Bivariate fractal interpolation functions on triangular domain for numerical integration and approximation

IF 1.6 4区工程技术 Q2 ENGINEERING, MULTIDISCIPLINARY

International Journal of Computational Methods Pub Date : 2022-08-08 DOI:10.1142/s0219876223500196

M. Aparna, P. Paramanathan

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引用次数: 0

Abstract

The primary objectives of this paper are to present the construction of bivariate fractal interpolation functions over triangular interpolating domain using the concept of vertex coloring and to propose a double integration formula for the constructed interpolation functions. Unlike the conventional constructions, each vertex in the partition of the triangular region has been assigned a color such that the chromatic number of the partition is 3. A new method for the partitioning of the triangle is proposed with a result concerning the chromatic number of its graph. Following the construction, a formula determining the vertical scaling factor is provided. With the newly defined vertical scaling factor, it is clearly observed that the value of the double integral coincides with the integral value calculated using fractal theory. Further, a relation connecting the fractal interpolation function with the equation of the plane passing through the vertices of the triangle is established. Convergence of the proposed method to the actual integral value is proven with sufficient lemmas and theorems. Sufficient examples are also provided to illustrate the method of construction and to verify the formula of double integration.

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三角域上二元分形插值函数的数值积分与逼近

本文的主要目的是利用顶点着色的概念在三角形插值域上构造二元分形插值函数，并给出构造的插值函数的二重积分公式。与传统结构不同，三角区域分区中的每个顶点都被赋予了一种颜色，使得分区的色数为3。提出了一种新的三角形划分方法，得到了一个关于三角形图的色数的结果。在构造之后，给出了确定垂直比例系数的公式。利用新定义的垂直尺度因子，可以清楚地观察到二重积分的值与分形理论计算的积分值吻合。进一步，建立了分形插值函数与经过三角形顶点的平面方程之间的关系。用充分的引理和定理证明了该方法对实际积分值的收敛性。文中还举例说明了二重积分公式的构造方法和验证。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

International Journal of Computational Methods ENGINEERING, MULTIDISCIPLINARY-MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

CiteScore

3.30

自引率

17.60%

发文量

审稿时长

15 months

期刊介绍： The purpose of this journal is to provide a unique forum for the fast publication and rapid dissemination of original research results and innovative ideas on the state-of-the-art on computational methods. The methods should be innovative and of high scholarly, academic and practical value. The journal is devoted to all aspects of modern computational methods including mathematical formulations and theoretical investigations; interpolations and approximation techniques; error analysis techniques and algorithms; fast algorithms and real-time computation; multi-scale bridging algorithms; adaptive analysis techniques and algorithms; implementation, coding and parallelization issues; novel and practical applications. The articles can involve theory, algorithm, programming, coding, numerical simulation and/or novel application of computational techniques to problems in engineering, science, and other disciplines related to computations. Examples of fields covered by the journal are: Computational mechanics for solids and structures, Computational fluid dynamics, Computational heat transfer, Computational inverse problem, Computational mathematics, Computational meso/micro/nano mechanics, Computational biology, Computational penetration mechanics, Meshfree methods, Particle methods, Molecular and Quantum methods, Advanced Finite element methods, Advanced Finite difference methods, Advanced Finite volume methods, High-performance computing techniques.