Fractals

Shale gas is produced by fracturing the reservoir to increase production. Quantifying the complex structure of the rock fracture network has a very important role in shale gas development. In order to quantify the complex structure of rock fracture network, in this paper, combined with the hierarchical structure of fractal figure, long tail distribution and fractal self-similarity, the current calculation method of fractal dimension is improved, and the basic fractal dimension calculation method is proposed based on the principle. By removing the limit calculation method which causes the deviation of numerical results, a reasonable and convenient calculation model for engineering application is proposed. A set of two-dimensional fracture pictures are used to test the model. In the 10 pictures, the more complex the figure is, the larger the new fractal dimension value is, which is in line with the characteristics of fractal dimension describing the complexity of graphics. Furthermore, lattice Boltzmann method (LBM) is used to simulate the velocity field, and the permeability of the fracture network is calculated according to Darcy’s law. The permeability and fractal dimension of the fracture network are compared and analyzed. The conclusion that the permeability is positively related to the fractal dimension of the fracture network is obtained, which provides theoretical support for the further study of shale mining problems such as hydraulic fracturing.

In recent years, fuzzy and fractional calculus are utilized for simulating complex models with uncertainty and memory effects. This study is focused on fuzzy-fractional modeling of (2+1)-dimensional Wu–Zhang (WZ) system. Caputo-type time-fractional derivative and triangular fuzzy numbers are employed in the model to observe uncertainties in the presence of non-local and memory effects. The extended He–Mohand algorithm is proposed for the solution and analysis of the current model. This approach is based on homotopy perturbation method along with Mohand transformation. Effectiveness of proposed methodology at upper and lower bounds is confirmed through residual errors. The theoretical convergence of proposed algorithm is proved alongside numerical computations. Existence and uniqueness of solution are also theoretically proved in the given paper. Current investigation considers three types of fuzzifications i.e. fuzzified equations, fuzzified conditions, and finally fuzzification in both model and conditions. Different physical aspects of WZ system profiles are analyzed through 2D and 3D illustrations at upper and lower bounds. The obtained results highlight the impact of uncertainty on WZ system in fuzzy-fractional space. Hence, the proposed methodology can be used for other fuzzy-fractional systems for better accuracy with lesser computational cost.

In this research, we use a novel version of the Extended Direct Algebraic Method (EDAM) namely generalized EDAM (gEDAM) to investigate periodic soliton solutions for nonlinear systems of fractional Schrödinger equations (FSEs) with conformable fractional derivatives. The FSEs, which is the fractional abstraction of the Schrödinger equation, grasp notable relevance in quantum mechanics. The proposed gEDAM technique entails creating nonlinear ordinary differential equations via a fractional complex transformation, which are then solved to acquire soliton solutions. Several 3D and contour graphs of the soliton solutions reveal periodicity in the wave profiles that offer crucial perspectives into the behavior of the system. The work sheds light on the dynamics of FSEs by displaying numerous families of periodic soliton solutions and their intricate relationships. These results hold significance not only for comprehending the dynamics of FSEs but also for nonlinear fractional partial differential equation applications.

This study investigates the nonlinear time-fractional Harry Dym ($𝕋𝔽ℍ𝔻$) equation, a model with significant applications in soliton theory and connections to various other nonlinear evolution equations. The Harry Dym ($ℍ𝔻$) equation describes the propagation of nonlinear waves in various physical contexts, including shallow water waves, nonlinear optics, and plasma physics. The fractional-order derivative introduces a memory effect, allowing the model to capture nonlocal interactions and long-range dependencies in the wave dynamics. The primary objective of this research is to obtain accurate analytical solutions to the $𝕋𝔽ℍ𝔻$ equation and explore its physical characteristics. We employ the Khater III method as the primary analytical technique and utilize the He’s variational iteration ($ℍ𝕀𝕍$) method as a numerical scheme to validate the obtained solutions. The close agreement between analytical and numerical results enhances the applicability of the solutions in practical applications of the model. This research contributes to a deeper understanding of the $ℍ𝔻$ equation’s behavior, particularly in the presence of fractional-order dynamics. The obtained solutions provide valuable insights into the complex interplay between nonlinearity and memory effects in the wave propagation phenomena described by the model. By shedding light on the physical characteristics of the $𝕋𝔽ℍ𝔻$ equation, this study paves the way for further investigations into its potential applications in diverse physical settings.

In this research, we introduce a new and generalized family of convex functions, entitled the $\alpha$-convex functions in the second sense with respect to a parameter and examine their important algebraic properties. Based on this novel convexity concept, we explore a new class of fractional integral inequalities for functions that are twice differentiable. These results are derived from fundamental identities obtained using classical Riemann–Liouville fractional integrals. To validate our findings, we provide 2D and 3D graphs of the main results. Furthermore, as an additional aspect of our study, we explore error estimates for differences of generalized means.

