The distributional Henstock-Kurzweil integral and applications II

Liu, Wei; Ye, Guoju; Zhao, Dafang

doi:10.22436/jnsa.010.01.27

Cited by 4 publications

(3 citation statements)

References 5 publications

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“…Walaupun integral HK dan integral Denjoy-Perrron ekuivalen, Integral HK lebih banyak digunakan karena dalam pendefinisian dan perhitungannya seperti integral Riemann. Dalam banyak literatur, integral HK masih dilakukan kajian lebih lanjut seperti [15], [16], [17].…”

Section: Pendahuluanunclassified

Perbandingan Penyelesaian Integral Riemann, Lebesgue dan HK Berdasarkan Definisi

Fikri

Samsurizal

Makkulau

2021

LIMITS

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Sejak integral diperkenalkan oleh Newton dan Leibniz pada abad ke-17, alat ukur ini terus dilakukan kajian dan perumuman hingga saat ini. Terdapat tiga integral yang dikenal secara luas yaitu integral Riemann, integral Lebesgue dan integral HK. Pada masanya integral Riemann diterapkan untuk menyelesaikan berbagai persoalan tetapi kemudian diketahui memiliki keterbatasan. Integral Lebesgue muncul untuk mengatasi keterbatasan ini. Integral Lebesgue pun kemudian diketahui memiliki keterbatasannya tersendiri sehingga muncul integral Henstock-Kurzweil (integral HK) untuk mengatasi keterbatasan integral Lebesgue. Dalam banyak literatur seringkali untuk menyelesaikan soal integral tidak menggunakan definisi/ kerangka acuan yang sama sehingga sangat sulit memahami substansi maupun keterkaitan ketiga integral ini. Oleh karena itu, pada penelitian ini dilakukan perbandingan penyelesaian persoalan integral menggunakan definisi. Adapun dari empat persoalan integral yang dibahas, integral Riemann dapat menyelesaikan satu persoalan, integral Lebesgue dapat menyelesaikan dua persoalan dan integral HK dapat menyelesaikan seluruh persoalan yang dibahas.

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Section: Pendahuluanunclassified

Perbandingan Penyelesaian Integral Riemann, Lebesgue dan HK Berdasarkan Definisi

Fikri

Samsurizal

Makkulau

2021

LIMITS

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“…Talvia in [Tal06], studied the Henstock-Kurzweil distributional integral and presented convergence theorems and lattice properties of the space of integrable Henstock-Kurzweil distributions. Guoju Ye et al in [YL16] and [LYZ17] continued the study of convergence theorems and lattice properties, whereas in [GMERMTMM13] Gutiérrez et al analysed the topological properties of such a space.…”

Section: Introductionmentioning

confidence: 99%

Topological properties of the space of integrable Henstock-Kurzweil vector distributions.

Escamilla-Rocha,

Alanís-López,

Escamilla-Reyna

2023

Preprint

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In this work, we prove that the space of integrable Henstock-Kurzweil vector distributions denoted as DHK([a,b],X) has a complemented copy of c0. We show conditions for DHK([a,b],X) to have a complemented copy of l1 and a copy of l∞. We also give conditions for DHK([a,b],X) to have property (V) and the Dieudonné property. Furthermore, we prove that DHK([a,b],X) does not have property (u). Mathematics Subject Classification (2010). Primary 26A39, 58A30; Secondary 46B03, 46B04.

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“…One example is shown to illustrate the application of these results, in which the solution of the problem presented belongs to ([ , ]) and this example is not covered by any result using its existence, continuity and di erentiation but not examples of applications were presented in this work. In [9], the Henstock-Kurzweil integral is considered from a distributional analysis and two xed-point theorems are presented. is work is organized as follows: In Section 2, the basic elements of the FEM are given; in Section 3, the de nition of the function and some basic results, which allow the application of the FEM, are given; in Section 4, some quadratures for functions are described; in Section 5, numerical examples are presented in order to validate the proposed methodology.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Some Differential Equations with Henstock–Kurzweil Functions

León-Velasco

Morín-Castillo

Óliveros

et al. 2019

Journal of Function Spaces

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In this work, the Finite Element Method is used for finding the numerical solution of an elliptic problem with Henstock–Kurzweil integrable functions. In particular, Henstock–Kurzweil high oscillatory functions were considered. The weak formulation of the problem leads to integrals that are calculated using some special quadratures. Definitions and theorems were used to guarantee the existence of the integrals that appear in the weak formulation. This allowed us to apply the above formulation for the type of slope bounded variation functions. Numerical examples were developed to illustrate the ideas presented in this article.

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The distributional Henstock-Kurzweil integral and applications II

Cited by 4 publications

References 5 publications

Perbandingan Penyelesaian Integral Riemann, Lebesgue dan HK Berdasarkan Definisi

Perbandingan Penyelesaian Integral Riemann, Lebesgue dan HK Berdasarkan Definisi

Topological properties of the space of integrable Henstock-Kurzweil vector distributions.

Numerical Solution of Some Differential Equations with Henstock–Kurzweil Functions

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