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On the integral transform of fractal interpolation functions

MATHEMATICS AND COMPUTERS IN SIMULATION 2024年 222卷 209-224页

作者： Agathiyan, A. Gowrisankar, A. Fataf, Nur Aisyah Abdul Vellore Inst Technol Sch Adv Sci Dept Math Vellore 632014 Tamil Nadu India Univ Pertahanan Nas Malaysia Cyber Secur & Digital Ind Revolut Ctr Kuala Lumpur Malaysia

This paper explores the integral transform of two distinct fractal interpolation functions, namely the linear fractal interpolation function and the hidden variable fractal interpolation function with variable scaling factors. Further, with a particular application of kernel functions, we investigate the integral transform of fractal functions, such as the Laplace transform and the Laplace Carson transform. Moreover, we show that the compositeness of two fractal interpolation functions, f1 in {t8, x8} and f2 in {x8, z8} remains a fractal interpolation function. It also generates iterated function system from given iterated function systems. In addition to this, the study is carried out on the composite linear fractal interpolation function of the integral transform, the Laplace transform, and the Laplace Carson transform. (c) 2023 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

关键词： fractal interpolation functions Composite fractal interpolation functions Integral transform Laplace transform Laplace carson transform

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Multifractal analysis of fractal interpolation functions

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PHYSICA SCRIPTA 2024年第11期99卷 115230-115230页

作者： Priyanka, T. M. C. Gowrisankar, A. Vellore Inst Technol Sch Adv Sci Dept Math Vellore 632014 Tamil Nadu India

This paper presents a novel algorithm to utilize multifractal spectrum as a quantitative measure for the fractal interpolation functions with respect to scaling factor and fractional order. As of yet, there were no error estimation techniques to interpret the fractal interpolation functions in the literature. To bridge this gap, this paper sketches multifractality as a quantitative measure for inquiring and comparing the effects of different scaling factors. The proposed algorithm for analyzing the multifractal measure depends on the probability measure of data points, which fractal function passes through, enabling to effectively discuss the heterogeneity of fractal interpolation functions. In addition, the impact of fractional orders on the fractional derivative (integral) of fractal interpolation functions is also discussed tailoring the multifractal measure.

关键词： multifractal spectrum fractal interpolation functions scaling factors fractional order

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Remarks on the integral transform of non-linear fractal interpolation functions

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CHAOS SOLITONS & fractalS 2023年第1期173卷

作者： Agathiyan, A. Gowrisankar, A. Fataf, Nur Aisyah Abdul Cao, Jinde Vellore Inst Technol Sch Adv Sci Dept Math Vellore 632014 Tamil Nadu India Univ Pertahanan Nas Malaysia Cyber Secur & Digital Ind Revolut Ctr Kuala Lumpur 57000 Malaysia Southeast Univ Sch Math Nanjing 210096 Peoples R China Yonsei Univ Yonsei Frontier Lab Seoul 03722 South Korea

This paper examines the integral transform of fractal interpolation functions with function scaling factors. Initially, the integral transform of quadratic fractal interpolation function, quadratic hidden variable fractal interpolation function and ��-fractal functions with function scaling factors is investigated. Using a specific application of kernel functions, we further explore the integral transform of fractal functions such as the Laplace transform and the Laplace-Carson transform.

关键词： fractal interpolation functions Integral transform Laplace transform Laplace-Carson transform

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Construction of New fractal interpolation functions Through Integration Method

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RESULTS IN MATHEMATICS 2022年第3期77卷 1-20页

作者： Agathiyan, A. Gowrisankar, A. Priyanka, T. M. C. Vellore Inst Technol Sch Adv Sci Dept Math Vellore 632014 Tamil Nadu India

This paper investigates the classical integral of various types of fractal interpolation functions namely linear fractal interpolation function, alpha-fractal function and hidden variable fractal interpolation function with function scaling factors. The integral of a fractal function is again a fractal function to a different set of interpolation data if the integral of fractal function is predefined at the initial point or end point of the given data. In this study, the selection of vertical scaling factors as continuous functions on the closed interval of R provides more diverse fractal interpolation functions compared to the fractal interpolations functions with constant scaling factors.

关键词： fractal interpolation functions classical integral function scaling factors

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CONSTRUCTION OF MONOTONOUS APPROXIMATION BY fractal interpolation functions AND fractal DIMENSIONS

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fractalS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY 2024年第2期32卷 2440006-2440006页

作者： Yu, Binyan Liang, Yongshun Nanjing Univ Sci & Technol Sch Math & Stat Nanjing 210094 Peoples R China

In this paper, we research on the dimension preserving monotonous approximation by using fractal interpolation techniques. A constructive result of the approximating sequence of self-affine continuous functions has been given, which can converge to the object continuous function of bounded variation on [0, 1] monotonously and unanimously, meanwhile their graphs can be any value of the Hausdorff and the Box dimension between one and two. Further, such approximation for continuous functions of unbounded variation or even general continuous functions with non-integer fractal dimension has also been discussed elementarily.

关键词： Monotonous Approximation fractal interpolation functions The Hausdorff Dimension The Box Dimension Self-Affine functions Variation

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fractal MULTIQUADRIC interpolation functions

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2024年第5期62卷 2349-2369页

作者： Kumar, D. Chand, A. K. B. Massopust, P. R. Indian Inst Technol Madras Dept Math Chennai 600036 India Tech Univ Munich Dept Math D-85748 Garching Germany

In this article, we impose fractal features onto classical multiquadric (MQ) functions. This generates a novel class of fractal functions, called fractal MQ functions, where the symmetry of the original MQ function with respect to the origin is maintained. This construction requires a suitable extension of the domain and similar partitions on the left side with the same choice of scaling parameters. Smooth fractal MQ functions are proposed to solve initial value problems via a collocation method. Our numerical computations suggest that fractal MQ functions offer higher accuracy and more flexibility for the solutions compared to the existing classical MQ functions. Some approximation results associated with fractal MQ functions are also presented.