The main aim of this paper is to analyze the behavior of time-fractional Emden–Fowler (EF) equation associated with Caputo–Katugampola fractional derivative occurring in mathematical physics and astrophysics. A powerful analytical approach, which is an amalgamation of q-homotopy analysis approach and generalized Laplace transform with homotopy polynomials, is implemented to obtain approximate analytical solution of the time-fractional EF equation. Main advantage of this research work is that the implemented technique contains an auxiliary parameter to control the convergence region of obtained series solution. Some examples are considered to illustrate the accuracy and efficiency of the applied technique. Numerical results are demonstrated in the form of tabular and graphical representations.

This paper employs the monotonicity analysis for non-negativity to derive a class of sequential fractional backward differences of Riemann–Liouville type $\left({}_{a+1}^{\mathrm{\text{RL}}}{\nabla }^{\nu }{}_a^{\mathrm{\text{RL}}}{\nabla }^{\alpha }u\right)(t)$ based on a certain subspace in the parameter space $(0,1)\times (0,1)$. Auxiliary and restriction conditions are included in the monotonicity results obtained in this paper and they confirm the monotonicity of the function on $\{a+2,a+3,\dots \}$. A non-monotonicity result is also established based on the main conditions together with further dual conditions, and this confirms that the main theorem is almost sharp. Furthermore, we recast the dual conditions in a sing condition, and then we represent the sharpness result in a new corollary. Finally, numerical results via MATLAB software are used to illustrate the main mathematical results for some special cases.

The fractional nonlinear generalized $(3+1)$-dimensional Chaffee–Infante (FGCI) equation is a nonlinear evolution equation that governs the propagation of waves in dispersive media, exhibiting both dissipative and dispersive effects. It finds applications in various fields such as fluid mechanics, plasma physics, and nonlinear optics, where the interplay between nonlinearity, dispersion, and dissipation plays a crucial role in determining the dynamics of the system. This study aims to obtain analytical and numerical solutions for the FGCI equation, which is a generalization of the Chaffee–Infante equation. The analytical approach employed is the Khater II (Khat II) method, an efficient technique for constructing approximate analytical solutions to nonlinear fractional differential equations. Additionally, He’s variational iteration (HVI) method is implemented as a numerical scheme to validate the accuracy of the analytical solutions obtained from the Khat II method. The results obtained through these methods demonstrate the effectiveness of the proposed techniques in solving the FGCI equation and provide valuable insights into the dynamics of wave propagation phenomena governed by the equation. The significance of this study lies in its contribution to the understanding of nonlinear wave propagation phenomena and the development of analytical and numerical techniques for solving fractional differential equations. The novelty of the research lies in the application of the Khat II method to the FGCI equation, which has not been previously explored, and the validation of the analytical solutions through the HVI method. The study is conducted in the field of applied mathematics, with a focus on nonlinear fractional differential equations and their applications in various physical systems.

The method of solving the vibration frequency of local fractional nonlinear oscillators under general initial conditions is studied by coupling the energy balance method with the Galerkin method. One test case of fractional vibration is given to prove the feasibility of this method.

This paper presents the following AB-Caputo fractional boundary value problem

In this work, the variational direct method (VDM) based on the variational theory and Ritz-like method, combined with the fractal two-scale transformation, is employed to construct the semi-domain solutions of the (3+1)-dimensional fractal Jimbo–Miwa (JM) equation, which is derived by He’s fractal derivative. The applicability and effectiveness of the acquired solutions on the Cantor sets are presented through the numerical results in the form of 3-dimensional graphs. In addition, as a comparison, the exact solutions of the classic (3+1)-dimensional JM equation for $\gamma =1$ are also illustrated. The obtained results in this work are expected to open new perspectives for the traveling wave theory.

This paper proposes a mathematical model to investigate the dynamic behaviors of a modified computer virus model using the Atangana–Baleanu fractional derivative in the Caputo sense, aiming to elucidate the connection between its parameters and network attributes. By introducing a relatively new numerical method, we address the memory-dependent and nonlocal features of the system. The existence and uniqueness of the model’s solution are confirmed. To explore solution trajectories and assess the impact of various input factors on computer virus dynamics, we employ an efficient numerical technique. Our simulations provide insights into the consequences of fractional order, anti-virus measures, asymptotic fraction, damage rate, and removal rate in the system. These findings illuminate the relationships between model parameters, facilitating the design of networks that minimize the risk of virus outbreaks and prevent future cyber threats. By identifying critical factors involved in the progression of viruses, these results enable the development of more effective defense mechanisms.

