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In this work, the Bogoyavlenskii system (BS) and fractal BS are investigated by variational method for the first time. An efficient and simple scheme is proposed to seek their exact solitary wave solutions, which is called variational analysis method. The novel scheme requires only two steps, making it much attractive in practical applications, and a good result is obtained. This paper cleans up the road to the exact solitions, and it sheds a new light on the soliton theory. Finally, the physical properties of solitary wave solutions obtained are analyzed by some simulation figures.

Based on the experimental measurement results of fluid particle transverse accelerations in fully developed pipe turbulence published in Nature (2001) by La Porta et al, the present authors recently develop a multiscale statistical model which considers both normal diffusion in molecular scale and anomalous diffusion in vortex scale. This model gives rise to a new probability density function, called Power-Stretched Gaussian Distribution model (PSGD). In this study, we make a further comparison of this statistical distribution model with the well-known Lévy distribution, Tsallis distribution and stretched-exponential distribution. Our model is found to have the following merits: 1) fewer parameters, 2) better fitting with experimental data, 3) more explicit physical interpretation.

A new calculus based on fractal subsets of the real line is formulated. In this calculus, an integral of order α, 0 < α ≤ 1, called F^α-integral, is defined, which is suitable to integrate functions with fractal support F of dimension α. Further, a derivative of order α, 0 < α ≤ 1, called F^α-derivative, is defined, which enables us to differentiate functions, like the Cantor staircase, "changing" only on a fractal set. The F^α-derivative is local unlike the classical fractional derivative. The F^α-calculus retains much of the simplicity of ordinary calculus. Several results including analogues of fundamental theorems of calculus are proved.

The integral staircase function, which is a generalization of the functions like the Cantor staircase function, plays a key role in this formulation. Further, it gives rise to a new definition of dimension, the γ-dimension.

Spaces of F^α-differentiable and F^α-integrable functions are analyzed. Analogues of Sobolev Spaces are constructed on F and F^α-differentiability is generalized using Sobolev-like construction.

F^α-differential equations are equations involving F^α-derivatives. They can be used to model sublinear dynamical systems and fractal time processes, since sublinear behaviors are associated with staircase-like functions which occur naturally as their solutions. As examples, we discuss a fractal-time diffusion equation, and one-dimensional motion of a particle undergoing friction in a fractal medium.

Calculus on fractals, or F^α-calculus, developed in a previous paper, is a calculus based fractals F ⊂ R, and involves F^α-integral and F^α-derivative of orders α, 0 < α ≤ 1, where α is the dimension of F. The F^α-integral is suitable for integrating functions with fractal support of dimension α, while the F^α-derivative enables us to differentiate functions like the Cantor staircase. Several results in F^α-calculus are analogous to corresponding results in ordinary calculus, such as the Leibniz rule, fundamental theorems, etc. The functions like the Cantor staircase function occur naturally as solutions of F^α-differential equations. Hence the latter can be used to model processes involving fractal space or time, which in particular include a class of dynamical systems exhibiting sublinear behaviour.

In this paper we show that, as operators, the F^α-integral and F^α-derivative are conjugate to the Riemann integral and ordinary derivative respectively. This is accomplished by constructing a map ψ which takes F^α-integrable functions to Riemann integrable functions, such that the corresponding integrals on appropriate intervals have equal values. Under suitable conditions, a restriction of ψ also takes F^α-differentiable functions to ordinarily differentiable functions such that their values at appropriate points are equal. Further, this conjugacy is generalized to one between Sobolev spaces in ordinary calculus and F^α-calculus.

This conjugacy is useful, among other things, to find solutions to F^α-differential equations: they can be mapped to ordinary differential equations, and the solutions of the latter can be transformed back to get those of the former. This is illustrated with a few examples.

In this paper, stability results of central concern for control theory are given for finite-dimensional linear and nonlinear local fractional or fractal differential systems. The main purpose of this paper is to provide some results on stability and asymptotic stability of conformable order systems, together with some illustrating examples.

In this paper, we mainly consider the Whitham–Broer–Kaup equations with unsmooth boundaries, and a new fractal derivative is adopted to construct the fractal model. The variational principles of the fractal Whitham–Broer–Kaup equations are successfully constructed by fractal Semi-inverse method, which is helpful to study the symmetry, discover the conserved quantity, and has a wide application prospect in numerical simulation. The results are discussed by the two-scale transform method. The approximate analytical solutions of the fractal Whitham–Broer–Kaup equations are obtained according to the variational iteration method and two-scale transform method.

Pull-in instability, as an inherent nonlinear problem, continues to become an increasingly important and interesting topic in the design of electrostatic Nano/Micro-electromechanical systems (N/MEMS) devices. Generally, the pull-in instability was studied in a continuous space, but when the electronic devices work in a porous medium, they need to be analyzed in a fractal partner. In this paper, we establish a fractal model for N/MEMS, and find a pull-in stability plateau, which can be controlled by the porous structure, and the pull-in instability can be finally converted to a stable condition. As a result, the pull-in instability can be completely eliminated, realizing the transformation of pull-in instability into pull-in stability.

