In this manuscript, we review Fractal calculus and the analogues of both local Fourier transform with its related properties and Fourier convolution theorem are proposed with proofs in Fractal calculus. The Fractal Dirac delta with its derivative and the Fractal Fourier transform of the Dirac delta is also defined. In addition, some important applications of the local Fractal Fourier transform are presented in this paper such as the Fractal electric current in a simple circuit, the Fractal second order ordinary differential equation, and the Fractal Bernoulli-Euler beam equation. All discussed applications are closely related to the fact that, in Fractal calculus, a useful local Fractal derivative is a generalized local derivative in the standard calculus sense. In addition, a comparative analysis is also carried out to explain the benefits of this Fractal calculus parameter on the basis of the additional alpha parameter, which is the dimension of the Fractal set, such that when α,= 1, we obtain the same results in the standard calculus.

In this paper, we introduce a generalized delta q−Mittag-Leffler function. Also, we solve some Caputo delta q−fractional dynamic equations and these solutions are expressed by means of the newly introduced delta q−Mittag-Leffler function.

Scatterings of electrons at quasiparticles or photons are very important for many topics in solid-state physics, e.g., spintronics, magnonics or photonics, and therefore a correct numerical treatment of these scatterings is very important. For a quantum-mechanical description of these scatterings, Fermi’s golden rule is used to calculate the transition rate from an initial state to a final state in a first-order time-dependent perturbation theory. One can calculate the total transition rate from all initial states to all final states with Boltzmann rate equations involving Brillouin zone integrations. The numerical treatment of these integrations on a finite grid is often done via a replacement of the Dirac delta distribution by a Gaussian. The Dirac delta distribution appears in Fermi’s golden rule where it describes the energy conservation among the interacting particles. Since the Dirac delta distribution is a not a function it is not clear from a mathematical point of view that this procedure is justified. We show with physical and mathematical arguments that this numerical procedure is in general correct, and we comment on critical points.

This paper explores the lateral vibration behavior of a micro cantilever beam with an open edge crack under axial load using the modified strain gradient theory (MSGT). A concentrated mass, incorporating its rotational inertia, is positioned at the beam's free end. The open edge crack is represented using the Dirac delta function. By employing MSGT, Hamilton's principle, and the Dirac delta function, the governing equations for system motion and relevant boundary conditions are derived to examine the size-dependence effects. Analytical solutions for the first and second natural frequencies of the cracked cantilever beam are provided, and validated through finite element modeling. The study further investigates the impact of various system parameters, including material length scale parameters, crack depth, crack location, cantilever beam length, axial load, and the presence of the concentrated mass, on the natural frequencies. The findings demonstrate that the crack depth, crack location, and material length scale parameters considerably influence the lateral vibration characteristics of the system. Notably, increasing the values of l_i/h from 0 to 0.25 leads to an approximate 40% rise in the natural frequency.

In this paper, we are introducing for the first time a generalized class named the class of $(\alpha,\beta,\gamma,\delta)-$convex functions of mixed kind. This generalized class contains many subclasses including the class of $(\alpha,\beta)-$convex functions of the first and second kind, $(s,r)-$convex functions of mixed kind, $s-$convex functions of the first and second kind, $P-$convex functions, quasi-convex functions and the class of ordinary convex functions. In addition, we would like to state the generalization of the classical Ostrowski inequality via fractional integrals, which is obtained for functions whose first derivative in absolute values is $(\alpha,\beta,\gamma,\delta)-$ convex function of mixed kind. Moreover, we establish some Ostrowski-type inequalities via fractional integrals and their particular cases for the class of functions whose absolute values at certain powers of derivatives are $(\alpha,\beta,\gamma,\delta)-$ convex functions of mixed kind using different techniques including H\"older's inequality and power mean inequality. Also, various established results would be captured as special cases. Moreover, the applications of special means will also be discussed.

example of a fully relativistic equation with both negative and positive energy solutions. The effect of the negative energy states was mitigated by maximizing the energy with respect to a relevant parameter while at the same time minimizing it with respect to another parameter in the wave function. The Cornell potential and a power-law scalar and vector potentials were used in our calculations for the quark confinement. Cares were taken to avoid the Klein paradox by the dominance of the scalar component over the vector part.