关键词： fractals fractal interpolation functions MQ functions initial value problems col location method

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REVIEW ON fractal interpolation functions

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fractalS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY 2025年

作者： Priyanka, T. M. C. Gowrisankar, A. Vellore Inst Technol Sch Adv Sci Dept Math Vellore 632014 Tamil Nadu India

The idea of fractal interpolation dates back to the 1980s, however several recent developments on its new types and generalization frameworks have made this domain ripe for extensions, further analyses, and reliable applications. Commencing from the background of Barnsley's fundamental fractal interpolation function, this review paper summarizes the state of the art encompassing significant contributions in the fractal literature. This paper begins with the review on types of fractal interpolation functions and discusses results on fractional calculus theory emphasizing the relation between scaling factor and fractional order with numerical examples. Special focus is shed on the fractal dimension of fractal interpolation functions and its linear connection with fractional order. Further, the discussion on parameter identification problems highlights the importance of right choice of scaling factors for effective approximation. The paper also reviews differentiable fractal interpolation functions, in addition, encompasses recent advancements related to new contraction maps, shape preserving properties and real-world applications in different domains.

关键词： fractals fractal interpolation functions fractal Dimension Fractional Calculus Parameter Identification fractal Splines Contractions

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fractal interpolation functions for random data sets

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CHAOS SOLITONS & fractalS 2018年 114卷 256-263页

作者： Luor, Dah-Chin I Shou Univ Dept Financial & Computat Math 1Sec 1Syuecheng Rd Kaohsiung 84001 Taiwan

Let x(0) < x(1) < x(2) < ... < x(N) and I = [x(0,) x(N)]. Let u be a continuous function defined on I and let Delta(mu) = {(x(k), mu(k)) : k = 0, 1,..., N}, where mu(k) = u(x(k)). We establish a fractal interpolation function f((T mu)) on I corresponding to the set of points Delta(mu). Let Y-k be a random perturbation of mu(k) and set Delta(Y) = {(x(k), Y-k) : k = 0, 1,..., N}. By a similar way, we construct a fractal interpolation function f((Ty)) on I corresponding to the set Delta(Y). f((Ty)) (x) is a random variable for any x is an element of I, and the function f((Ty)) can be treated as a fractal perturbation of u under some random noise in the set of interpolation points Delta mu. In this article we investigate some statistical properties of f((Ty)) and give estimations of the difference between f((Ty)) and u. (C) 2018 Elsevier Ltd. All rights reserved.

关键词： fractals interpolation fractal interpolation functions Random data sets

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fractal interpolation functions with partial self similarity

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2018年第1期464卷 911-923页

作者： Luor, Dah-Chin I Shou Univ Dept Financial & Computat Math 1Sec 1Syuecheng Road Kaohsiung 84001 Taiwan

Let a data set Delta = {(t(i), y(i)) is an element of R x Y : i = 0,1, . . . ,N} be given, where t(0) I-k is a homeomorphism and M-k : J(k) x Y -> Y is continuous. Here I-k = [t(k-1),t(k)] and J(k) = [t(j(k)),t(1(... 详细信息

Let a data set Delta = {(t(i), y(i)) is an element of R x Y : i = 0,1, . . . ,N} be given, where t(0) < t(1) < t(2) < . . . < t(N) and Y is a complete metric space. In this article, fractal interpolation functions (FIFs) on I = [t(0),t(N)] corresponding to the set Delta are constructed by mappings W-1, . . . ,W-N. Each W-k is of the form W-k = (L-k, M-k), where L-k : J(k) -> I-k is a homeomorphism and M-k : J(k) x Y -> Y is continuous. Here I-k = [t(k-1),t(k)] and J(k) = [t(j(k)),t(1(k))], j(k),l(k) is an element of{0,1, . . . , N}, are subintervals of I which depend on k. In this construction, the length of J(k) is not assumed to be larger than the length of I-k, and each L-k is not supposed to be a contraction. A FIF established by this method has a property of self similarity between its graph on J(k) and on I-k. In this paper we give a construction of FIFs with locally self similar graphs. The stability and sensitivity of FIFs established in this way are also discussed. (C) 2018 Elsevier Inc. All rights reserved.

关键词： fractals interpolation fractal interpolation functions Fixed points Self similarity

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BOX DIMENSION OF WEYL-MARCHAUD FRACTIONAL DERIVATIVE OF LINEAR fractal interpolation functions

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fractalS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY 2019年第4期27卷

作者： Peng, Wen Liang Yao, Kui Zhang, Xia Yao, Jia Army Univ Engn PLA Nanjing 211101 Jiangsu Peoples R China Nanjing Univ Sci & Technol Sch Sci Nanjing 210094 Jiangsu Peoples R China

This paper mainly explores Weyl-Marchaud fractional derivative of linear fractal interpolation functions (FIFs). We prove that Weyl-Marchaud fractional derivative of a linear FIF is still a linear FIFs. More generally, we get a conclusion that there exists some linear relationship between the order of Weyl-Marchaud fractional derivative and box dimension of linear FIFs.

关键词： Weyl-Marchaud Fractional Derivative Box Dimension fractal interpolation functions

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