On the convergence of fixed point iterations for the moving geometry in a fluid-structure interaction problem

Hundertmark-Zaušková, Anna

doi:10.1016/j.jde.2019.07.009

Cited by 8 publications

(5 citation statements)

References 25 publications

(97 reference statements)

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“…Fixed point theory provides a suitable framework to investigate various nonlinear phenomena arising in the applied sciences including complex graphics, geometry, biology and physics [1][2][3][4]. Complex graphical shapes such as fractals, were discovered as fixed points of certain set maps [1].…”

Section: Introductionmentioning

confidence: 99%

Generation of Julia and Mandelbrot Sets via Fixed Points

2020

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The aim of this paper is to present an application of a fixed point iterative process in generation of fractals namely Julia and Mandelbrot sets for the complex polynomials of the form T ( x ) = x n + m x + r where m , r ∈ C and n ≥ 2 . Fractals represent the phenomena of expanding or unfolding symmetries which exhibit similar patterns displayed at every scale. We prove some escape time results for the generation of Julia and Madelbrot sets using a Picard Ishikawa type iterative process. A visualization of the Julia and Mandelbrot sets for certain complex polynomials is presented and their graphical behaviour is examined. We also discuss the effects of parameters on the color variation and shape of fractals.

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Section: Introductionmentioning

confidence: 99%

Generation of Julia and Mandelbrot Sets via Fixed Points

2020

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“…One of these generalizations is the utilization of iteration procedures from the fixed point theory. Fixed point theory gives an appropriate framework to research different nonlinear phenomena emerging in the applied sciences including complex graphics, biology, geometry and material science [11], [12]. Complex graphical shapes such as fractals, were discovered as fixed points of certain set maps [11].…”

Section: Introductionmentioning

confidence: 99%

CR Iteration in Generation of Antifractals With s-Convexity

Shahid

Tassaddiq

et al. 2020

IEEE Access

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Complex graphics of nonlinear dynamical systems perform vital role in many fields, e.g., image compression or encryption, art, science, and so on. Antifractals are generated by applying a map recursively to an initial point in complex plane that have become a significant area of research these days. The purpose of this paper is to investigate the dynamics of antifractals like anti-Julia sets, tricornsand multicorns of antiholomorphic polynomials via CR iteration scheme with s-convexity. Many beautiful aesthetic patterns are visualized for antipolynomial z → z m + c of complex polynomial z m + c, for m ≥ 2 to explore the geometry of antifractals.

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“…The branch of mathematics known as fixed point theory is a powerful tool to study natural phenomena that are usually nonlinear and has many applications in almost every area of research including biology, computer sciences, image generations, complex graphics, etc [1], [2]. Fractal and antifractal images can be generated by using iterative algorithms for finding fixed points of particular mappings [1]. Very first time, Julia in the year 1918 [3] studied graphics of the following complex function (1) and lead the foundation of Fractal geometry:…”

Section: Introductionmentioning

confidence: 99%

“…Fractal and antifractal images can be generated by using iterative algorithms for finding fixed points of particular mappings [1]. Very first time, Julia in the year 1918 [3] studied graphics of the following complex function (1) and lead the foundation of Fractal geometry:…”

Section: Introductionmentioning

confidence: 99%

“…The word Fractal was introduced by Mandelbrot and introduced Mandelbrot set [6]. He took c as a complex variable in (1) and utilized the idea of Gostan Julia to observe the behaviour of Julia sets that are connected sets [6]. Fractals don't have a conventional definition, anyway they are distinguished through their irregular structure that can't be found in Euclidean geometry.…”

Section: Introductionmentioning

confidence: 99%

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Generation of Antifractals via Hybrid Picard-Mann Iteration

Wang

Shahid

et al. 2020

IEEE Access

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The aim of this paper is to generate antifractals using fixed point iterative algorithms, i.e., we aim to generate anti Julia sets, tricorns and multicorns for the anti-polynomial z → z k + c of the complex polynomial z k +c, for k ≥ 2. A hybrid Picard-Mann iterative procedure used to establish escape criterion and explore the geometry of antifractals. A visualization of the antifractals for certain complex antipolynomials is presented and their graphical behavior is compared with antifratals generated via Mann iteration. We also explain the effects of parameters on shape of antifractals. INDEX TERMS Antifractals, fixed point iterative schemes, escape criterion, tricorns and multicorns.

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On the convergence of fixed point iterations for the moving geometry in a fluid-structure interaction problem

Cited by 8 publications

References 25 publications

Generation of Julia and Mandelbrot Sets via Fixed Points

Generation of Julia and Mandelbrot Sets via Fixed Points

CR Iteration in Generation of Antifractals With s-Convexity

Generation of Antifractals via Hybrid Picard-Mann Iteration

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