This paper is devoted to introducing a new $q$-fractional calculus in the framework of Atangana–Baleanu (${𝒜}{ℬ}$) and Caputo–Fabrizio (${𝒞}{ℱ}$) operators. First, an appropriate $q$-Mittag-Leffler function is defined, and then $q$-analogues of fractional derivatives of Atangana–Baleanu–Riemann (${𝒜}{ℬ}{ℛ}$) and Atangana–Baleanu–Caputo (${𝒜}{ℬ}{𝒞}$) are derived. Next, the $q$-analogues of proper fractional integrals in the ${𝒜}{ℬ}$ sense are proved. Several important properties of these definitions are investigated by using the $q$-Laplace transform. Additionally, a suitable $q$-exponential function is defined, and the $q$-analogues of ${𝒞}{ℱ}$ fractional derivatives with their inverse operators are introduced. The higher-order extension of the $q$-analogues of ${𝒜}{ℬ}$ and ${𝒞}{ℱ}$ fractional operators is discussed. Finally, a demonstrative example is enhanced to check the effectiveness of $q$-${𝒜}{ℬ}{𝒞}$ calculus. We believe that these outcomes will be the care of many researchers in the field of fractional calculus.

This paper is devoted to studying some new reverse Hölder-type inequalities associated with the dynamic integral called diamond on time scale, which is expressed as an ‘approximate’ symmetric integral on time scales. Some related inequalities are also presented. The obtained inequalities extend known results related with discrete and continuous forms.

Fins serve the purpose of enhancing heat transfer between their surroundings and the underlying surface, resembling the function of heat exchangers in various scientific domains. This research undertakes an investigation into the temperature distribution within convective straight fins, considering the impact of both constant and thermally conductive temperature-dependent properties. The chosen approach utilizes a numerical technique known as the Laguerre operational matrix collocation method, which relies on the Laguerre operational matrix and Laguerre collocation method for Caputo fractional derivatives of arbitrary orders. This methodology effectively converts the issue into a set of nonlinear algebraic equations (NLAEs). To evaluate this arbitrary order model, we employed the collocation method to obtain an approximate solution. The numerical results obtained through this approach demonstrate its ease of application and computational efficiency. In addition, we performed a comparative analysis with the Homotopy Perturbation Sumudu Transform Method (HPSTM) and the Variational Iteration Method (VIM). The numerical and graphical presentation of the results underscores the efficacy and precision of the methodologies employed within the scope of this investigation.

The netted Collocated Multiple-Input Multiple-Output (C-MIMO) radar systems have attracted widespread attention due to their high resolution and excellent target detection capabilities. The self-similarity and spatial filling properties of fractal geometry enable antenna layouts to function effectively across multiple scales and frequency bands, which helps improve the overall performance and resource utilization of systems. In this paper, a radar array element model based on the Fudgeflake fractal is used to establish a target localization performance model directly related to power bandwidth allocation. Based on the model, a three-step solving strategy was designed with the Sequence Quadratic Program (SQP) algorithm as the foundation to achieve optimized allocation of power and bandwidth while maximizing overall performance. Experimental results demonstrate that under this algorithm, the use of power bandwidth resources is optimized through fractal geometry-based design.

In this paper, we investigate the fractal nature of the local fractional Landau–Ginzburg–Higgs Equation (LFLGHE) describing nonlinear waves with weak scattering in a fractal medium. The main goal of the paper is to introduce and apply the Local Fractional Elzaki Variational Iteration Method (LFEVIM) for solution of LFLGHE. Convergence analysis of LFEVIM solution for general nonlinear local fractional partial differential equation is also provided. Two examples of the local fractional LFLGHE are considered to demonstrate the applicability of the proposed technique with numerical simulations on Cantor set.

In this paper, a three-dimensional mathematical model of cancer, incorporating tumor cells, host normal cells, and effector immune cells, is examined. The chaotic nature of this cancer model is explored within the framework of Caputo fractional calculus. The analysis employs local stability assessment using the Matignon criteria, Lyapunov exponents (LEs) for various fractional orders of derivatives, bifurcation plots reflecting changes in the fractional derivative order, simulation outcomes of a novel chaotic attractor, Kaplan–Yorke dimension, and sensitivity to initial conditions. The results demonstrate that the Caputo fractional cancer model exhibits chaotic behavior for selected parameters. Moreover, the fractional derivative orders significantly influence the extent of chaotic behavior. Specifically, as the fractional derivative order increases from zero to one, the intensity of chaos diminishes, evidenced by a decrease in both the LE and the Kaplan–Yorke dimension. A piecewise modeling approach is applied to the cancer model, incorporating deterministic, stochastic, and power-law behaviors. Three distinct scenarios are developed within this framework, revealing several novel crossover behaviors through simulation. Notably, it is observed that when cancer dynamics initially follow a power-law behavior and subsequently transition into a stochastic process, the disease dynamics can stabilize, suggesting a positive outlook for disease control. Conversely, if the cancer dynamics begin with a deterministic process and then shift to a stochastic process, the system may enter a chaotic state, complicating disease management efforts.