A NOVEL VARIATIONAL PERSPECTIVE TO FRACTAL WAVE EQUATIONS WITH VARIABLE COEFFICIENTS

This paper aims at establishing two different types of wave models with unsmooth boundaries by the fractal calculus, and their fractal variational principles are successfully designed by employing the fractal semi-inverse transform method. A new approximate technology is proposed to solve the two fractal models based on the variational principle and fractal two-scale transform method. Finally, two numerical examples show that the proposed method is efficient and accurate, which can be extended to solve different types of fractal models.

Nonlocal fractal calculus based analyses of electrical circuits on fractal set

Purpose The purpose of this paper is to present the analyses of electrical circuits with arbitrary source terms defined on middle b cantor set by means of nonlocal fractal calculus and to evaluate the appropriateness of such unconventional calculus. Design/methodology/approach The nonlocal fractal integro-differential equations describing RL, RC, LC and RLC circuits with arbitrary source terms defined on middle b cantor set have been formulated and solved by means of fractal Laplace transformation. Numerical simulations based on the derived solutions have been performed where an LC circuit has been studied by means of Lagrangian and Hamiltonian formalisms. The nonlocal fractal calculus-based Lagrangian and Hamiltonian equations have been derived and the local fractal calculus-based ones have been revisited. Findings The author has found that the LC circuit defined on a middle b cantor set become a physically unsound system due to the unreasonable associated Hamiltonian unless the local fractal calculus has been applied instead. Originality/value For the first time, the nonlocal fractal calculus-based analyses of electrical circuits with arbitrary source terms have been performed where those circuits with order higher than 1 have also been analyzed. For the first time, the nonlocal fractal calculus-based Lagrangian and Hamiltonian equations have been proposed. The revised contradiction free local fractal calculus-based Lagrangian and Hamiltonian equations have been presented. A comparison of local and nonlocal fractal calculus in terms of Lagrangian and Hamiltonian formalisms have been made where a drawback of the nonlocal one has been pointed out.

Fractal Calculus on Fractal Interpolation Functions

In this paper, fractal calculus, which is called Fα-calculus, is reviewed. Fractal calculus is implemented on fractal interpolation functions and Weierstrass functions, which may be non-differentiable and non-integrable in the sense of ordinary calculus. Graphical representations of fractal calculus of fractal interpolation functions and Weierstrass functions are presented.

Fractal Stochastic Processes on Thin Cantor-Like Sets

We review the basics of fractal calculus, define fractal Fourier transformation on thin Cantor-like sets and introduce fractal versions of Brownian motion and fractional Brownian motion. Fractional Brownian motion on thin Cantor-like sets is defined with the use of non-local fractal derivatives. The fractal Hurst exponent is suggested, and its relation with the order of non-local fractal derivatives is established. We relate the Gangal fractal derivative defined on a one-dimensional stochastic fractal to the fractional derivative after an averaging procedure over the ensemble of random realizations. That means the fractal derivative is the progenitor of the fractional derivative, which arises if we deal with a certain stochastic fractal.

Equilibrium and non-equilibrium statistical mechanics with generalized fractal derivatives: A review

Fractal calculus generalizes ordinary calculus, offering a way to differentiate otherwise non-differentiable domains and phenomena. This paper discusses the equilibrium and non-equilibrium statistical mechanics involving fractal structure, as well as fractal temperature in the partition function.

On fractional and fractal Einstein’s field equations

In this study, Einstein’s field equations are derived based on two dissimilar frameworks: the first is based on the concepts of “fractional velocity” and “fractal action” motivated by Calcagni’s approach to fractional spacetime while the second is derived based on fractal calculus which is a generalization of ordinary calculus that include fractal sets and curves. The fractional theory displays a breakdown of Lorentz invariance. It was observed that a spatially dependent cosmological constant emerges in the fractional theory. A connection between the fractional order parameter and the dimensionless parameter [Formula: see text] arising in the parameterized post-Newtonian (PPN) formalism is observed. A confrontation with very long-baseline radio interferometry targeting quasars 3C273 and 3C279 is done which proves that the fractional order parameter is within the range [Formula: see text]. Moreover, emergence of quantum Hawking radiation is realized in the theory supporting Hawking’s best calculations that black holes are not black. Nevertheless, based on the fractal calculus approach, there is a conservation of the Lorentz invariance and absence of spatially-dependent cosmological constant. The theory depends on the fractal order [Formula: see text] and gives rise to a fractal Schwarzschild radius of the massive body greater than the conventional radius besides a fractal Hawking’s temperature less than the standard one. However, the confrontation with radio interferometry targeting quasars 3C273 and 3C279 gives [Formula: see text].