PREFACE — SPECIAL ISSUE ON FRACTALS AND LOCAL FRACTIONAL CALCULUS: RECENT ADVANCES AND FUTURE CHALLENGES

Xiao-Jun Yang, Dumitru Baleanu, J. A. Tenreiro Machado () and Carlo Cattani ()
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Xiao-Jun Yang: State Key Laboratory of Intelligent Construction and Healthy Operation and Maintenance of Deep, Underground Engineering, School of Mathematics, China University of Mining and Technology, Xuzhou 221116, Jiangsu, P. R. China2Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80257, Jeddah 21589, Saudi Arabia3Department of Mathematics, College of Science, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Republic of Korea
Dumitru Baleanu: Lebanese American University, Beirut 11022801, Lebanon5Institute of Space Sciences, Magurele-Bucharest 077125, Romania
J. A. Tenreiro Machado: Instituto Superior de Engenharia do Porto (ISEP), Porto, Portugal
Carlo Cattani: Department of Economics, Engineering, Society and Enterprise, University of Tuscia, Viterbo, Italy

FRACTALS (fractals), 2024, vol. 32, issue 04, 1-13

Abstract: Fractal geometry plays an important role in the description of the characteristics of nature. Local fractional calculus, a new branch of mathematics, is used to handle the non-differentiable problems in mathematical physics and engineering sciences. The local fractional inequalities, local fractional ODEs and local fractional PDEs via local fractional calculus are studied. Fractional calculus is also considered to express the fractal behaviors of the functions, which have fractal dimensions. The interesting problems from fractional calculus and fractals are reported. With the scaling law, the scaling-law vector calculus via scaling-law calculus is suggested in detail. Some special functions related to the classical, fractional, and power-law calculus are also presented to express the Kohlrausch–Williams–Watts function, Mittag-Leffler function and Weierstrass–Mandelbrot function. They have a relation to the ODEs, PDEs, fractional ODEs and fractional PDEs in real-world problems. Theory of the scaling-law series via Kohlrausch–Williams–Watts function is suggested to handle real-world problems. The hypothesis for the tempered Xi function is proposed as the Fractals Challenge, which is a new challenge in the field of mathematics. The typical applications of fractal geometry are proposed in real-world problems.

Keywords: Fractal Geometry; Local Fractional Derivative; Local Fractional Integral; Local Fractional Calculus; Fractional Derivative; Fractional Integral; Fractional Calculus; Fractal Scaling Law; Fractal Dimension; General Calculus; Power-Law Calculus; Scaling-Law Calculus; Scaling-Law Vector Calculus; ODEs; PDEs; Fractional ODEs; Fractional PDEs; Local Fractional ODEs; Local Fractional PDEs; Kohlrausch–Williams–Watts Function; Mittag-Leffler Function; Weierstrass–Mandelbrot Function; Scaling-Law Series; Special Function; Local Fractional Inequalities; Hausdorff Vector Calculus; Mandelbrot Scaling Law; Number Theory; Riemann Hypothesis; Jensen Conjecture; Completed Riemann Zeta Function; Completed Ramanujan Zeta Function; Completed Dedekind Zeta Function; Completed Automorphic L-Function; Completed Hasse–Weil L-Function; Generalized Riemann Hypothesis; Extended Riemann Hypothesis; Ramanujan Xi Function; Automorphic Xi Function; Quadratic Dirichlet Xi Function; Ramanujan Xi Function; Hasse–Weil Xi Function; Generalized Mixed Weighted Fractional Brownian Motion; Cantor Sets; Mandelbrot Set; Strong Conjecture; Weak Conjecture; Local Fractional Ostrowski-Type Inequality; Local Fractional Hilbert-Type Inequality; Local Fractional Hermite–Hadamard-Type Inequality; Local Fractional Pompeiu-Type Inequality; Local Fractional Newton-Type Inequality; Local Fractional Lyapunov-Type Inequality; Local Fractional Gronwall–Bellman-Type Inequality (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1142/S0218348X24020031

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