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

Fractals Pub Date : 2024-05-15 DOI:10.1142/s0218348x24020031
Xiao-Jun Yang, D. Baleanu, J. A. TENREIRO MACHADO, CARLO CATTANI
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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.
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序言--分形与局部分形微积分特刊:最新进展与未来挑战
分形几何在描述自然特征方面发挥着重要作用。局部分数微积分是数学的一个新分支,用于处理数学物理和工程科学中的不可分问题。通过局部分数微积分研究了局部分数不等式、局部分数 ODE 和局部分数 PDE。分数微积分还被用来表达具有分数维度的函数的分数行为。报告了分数微积分和分形的有趣问题。通过缩放律微积分,详细提出了缩放律向量微积分。此外,还介绍了一些与经典、分数和幂律微积分相关的特殊函数,以表达 Kohlrausch-Williams-Watts 函数、Mittag-Leffler 函数和 Weierstrass-Mandelbrot 函数。它们与实际问题中的 ODE、PDE、分式 ODE 和分式 PDE 有关。通过 Kohlrausch-Williams-Watts 函数提出了处理实际问题的缩放律序列理论。提出了 "分形挑战"(Fractals Challenge),即回火 Xi 函数的假设,这是数学领域的一项新挑战。提出了分形几何在实际问题中的典型应用。
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