Narrow Results

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.

This paper provides a summary of the fractal calculus framework. It presents higher-order homogeneous and nonhomogeneous linear fractal differential equations with $\alpha$-order. Solutions for these equations with constant coefficients are obtained through the method of variation of parameters and the method of undetermined coefficients. The solution space for higher $\alpha$-order linear fractal differential equations is defined, showcasing its non-integer dimensionality. The solutions to $\alpha$-order linear fractal differential equations are graphically depicted to illustrate their non-differentiability. Additionally, equations of motion governing the behavior of two masses in fractal time are proposed and solved.

In this paper, we present a summary of fractal calculus and propose the use of Lagrange multipliers for both fractal calculus and fractal variational calculus with constraints. We examine the application of these methods across various branches of physics. By employing fractal variational calculus with constraints, we derive fundamental equations such as the fractal mechanical wave equation, the fractal Schrödinger equation in quantum mechanics, Maxwell’s equations in fractal electromagnetism, and the Lagrange equation for constraints in fractal classical mechanics. Several examples are provided to illustrate these concepts in detail.

This paper introduces the fractal form of the generalized nonlinear Schrödinger equation, newly named as the Biswas–Milovic model (BM). The BM equation theoretically explains the transmission of solitons for transatlantic and transcontinental distances utilizing optical fibers. The BM equation relating to Kerr law, parabolic law and nonlinearity quadratic law was studied using a variational approach for optical soliton solutions. Essential novel conditions are presented that guarantee the existence of the appropriate solitons. Besides, the physical action of the solution obtained was recorded in terms of 3D and contour plots for distinct parameters for the three different nonlinearities. This study shows the relevance and huge potential of the variational approach to the generalized nonlinear Schrödinger equation.

The Fokas–Lenells (FL) equation is analyzed in this paper as an ironic physical function in optical fibers. A class of FL-equation soliton solutions is constructed by He’s variational principle. Besides, the fractal model of FL and its theory of variation are established. This paper focuses on the innovative research frontiers of FL equation.

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.

Tsunami traveling in an unsmooth boundary is studied, where the boundary is considered as a fractal surface, and mass conservation and moment conservation in a fractal space are established, and a fractal derivative-based tsunami model is established. Other discontinuous boundaries can be treated in a similar way.

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.

The semi-inverse method is adopted to establish a family of fractal variational principles of the one-dimensional compressible flow under the microgravity condition, and Cauchy–Lagrange integral is successfully derived from the obtained variational formulation. A suitable application of the Lagrange multiplier method is also elucidated.

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.

The nonlinear Schrödinger equation (NLSE) can well identify the development of waves in deep water and optical fibers towards the least-order approximation. This study addresses the Lakshmanan–Porsezian–Daniel (LPD) fractal model which emerges from the application of the Heisenberg spin chain and fiber optics. This paper analyzes three types of nonlinear rules, namely Kerr law, quadratic law, and parabolic law. The variational approach to the combination of the Ritz idea is used to discover the new optical soliton solutions for the LPD-equation. It poses the requisite novel conditions for ensuring the existence of valid solitons. Three- and two-dimensional configurations are demonstrated by choosing the correct values for the parameters. This study focused on the pioneering research boundaries of the LPD-equation and other associated nonlinear evolution models in the field of communications network technology and optical fiber.

In this study, the fractal nonlinear dispersive Boussineq-like equation is investigated by variational perspective for the first time. The fractal variational principle of the fractal Boussineq-like equation is established via fractal semi-inverse method (FSM). Based on the fractal variational principle, variational transform method (VTM) is proposed to find the exact fractal solitary wave solution for the fractal nonlinear dispersive Boussineq-like equation. The numerical examples illustrate the novel scheme is very simple and efficient. Moreover, some physical properties of fractal solitary wave solutions are illustrated by some simulation graphics.

This paper probes into change of box dimension for an arbitrary fractal continuous function after Hadamard fractional integration. For classic calculus, we know that a function’s differentiability increases one-order after integration, and decreases one-order after differentiation. But for fractional integro-differentiation, this similar fundamental problem is still open [F. B. Tatom, The relationship between fractional calculus and fractals, Fractals2 (1995) 217–229; M. Zähle and H. Ziezold, Fractional derivatives of Weierstrass-type functions, J. Comput. Appl. Math.76 (1996) 265–275; Y. S. Liang and W. Y. Su, Riemann–Liouville fractional calculus of 1-dimensional continuous functions, Sci. Sin. Math.4 (2016) 423–438 (in Chinese); Y. S. Liang, Progress on estimation of fractal dimensions of fractional calculus of continuous functions, Fractals26 (2019), doi:10.1142/S0218348X19500841] (see conjecture 1.1), although it is believed that the roughness of a fractal function is increased or decreased accounts $\alpha$ after $\alpha$-order fractional differentiation or integration. This paper partly answers this problem. Particularly, the estimation of box dimension of Hadamard fractional integral is of methodology for considering similar fractional integration. In this paper, it is proved that the upper box dimension of the graph of $f(x)$ does not increase after $\alpha$-order Hadamard fractional integral $D_H^{-\alpha }f(x)$, i.e.

