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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.

Every shallow-water wave propagates along a fractal boundary, and its mathematical model cannot be precisely represented by integer dimensions. In this study, we investigate a coupled fractal–fractional KdV system moving along an irregular boundary within the framework of variational theory, which is commonly employed to derive governing equations. However, not every fractal–fractional differential equation can be formulated using variational principles. The semi-inverse method proves to be challenging in finding an appropriate variational principle for nonlinear problems and eliminating extraneous components from the studied model. We consider the coupled fractal–fractional KdV system with arbitrary coefficients and establish its variational formulation to unveil the remarkable insights into the energy structure of the model and interrelationships among coefficients. Encouraging results are obtained for this coupled KdV system.

Denote by J_ν the Bessel function of the first kind of order ν and μ_ν,k is its kth positive zero. For ν > ½, a theorem of Lorch, Muldoon and Szegö states that the sequence is decreasing, another theorem of theirs states that the sequence has higher monotonicity properties. In the present paper, we proved that when ν > ½ the sequence has higher monotonicity properties and the properties imply those of the sequence of the local maxima of the function x^-ν+1|J_ν-1(x)|, x ∈ (0, ∞), i.e. the sequence has higher monotonicity properties.

In this work, we study the relativistic quantum motions of spin-zero scalar massive charged bosons in a topologically nontrivial four-dimensional rotating space-time in the presence of a uniform magnetic field and quantum flux. Afterwards, a spin-zero relativistic quantum oscillator model is also studied and determines the bound-state eigenvalue solutions of the quantum system. We see that the energy eigenvalue and the wave function get modified by the nontrivial topology of the geometry, the rotating frame of reference, and the magnetic field. We also observed the gravitational analog of the Aharonov–Bohm (AB) effect due to the dependence of the energy eigenvalue on the geometric quantum phase.

In this paper, we have attempted a fresh method to demonstrate how special functions and fractional calculus are used in real-world problems. Here, we have examined the glucose supply in human blood using the incomplete Aleph function (IAF) and the Caputo fractional operator. In this study, we used the incomplete Aleph function to find the blood glucose equation that is supplied to human blood. In terms of various hyper-geometric functions, we have also obtained several significant and unique results, and we have defined the blood glucose function in terms of IAF.

This paper’s objective is to arrange fractional active conditions while taking into account the Bessel–Maitland capability reported by Khan et al. The solution’s final form is expressed in terms of the Wright hypergeometric function, which allows us to define numerous additional fractional kinetic equation solutions. Generally, fractional kinetic condition is applied to many problems in astronomy and material science.