Under the non-smooth condition, many theories obtained by the assumption on the smooth condition become invalid, so a generalized Burgers–Huxley equation (GBHE) with fractal derivative is introduced in this work. The fractal variational formulation for the problem is established by using the semi-inverse method, which provides conservation laws in an energy form and possible solution structures of the equation. The two-scale transform method and variational iteration method (VIM) are used to solve the fractal GBHE. The obtained results show a great agreement with the existed results.

In this paper, the Burgers–Benjamin–Bona–Mahony (Burgers–BBM) equation is described according to the use of the fractal derivative. The variational principle of the fractal Burgers–BBM equation is successfully established by adopting the fractal semi-inverse method, which is very helpful to research the structures of fractal solution for the solitary waves and construct the conservation law. Based on the proposed variational theory, a novel approximate method is successfully proposed for the fractal Burgers–BBM equation, which is called Fractal Variational Iteration Transform Method (FVITM). The example illustrates the FVITM is an excellent and fascinating tool for fractal models.

In this study, a new fractal derivative is employed to describe the nonlinear oscillator model. A variational principle of the fractal model is successfully established by using the fractal semi-inverse transform method, and its approximate frequency is obtained by a new fractal transform frequency formulation. The numerical example shows that the proposed technique is simple and accurate for fractal nonlinear oscillators.

Fractional operators are the main tools to describe diffusion problems on fractal objects. This paper first shows the equivalence between conformable fractional derivative and fractal derivative, and then investigates various models for the diffusion on fractal objects by the conformable fractional derivative and its generalized form. The recommended models are solved by the mesh-free physics-informed neural network method with high computational effectiveness and universality.

The well-known generalized Benjamin–Bona–Mahony–Burgers (gBBMB) equation is widely used in the fluid mechanics, but it becomes invalid under the non-smooth boundary. So this paper, for the first time ever, extends the gBBMB equation into the fractal form that still works under the non-smooth boundary. By using the semi-inverse method, we develop the fractal variational formulations for the problem, which can provide the conservation laws in an energy form, and reveal the possible solution structures of the equation. Furthermore, the two-scale transform method combined with the variational iteration method is used to solve the fractal gBBMB equation. The obtained results show a good agreement with the existed results.

In this work, the Schrödinger equation is described by the fractal derivative, and its variational principle is obtained by using the fractal semi-inverse method. The variational principle is helpful to research the construction of the solution. The approximate analytical solution of the fractal Schrödinger equation is obtained based on the proposed variational approach and the fractal two-scale transform method. Finally, an example shows that the proposed approach is very fascinating in deal with nonlinear fractal models.

The unsmooth boundary will greatly affect the motion morphology of ion-acoustic waves in plasma, so a modified equal width equation with fractal derivatives is proposed. The fractal variational formulation of the problem is established by using the semi-inverse method, which provides the conservation laws in an energy form in the fractal space and reveals the possible solution structures of the equation. Then He’s variational method based on the variational theory and Ritz-like method, combined with the two-scale transform is used to seek the periodic and solitary wave solutions of the fractal modified equal width equation. The obtained results show that the variational method is simple but powerful, which is expected to open some new perspectives toward the study of traveling wave theory in fractal space.

In this work, the nonlinear fractal $K(p,q)$ model is defined via the fractal derivative. The variational principle of the fractal model is constructed by using the fractal semi-inverse method, and it is of great significance to study the structure and properties of its soliton solution. Based on the established variational principle, we successfully proposed a new method to solve the fractal model. The efficiency and accuracy of the proposed approach are illustrated by an example.

Under a microgravity condition, the microgravity-induced motion of a rigid pendulum has many special properties, which cannot be successfully revealed by any known theory. This work suggests a new fractal model for this purpose to describe the microgravity-induced motion by using the fractal derivative, and its variational principle is successfully established according to the semi-inverse method. The variational principle is very helpful to construct conservation laws and to suggest solution structures of the new fractal model. The approximate analytical solution of the model is gained by the two-scale transform method and the homotopy perturbation method.

In this work, two different varieties of Zakharov–Kuznetsov models are defined by the fractal derivative, and their variational principles are favorably designed by using the fractal semi-inverse transform method. A novel approximate technology is suggested to elucidate the two fractal models which show that the obtained results are in good agreement. Finally, an example is considered to present the approximate solution which admits that the present method is a very powerful tool for various types of fractal models.

In this paper, the fractal derivative is employed to define the fractal Blasius equation in a microgravity space arising in fluid dynamics, and its fractal variational principle is successfully established by the fractal semi-inverse transform method. The approximate analytical solution of the fractal Blasius equation is obtained by the variational iteration method. Our results have theoretical significance and practical application value. The example shows the approximate approach is reliable and accurate.

In this work, the fractal variational principle of the fractal Ginzburg–Landau equation is established by employing fractal semi-inverse method, and its approximate analytical solution is obtained by fractal homotopy perturbation transform method. An example illustrates that the proposed technique is an excellent and fascinating method to deal with complex fractal models.

This paper argues that in the microgravity space, the fractal nonlinear oscillators’ models with discontinuities are established by the fractal calculus, and their fractal variational principles are obtained via using the fractal semi-inverse method. Finally, a novel technology is proposed to find the frequency of the fractal nonlinear oscillators’ model by using the Fourier series. The results illustrate that the proposed method is efficient and accurate.