The problem of the fractal Swift–Hohenberg model (FSHM) with variable coefficient is considered in this work based on the fractal derivative. First, the fractal variational principle (FVP) of the FSHM with variable coefficient is successfully established by employing the fractal semi-inverse method (FSIM), which is very helpful to investigate the structure of the analytical solution. Second, the fractal two-scale variational method (FTSVM) is established by combining the FVP and fractal two-scale transform method (FTSTM). Finally, an example is presented to illustrate the proposed method which is efficient and accurate. The proposed fractal two-scale variational method sheds new light on the nonlinear fractal models.

The purpose of this paper is to establish a novel algorithm to find the fractal solitary wave solution for fractal nonlinear Pochhammer–Chree model based on fractal variational theory, which is called variational transform method (VTM). The numerical examples are given to reveal the effectiveness and simplicity of the proposed method. Finally, some physical properties of the obtained fractal traveling wave solutions are illustrated by some graphs. The novel method sheds a new light on the fractal soliton theory.

In this paper, the fractal nonlinear dispersive model (FNDM) with variable coefficients is investigated by variational perspective, and its fractal variational principle is constructed by using the He’s fractal semi-inverse method. Based on the fractal variational principle, a novel analytical algorithm is successfully established to solve the fractal model. Two examples are given to illustrate the efficiency and accuracy of the proposed algorithm.

Non-local operators of differentiation are bestowed with capabilities of encompassing complex natural into mathematical equations. Symmetry as invariance under a specified group of transformations can allow for the concept to be applied extensively not only to spatial figures but also to abstract objects like mathematical expressions which can be said to be expressions of physical relevance, in particular dynamical equations. Derived from this point of view, it can be noted that the more complex physical problems are, the more complex mathematical operators of differentiation are required. Accordingly, the fractal–fractional operators (FFOs) are expanded into the complex plane in our research which revolves around a unique class of normalized analytic functions in the open unit disk. To bring FFOs (differential and integral) into the normalized class, the study aims to expand and modify them along with the investigation of the FFOs geometrically. The qualities of convexity and starlikeness are implicated in this study where the differential subordination technique serves as the foundation for the inquiry under consideration. Furthermore, a collection of differential FFO inequalities is taken into account, demonstrating that the normalized Fox–Wright function can contain all FFOs. Besides these steps, the concept of Grunsky factors is applied to investigate symmetry, while boundary value issues involving FFOs are probed. Consequently, the related properties and applications can be further developed, which requires the devotion to differential fractional problems and diverse complex problems in relation to viable applications, pointing out the room to modify and upgrade the existing methods for more optimal outcomes in challenging real-world problems.

In this work, the $(2+1)$-dimensional Kadomtsev–Petviashvili model is investigated. A novel variational scheme, namely, the variational transform wave method (VTWM), is successfully established to seek the solitary wave solution of the Kadomtsev–Petviashvili model. Furthermore, the fractal solitary solution of fractal Kadomtsev–Petviashvili model is also studied based on the local fractional derivative. Numerical examples are given to fully demonstrate that the VTWM is straightforward, efficient and attractive. Finally, the physical properties of solitary wave solutions are demonstrated by some 3D graphs.

This paper introduces the fractal version of the higher-order dispersion model for the construction of novel soliton solutions through fractal variational technology. Higher-order dispersion model theoretical study of the soliton propagation dynamics is known in the absence of self-phase modulation. In the context of negligibly small group velocity dispersion, this model involves higher-order spatio-temporal dispersion and can be a core component of the telecommunications industry. Using the variational approach, the model effectively produces bright and dark soliton solutions. Essential novel conditions guaranteeing the existence of suitable solitons have been developed. The 3D, 2D and contour graphs of the computed effects are seen in the collection of the relevant parameter values. This study shows the significance and immense latency of variational technologies to the derivative nonlinear Schrödinger equation (DNLSE).

To explain the origin of quantum behavior, we propose a fractal calculus to describe the non-local property of the fractal curve [Y. Tao, J. Appl. Math.2013 (2013) 308691]. This study demonstrates that if the dimension of time axis is slightly less than 1, then Planck’s energy quantum formula will naturally emerge. In this paper, we further show that if the dimension of time axis is less than 1, Heisenberg’s Principle of Uncertainty will emerge as well. Our finding implies that fractal calculus may be an intrinsic way of describing quantum behavior. To test our theory, we also provide an experimental proposal for measuring the dimension of time axis.