Any physical laws are scale-dependent, the same phenomenon might lead to debating theories if observed using different scales. The two-scale thermodynamics observes the same phenomenon using two different scales, one scale is generally used in the conventional continuum mechanics, and the other scale can reveal the hidden truth beyond the continuum assumption, and fractal calculus has to be adopted to establish governing equations. Here basic properties of fractal calculus are elucidated, and the relationship between the fractal calculus and traditional calculus is revealed using the two-scale transform, fractal variational principles are discussed for 1-D fluid mechanics. Additionally planet distribution in the fractal solar system, dark energy in the fractal space, and a fractal ageing model are also discussed.

http://www.doiserbia.nb.rs/ft.aspx?id=0354-98362000065H

New promises and future challenges of fractal calculus: From two-scale thermodynamics to fractal variational principle

A brief introduction to the development of the homotopy perturbation method is given, and the main milestones are elucidated with more than 90 references. This paper further improves the method by ...

https://journals.sagepub.com/doi/pdf/10.1177/1461348418811028

Homotopy perturbation method with an auxiliary parameter for nonlinear oscillators

In this paper, we develop a new extended Kadomtsev–Petviashvili (eKP) equation We use the Painleve analysis to confirm the integrability of the eKP equation We derive the bilinear form, multiple soliton solutions and lump solutions via using the Hirota’s direct method Moreover, the soliton, breather and lump interaction solutions for this model are also obtained as well Graphs are drawn to illustrate the abundant dynamical behaviors of the obtained solutions

New extended Kadomtsev–Petviashvili equation: multiple soliton solutions, breather, lump and interaction solutions

On novel nondifferentiable exact solutions to local fractional Gardner's equation using an effective technique

In this manuscript, we solve a model of the novel coronavirus (COVID-19) epidemic by using Corrector-predictor scheme. For the considered system exemplifying the model of COVID-19, the solution is established within the frame of the new generalized Caputo type fractional derivative. The existence and uniqueness analysis of the given initial value problem are established by the help of some important fixed point theorems like Schauder’s second and Weissinger’s theorems. Arzela-Ascoli theorem and property of equicontinuity are also used to prove the existence of unique solution. A new analysis with the considered epidemic COVID-19 model is effectuated. Obtained results are described using figures which show the behaviour of the classes of projected model. The results show that the used scheme is highly emphatic and easy to implementation for the system of non-linear equations. The present study can confirm the applicability of the new generalized Caputo type fractional operator to mathematical epidemiology or real-world problems. The stability analysis of the projected scheme is given by the help of some important lemma or results.

Solution of a COVID-19 model via new generalized Caputo-type fractional derivatives

Under study in this paper is a time fractional generalized nonlinear diffusion equation which can be better to express diffusion phenomena than diffusion equation of integer order. Firstly, we appl...

On the generalized time fractional diffusion equation: Symmetry analysis, conservation laws, optimal system and exact solutions

In this paper, we mainly investigate the time fractional nonlinear wave equation which can be usually used to express nonlinear phenomena appearing in shallow water waves by using group analysis sc...

Jian-Gen Liu

Papers

On the generalized time fractional diffusion equation: Symmetry analysis, conservation laws, optimal system and exact solutions

Group analysis to the time fractional nonlinear wave equation

On integrability of the extended (3+1)‐dimensional Jimbo‐Miwa equation

On the (N+1)‐dimensional local fractional reduced differential transform method and its applications

On overall behavior of Maxwell mechanical model by the combined Caputo fractional derivative