Analysis of Fast Convergent Iterative Scheme with Fractal Generation

Nisa, Zaib Un; Ishtiaq, Umar; Kamran, Tayyab; Akram, Mohammad; Popa, Ioan-Lucian

doi:10.3390/fractalfract9090575

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Open AccessArticle

Analysis of Fast Convergent Iterative Scheme with Fractal Generation

by

Zaib Un Nisa

¹,

Umar Ishtiaq

²

,

Tayyab Kamran

^1,*

,

Mohammad Akram

³ and

Ioan-Lucian Popa

^4,5

¹

Department of Mathematics, Quaid-i-Azam University Islamabad, Islamabad 45320, Pakistan

²

Office of Research, Innovation and Commercialization, University of Management and Technology, Lahore 57770, Pakistan

³

Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah 42351, Saudi Arabia

⁴

Department of Computing, Mathematics, and Electronics, “1 Decembrie 1918” University of Alba Iulia, 510009 Alba Iulia, Romania

⁵

Faculty of Mathematics and Computer Science, Transilvania University of Brasov, Iuliu Maniu Street 50, 500091 Brasov, Romania

^*

Author to whom correspondence should be addressed.

Fractal Fract. 2025, 9(9), 575; https://doi.org/10.3390/fractalfract9090575 (registering DOI)

Submission received: 12 July 2025 / Revised: 21 August 2025 / Accepted: 27 August 2025 / Published: 30 August 2025

(This article belongs to the Special Issue Mathematical and Computational Approaches to Fractal and Fractional Systems Using Fixed Point Theory)

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Abstract

In this paper, a pattern for visualizing fractals, namely Julia and Mandelbrot sets for complex functions of the form

T (u) = u^{a} - ξ u^{2} + r u + sin ρ^{σ} for all u \in C

and

a \in N ∖ {1}

,

ξ \in C

,

r, ρ \in C ∖ {0}

are created using novel fast convergent iterative techniques. The new iteration scheme discussed in this study uses s-convexity and improves earlier approaches, including the Mann and Picard–Mann schemes. Further, the proposed approach is amplified by unique escape conditions that regulate the convergence behavior and generate Julia and Mandelbrot sets. This new technique allows greater versatility in fractal design, influencing the shape, size, and aesthetic structure of the designs created. By modifying various parameters in the suggested scheme, a significant number of visually interesting fractals can be generated and evaluated. Furthermore, we provide numerical examples and graphic demonstrations to demonstrate the efficiency of this novel technique.

Keywords: fixed point; fast convergent iterative scheme; fractals; computation; Julia and Mandelbrot sets

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MDPI and ACS Style

Nisa, Z.U.; Ishtiaq, U.; Kamran, T.; Akram, M.; Popa, I.-L. Analysis of Fast Convergent Iterative Scheme with Fractal Generation. Fractal Fract. 2025, 9, 575. https://doi.org/10.3390/fractalfract9090575

AMA Style

Nisa ZU, Ishtiaq U, Kamran T, Akram M, Popa I-L. Analysis of Fast Convergent Iterative Scheme with Fractal Generation. Fractal and Fractional. 2025; 9(9):575. https://doi.org/10.3390/fractalfract9090575

Chicago/Turabian Style

Nisa, Zaib Un, Umar Ishtiaq, Tayyab Kamran, Mohammad Akram, and Ioan-Lucian Popa. 2025. "Analysis of Fast Convergent Iterative Scheme with Fractal Generation" Fractal and Fractional 9, no. 9: 575. https://doi.org/10.3390/fractalfract9090575

APA Style

Nisa, Z. U., Ishtiaq, U., Kamran, T., Akram, M., & Popa, I.-L. (2025). Analysis of Fast Convergent Iterative Scheme with Fractal Generation. Fractal and Fractional, 9(9), 575. https://doi.org/10.3390/fractalfract9090575

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Analysis of Fast Convergent Iterative Scheme with Fractal Generation

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