Narrow Results

It is extremely difficult to establish the variational structure (VS) of the PDEs for fluids. This paper is concerned with the Benjamin–Ono equation for the fluids and develops the (VS)by applying the semi-inverse method (SIM). A detailed derivation process is given. The VS established in this paper is expected to bring some new inspiration for the study and application of the variational method.

In this paper, the shallow water wave with unsmooth boundaries is depicted by fractal calculus, and its exact fractal solitary wave solution is successfully found by He’s variational method. The numerical examples demonstrate that the method is simple, efficient and convenient. Finally, we elaborate the physical properties of the fractal solitary solution by some graphs. The novel scheme clears up the road to the exact fractal soliton, and it sheds a new light on the fractal soliton theory.

This paper is considered as a second-order redshift-based corrections in derivative of modified gravitational action, f(R), to explain the late time acceleration which is appeared by Supernova Type Ia (SNeIa) without considering the dark components. Here, we obtained the cosmological dynamic parameters of universe for this redshift depended corrections. Next, we used the recent data of SNeIa Union2, shift parameter of the cosmic background radiation, Baryon acoustic oscillation from sloan digital sky survey (SDSS), and combined analysis of these observations to put constraints on the parameters of the selected F(z) model. It is very interesting that the well-known age problem of the three old objects for combined observations can be alleviated in this model. Finally, the reference action will be constructed in terms of its Taylor expansion. Also, we show that the reconstructed action definitely pass the solar system and stability of the cosmological solution tests.

An anisotropic wave in a porous medium is a hot topic in the coastal protection. A fractal derivative model is established, and a variational principle is established for the anisotropic wave traveling. The variational principle reveals an energy conservation law during the traveling process.

Electrospinning is a complex process, and it can be modeled by a Bratu-type equation with fractal derivatives by taking into account the solvent evaporation. Though there are many analytical methods available for such a problem, e.g. the variational iteration method and the homotopy perturbation method, a straightforward method with a simple solution process and high accurate results is much needed. This paper applies the Taylor series technology to fractal calculus, and an analytical approximate solution is obtained. A fractal variational principle is also discussed. As the Taylor series is accessible to all non-mathematicians, this paper sheds a bright light on practical applications of fractal calculus.

A generalized KdV equation with fractal derivatives is suggested, and a special function is introduced to establish a fractal variational principle. A detailed derivation is elucidated step by step by the semi-inverse method, and some special cases are discussed.

This paper extends the $(2+1)$-dimensional Broer–Kaup equations in continuum mechanics to its fractional partner, which can model a lot of nonlinear waves in fractal porous media. Its derivation is demonstrated in detail by applying He’s fractional derivative. Using the semi-inverse method, two variational principles are established for the nonlinear coupled equations, which up to now are not discovered. The variational formulations can help to study the symmetries and find conserved quantities in the fractal space. The obtained variational principles are proved correct by minimizing the functionals with the calculus of variations, and might find potential applications in numerical simulation. The procedure reveals that the semi-inverse method is highly efficient and powerful, and can be generalized to other nonlinear evolution equations with fractal derivatives.

This paper gives a short remark on variational principle for the fractal Telegraph equation [K. L. Wang, S. W. Yao, Y. P. Liu et al., Fractals28(4) (2020) 2050058], the emphasis is put on temporal and spatial fractal derivatives and the derivation process of the fractal variational principle.

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.

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.

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.

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.

The unsmooth boundary has a great influence on the solitary wave form of a nonlinear wave equation. It this work, we for the first time ever propose the fractal regularized long-wave equation which can describe the shallow water waves under the unsmooth boundary (such as the fractal seabed). The fractal variational principle is established and is proved to have a strong minimum condition by the He–Weierstrass theorem. Then, the solitary wave solution is obtained by the fractal variational method which can reduce the order of differential equation and obtain the optimal solution by the stationary condition. Finally, the impact of the unsmooth boundary on the solitary wave is presented. It shows that the fractal order can affect the wave morphology, but cannot affect its peak value. The finding in this paper is important for the coast protection and expected to bring a light to the study of the fractal theoretical basis in the geosciences.

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.

The vibration of carbon nanotubes embedded in elastic medium is a hot topic and many research findings are available. But when the medium is a fractal medium such as chips, ceramics, etc., its vibration characteristics are difficult to explain with the existing theories. So in this paper, we mainly study the nonlinear vibration of carbon nanotube embedded in a fractal medium. The fractal variational principle of the problem is successfully established, and is proved to have a strong minimum condition according to the He–Weierstrass Theorem. Then the Hamiltonian theory combined with the residual equation is implemented to find the solution of the fractal nonlinear vibration. Finally, the impacts of different fractal orders on the vibration characteristics are also presented.

In this paper, two different types of plasma models with variable coefficients are described by using the fractal derivative. This paper aims at establishing fractal variational principles for two different types of discontinuous plasma physics by the fractal semi-inverse method, so that it can be applied to numerical analysis of the effect of the discontinuous time and discontinuous space on the solution properties.

In this work, the Kadomtsov–Petviashvili–Benjamin–Bona–Mahony model is described by the local fractional derivative (LFD) on Cantor sets. A novel algorithm is presented to seek the exact traveling wave solution of the nondifferentiable type for the local fractional Kadomtsov–Petviashvili–Benjamin–Bona–Mahony model based on the variational theory, which is called variational wave transform method (VWTM). This new algorithm provides a new idea for seeking the exact traveling wave solutions in fractal space with simplicity and efficiency. The physical properties of traveling wave solutions are described by some 3D simulation figures.

The Sharma–Tasso–Olver equation can well describe the wave motion in physics, however, it becomes ineffective when the boundary is non-smooth, so a modification of the equation is urgently needed. In this study, we derive a new fractal Sharma–Tasso–Olver equation that can model the wave motion with the non-smooth boundary by applying He’s fractal derivative. By means of the semi-inverse method, we successfully establish its fractal generalized variational principle, which provides the conservation laws in an energy form in the fractal space and reveals the possible solution structures of the equation. The obtained generalized variational principle can be used for the numerical and analytical studies of the solitary wave properties in the fractal PDEs.

The coupled Boussinesq equation (CBE) acts a key role in modeling the shallow water waves of two-layer fluid flow. However, it becomes powerless for the nonsmooth boundary. So, a fractal modification of the CBE is suggested in this paper. Aided by the semi-inverse method, we successfully construct its fractal generalized variational principle. In addition, three recent techniques, namely, the Sardar-subequation method, He’s frequency formulation method and subequation method, combined with the two-scale fractal dimension transform, are utilized to find abundant wave solutions. By these methods, various solutions expressed by the generalized hyperbolic functions, generalized trigonometric functions, hyperbolic functions and cosine function are obtained. With the aid of the mathematical software, the three-dimensional contours and two-dimensional curves are plotted to present the dynamic behaviors of the solutions. The results in this paper demonstrate that the proposed methods are powerful and effective to construct the traveling wave solutions of the fractal nonlinear evolution equations.

In this study, we present a fractal generalized fourth-order Boussinesq equation which can describe the shallow water waves with the non-smooth boundary (such as the fractal boundary). Aided by the semi-inverse method, we establish its variational principle, which is proved to have a strong minimum condition via the He–Weierstrass theorem. Then, two powerful approaches namely the variational method (VM) and energy balance theory (EBT) are utilized to search for the periodic wave solutions. As expected, the results obtained by the two methods are almost the same. Furthermore, the impact of the fractal orders on the periodic wave structure is illustrated via the 3D plot and 2D curve. The results of this paper are expected to provide a reference for the study of periodic wave theory in fractal space.