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.

分形几何在描述自然特征方面发挥着重要作用。局部分数阶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微问题。通过局部分数阶微积分研究局部分数不等式、局部分数常微分方程和局部分数偏微分方程。分数阶微积分也被认为是表达具有分形维数的函数的分形行为。报告了分数阶微积分和分形的有趣问题。结合标度律，详细提出了通过标度律演算的标度律矢量演算。还提出了一些与经典、分数和幂律微积分相关的特殊函数来表达 Kohlrausch-Williams-Watts 函数、Mittag-Leffler 函数和 Weierstrass-Mandelbrot 函数。它们与现实问题中的 ODE、PDE、分数 ODE 和分数 PDE 相关。建议通过 Kohlrausch-Williams-Watts 函数的标度律级数理论来处理现实世界的问题。调和 Xi 函数的假设被提出为分形挑战，这是数学领域的一个新挑战。提出了分形几何在现实问题中的典型应用。