•Definition of fractional derivative is such a mess that a new replacement is needed.•Fractal calculus is tutorially introduced from very beginning, and it is accessible to all audience.•Its ...geometrical explanation and applications are elucidated.
Fractal calculus is very simple but extremely effective to deal with phenomena in hierarchical or porous media. Its operation is almost same with that by the advanced calculus, making it much accessible to all non-mathematicians. This paper begins with the basic concept of fractal gradient of temperature, i.e., the temperature change between two points in a fractal medium, to reveal the basic properties of fractal calculus. The fractal velocity and fractal material derivative are then introduced to deduce laws for fluid mechanics and heat conduction in fractal space. Conservation of mass in a fractal space is geometrically explained, and an approximate transform of a fractal space on a smaller scale into its continuous partner on a larger scale is illustrated by a nanofiber membrane, which is smooth on any observable scales, but its air permeability has to studied in a nano scale, under such a small scale, the nanofiber membrane becomes a porous one. Finally an example is given to explain cocoon’s heat-proof property, which cannot be unveiled by advanced calculus.
In engineering, a fast estimation of the periodic property of a nonlinear oscillator is much needed. This paper reviews some simplest methods for nonlinear oscillators, including He’s frequency ...formulation, the max-min approach and the homotopy perturbation method. A mathematical insight into He’s frequency formulation is given, and the weighted average is introduced to further improve the estimated accuracy of the frequency. Fractional oscillators are also discussed.
Purpose
A three-dimensional (3D) unsteady potential flow might admit a variational principle. The purpose of this paper is to adopt a semi-inverse method to search for the variational formulation ...from the governing equations.
Design/methodology/approach
A suitable trial functional with a possible unknown function is constructed, and the identification of the unknown function is given in detail. The Lagrange multiplier method is used to establish a generalized variational principle, but in vain.
Findings
Some new variational principles are obtained, and the semi-inverse method can easily overcome the Lagrange crisis.
Practical implications
The semi-inverse method sheds a promising light on variational theory, and it can replace the Lagrange multiplier method for the establishment of a generalized variational principle. It can be used for the establishment of a variational principle for fractal and fractional calculus.
Originality/value
This paper establishes some new variational principles for the 3D unsteady flow and suggests an effective method to eliminate the Lagrange crisis.
Purpose
It is extremely difficult to establish a variational principle for plasma. Kalaawy obtained a variational principle by using the semi-inverse method in 2016, and Li and He suggested a ...modification in 2017. This paper aims to search for a generalized variational formulation with a free parameter.
Design/methodology/approach
The semi-inverse method is used by suitable construction of a trial functional with some free parameters.
Findings
A modification of Li-He’s variational principle with a free parameter is obtained.
Originality/value
This paper suggests a new approach to construction of a trial-functional with some free parameters.
This paper gives the simplest approach to the cubic-quintic Duffing equation (M.S.H. Chowdhury et al., Results in Physics 7(2017): 3962–3967), providing an extremely fast and relatively accurate ...estimation of the frequency of a nonlinear conservative oscillator.
•The Kundu–Mukherjee–Naskar equation admits both solitary solutions and periodic solution.•A variational formulation is established.•An approximate periodic solution is obtained.
...Kundu–Mukherjee–Naskar equation admits many attractive solitary wave solutions as discussed by Biswas et al. (Results in Physics, Volume16, 2020, 102850). This paper shows that a variational formulation can be established by the semi-inverse method, and its periodic solution is elucidated. This short note sheds a light on new research frontiers of solitary wave theory.
Purpose
This paper aims to review some effective methods for fully fourth-order nonlinear integral boundary value problems with fractal derivatives.
Design/methodology/approach
Boundary value ...problems arise everywhere in engineering, hence two-scale thermodynamics and fractal calculus have been introduced. Some analytical methods are reviewed, mainly including the variational iteration method, the Ritz method, the homotopy perturbation method, the variational principle and the Taylor series method. An example is given to show the simple solution process and the high accuracy of the solution.
Findings
An elemental and heuristic explanation of fractal calculus is given, and the main solution process and merits of each reviewed method are elucidated. The fractal boundary value problem in a fractal space can be approximately converted into a classical one by the two-scale transform.
Originality/value
This paper can be served as a paradigm for various practical applications.
A generalized variational principle and a parameterized generalized variational principle are obtained for large deformation analysis of circular cylinders by the semi-inverse method; all known ...variational principles in the literature are special cases of the obtained parameterized functional. In this approach, a trial functional is constructed with an energy-like integral involving an unknown function, which is identified step by step. The present paper provides a quite straightforward but rigorous tool to the construction of a variational principle for the shell or plate buckling.
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.
nema
Most patients with triple negative breast cancer (TNBC) do not respond to anti-PD1/PDL1 immunotherapy, indicating the necessity to explore immune checkpoint targets. B7H3 is a highly glycosylated ...protein. However, the mechanisms of B7H3 glycosylation regulation and whether the sugar moiety contributes to immunosuppression are unclear. Here, we identify aberrant B7H3 glycosylation and show that N-glycosylation of B7H3 at NXT motif sites is responsible for its protein stability and immunosuppression in TNBC tumors. The fucosyltransferase FUT8 catalyzes B7H3 core fucosylation at N-glycans to maintain its high expression. Knockdown of FUT8 rescues glycosylated B7H3-mediated immunosuppressive function in TNBC cells. Abnormal B7H3 glycosylation mediated by FUT8 overexpression can be physiologically important and clinically relevant in patients with TNBC. Notably, the combination of core fucosylation inhibitor 2F-Fuc and anti-PDL1 results in enhanced therapeutic efficacy in B7H3-positive TNBC tumors. These findings suggest that targeting the FUT8-B7H3 axis might be a promising strategy for improving anti-tumor immune responses in patients with TNBC.
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