Mathematical and Computational Applications

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Open AccessArticle

A Novel Method of Offset Approximation along the Normal Direction with High Precision

by Hongyan Zhao and Lian Zhou

Math. Comput. Appl. 2017, 22(3), 39; https://doi.org/10.3390/mca22030039 - 4 Sep 2017

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A new algorithm is proposed for polynomial or rational approximation of the planar offset curve. The best rational Chebyshev approximation could be regarded as a kind of geometric approximation along the fixed direction. Based on this idea, we developed a wholly new offset approximation method by changing the fixed direction to the normal directions. The error vectors follow the direction of normal, and thus could reflect the approximate performance more properly. The approximation is completely independent of the original curve parameterization, and thus could ensure the stability of the approximation result. Experimental results show that the proposed algorithm is reasonable and effective. Full article

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1615 KiB

Open AccessArticle

ICA and ANN Modeling for Photocatalytic Removal of Pollution in Wastewater

by Sina Razvarz and Raheleh Jafari

Math. Comput. Appl. 2017, 22(3), 38; https://doi.org/10.3390/mca22030038 - 11 Aug 2017

Cited by 9 | Viewed by 3443

Abstract

This paper discusses the elimination of Colour Index Acid Yellow 23 (C.I. AY23) using the ultraviolet (UV)/Ag-TiO₂ process. To anticipate the photocatalytic elimination of AY23 with the existence of Ag-TiO₂ nanoparticles processed under desired circumstances, two computational techniques, namely artificial neural network (ANN) and imperialist competitive algorithm (ICA) modeling are developed. A sum of 100 datasets are used to establish the models, wherein the introductory concentration of dye, UV light intensity, initial dosage of nano Ag-TiO₂, and irradiation time are the four parameters expressed in the form of input variables. Additionally, the elimination of AY23 is considered in the form of the output variable. Out of the 100 datasets, 80 are utilized in order to train the models. The remaining 20 that were not included in the training are used in order to test the models. The comparison of the predicted outcomes extracted from the suggested models and the data obtained from the experimental analysis validates that the performance of the ANN scheme is comparatively sophisticated when compared with the ICA scheme. Full article

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220 KiB

Open AccessReview

Analytical Solution to Normal Forms of Hamiltonian Systems

by Ali Allahem

Math. Comput. Appl. 2017, 22(3), 37; https://doi.org/10.3390/mca22030037 - 27 Jul 2017

Cited by 3 | Viewed by 3048

Abstract

The idea of the normalisation of the Hamiltonian system is to simplify the system by transforming Hamiltonian canonically to an easy system. It is under symplectic conditions that the Hamiltonian is preserved under a specific transformation—the so-called Lie transformation. In this review, we will show how to compute the normal form for the Hamiltonian, including computing the general function analytically. A clear example has been studied to illustrate the normal form theory, which can be used as a guide for arbitrary problems. Full article

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Math. Comput. Appl., Volume 22, Issue 3 (September 2017) – 3 articles

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