kantorovich Operators

by Faruk Özger, Ekrem Aljimi and Merve Temizer Ersoy

Mathematics 2022, 10(12), 2027; https://doi.org/10.3390/math10122027 - 11 Jun 2022

Cited by 28 | Viewed by 2478

Abstract

An alternative approach, known today as the Bernstein polynomials, to the Weierstrass uniform approximation theorem was provided by Bernstein. These basis polynomials have attained increasing momentum, especially in operator theory, integral equations and computer-aided geometric design. Motivated by the improvements of Bernstein polynomials [...] Read more.

$λ$ ,

$α$ and a positive integer as an original extension of Bernstein–Kantorovich operators. The statistical approximation properties and the statistical rate of convergence are also obtained by means of a regular summability matrix. Using the Lipschitz-type maximal function, the modulus of continuity and modulus of smoothness, certain local approximation results are presented. Some approximation results in a weighted space are also studied. Finally, illustrative graphics that demonstrate the approximation behavior and consistency of the proposed operators are provided by a computer program. Full article

(This article belongs to the Special Issue Polynomial Sequences and Their Applications)

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