Fractal and Fractional

We present a finite difference/spectral method for the two-dimensional generalized time fractional cable equation by combining the second-order backward difference method in time and the Galerkin spectral method in space with Legendre polynomials. Through a detailed analysis, we demonstrate that the scheme is unconditionally stable. The scheme is proved to have $\min \{2-\alpha ,2-\beta \}$-order convergence in time and spectral accuracy in space for smooth solutions, where $\alpha ,\beta$ are two exponents of fractional derivatives. We report numerical results to confirm our error bounds and demonstrate the effectiveness of the proposed method. This method can be applied to model diffusion and viscoelastic non-Newtonian fluid flow. Full article

The time-fractional Cattaneo equation is an equation where the fractional order $\alpha \in (1,2)$ has the capacity to model the anomalous dynamics of physical diffusion processes. In this paper, we consider an efficient scheme for solving such an equation in two space dimensions. First, we obtain the space’s semi-discrete numerical scheme by using the compact difference operator in the spatial direction. Then, the semi-discrete scheme is converted to a low-order system by means of order reduction, and the fully discrete compact difference scheme is presented by applying the L2-1${}_{\sigma }$ formula. To improve the computational efficiency, we adopt the fast discrete Sine transform and sum-of-exponentials techniques for the compact difference operator and L2-1${}_{\sigma }$ difference operator, respectively, and derive the improved scheme with fast computations in both time and space. That aside, we also consider the graded meshes in the time direction to efficiently handle the weak singularity of the solution at the initial time. The stability and convergence of the numerical scheme under the uniform meshes are rigorously proven, and it is shown that the scheme has second-order and fourth-order accuracy in time and in space, respectively. Finally, numerical examples with high-dimensional problems are demonstrated to verify the accuracy and computational efficiency of the derived scheme. Full article

In our present investigation, a subclass of starlike function ${\mathcal{S}}_{n-1,\mathcal{L}}^{*}$ connected with a domain bounded by an epicycloid with $n-1$ cusps was considered. The main work is to investigate some coefficient inequalities, and second and third Hankel determinants for functions belonging to this class. In particular, we calculate the sharp bounds of the third Hankel determinant for $f\in {\mathcal{S}}_{4\mathcal{L}}^{*}$ with $\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}$ bounded by a four-leaf shaped domain under the unit disk $\mathbb{D}$. Full article

Motivated by the recent interest in generalized fractional order operators and their applications, we consider some classes of integro-differential initial value problems based on derivatives of the Riemann–Liouville and Caputo form, but with non-singular kernels. We show that, in general, the solutions to these initial value problems possess discontinuities at the origin. We also show how these initial value problems can be re-formulated to provide solutions that are continuous at the origin but this imposes further constraints on the system. Consideration of the intrinsic discontinuities, or constraints, in these initial value problems is important if they are to be employed in mathematical modelling applications. Full article

A new auxiliary result pertaining to twice (${\mathrm{q}}_1,{\mathrm{q}}_2$)-differentiable functions is derived. Using this new auxiliary result, some new versions of Hermite–Hadamard’s inequality involving the class of generalized 𝔪-convex functions are obtained. Finally, to demonstrate the significance of the main outcomes, some applications are discussed for hypergeometric functions, Mittag–Leffler functions, and bounded functions. Full article

We study a class of nonlinear fractional differential equations with multiple delays, which is represented by the Voigt creep fractional model of viscoelasticity. We discuss two Voigt models, the first being linear and the second being nonlinear. The linear Voigt model give us the physical interpretation and is associated with important results since the creep function characterizes the viscoelastic behavior of stress and strain. For the nonlinear model of Voigt, our theoretical study and analysis provides existence and stability, where time delays are expressed in terms of Boltzmann’s superposition principle. By means of the Banach contraction principle, we prove existence of a unique solution and investigate its continuous dependence upon the initial data as well as Ulam stability. The results are illustrated with an example. Full article

This paper mainly considers the parameter estimation problem for several types of differential equations controlled by linear operators, which may be partial differential, integro-differential and fractional order operators. Under the idea of data-driven methods, the algorithms based on Gaussian processes are constructed to solve the inverse problem, where we encode the distribution information of the data into the kernels and construct an efficient data learning machine. We then estimate the unknown parameters of the partial differential Equations (PDEs), which include high-order partial differential equations, partial integro-differential equations, fractional partial differential equations and a system of partial differential equations. Finally, several numerical tests are provided. The results of the numerical experiments prove that the data-driven methods based on Gaussian processes not only estimate the parameters of the considered PDEs with high accuracy but also approximate the latent solutions and the inhomogeneous terms of the PDEs simultaneously. Full article

This paper proposes an algorithm and hardware realization of generalized chaotic systems using fractional calculus and rotation algorithms. Enhanced chaotic properties, flexibility, and controllability are achieved using fractional orders, a multi-scroll grid, a dynamic rotation angle(s) in two- and three-dimensional space, and translational parameters. The rotated system is successfully utilized as a Pseudo-Random Number Generator (PRNG) in an image encryption scheme. It preserves the chaotic dynamics and exhibits continuous chaotic behavior for all values of the rotation angle. The Coordinate Rotation Digital Computer (CORDIC) algorithm is used to implement rotation and the Grünwald–Letnikov (GL) technique is used for solving the fractional-order system. CORDIC enables complete control and dynamic spatial rotation by providing real-time computation of the sine and cosine functions. The proposed hardware architectures are realized on a Field-Programmable Gate Array (FPGA) using the Xilinx ISE 14.7 on Artix 7 XC7A100T kit. The Intellectual-Property (IP)-core-based implementation generates sine and cosine functions with a one-clock-cycle latency and provides a generic framework for rotating any chaotic system given its system of differential equations. The achieved throughputs are $821.92$ Mbits/s and $520.768$ Mbits/s for two- and three-dimensional rotating chaotic systems, respectively. Because it is amenable to digital realization, the proposed spatially rotating translational fractional-order multi-scroll grid chaotic system can fit various secure communication and motion control applications. Full article

In this paper, by considering the notions of the invex set, Fréchet differentiability, invexity and pseudoinvexity for the involved functionals of curvilinear integral type, we establish some relations between the solutions of a class of weak vector variational inequalities and (weak) efficient solutions of the associated control problem. Full article

Langmuir waves propagate in fractal complex plasma with fractal characteristics, which may cause some plasma particles to be trapped or causes wave turbulences. This phenomenon appears in the form of fractional order equations. Using an effective unified solver, some new solitary profiles such as rational, trigonometrically and hyperbolical functions forms are discussed, using fractional derivatives in conformable sense. The fractional order modulates the solitary properties, such as amplitudes and widths. The proposition technique can be executed to study many applied science models. Full article

This paper addresses the problems of the stability of a nabla discrete distributed-order dynamical system (NDDS). Firstly, based on a proposed generalized definition of discrete integral, some related definitions of nabla discrete distributed-order calculus are given. Then, several useful inequalities in sense of nabla discrete fractional-order difference are extended to distributed-order cases. Meanwhile, on basis of the proposed inequalities and Lyapunov direct method, some sufficient conditions guaranteeing the asymptotic stability of the origin of NDDS are established under both the Caputo and Riemann–Liouville sense. Finally, some designed simulation examples are given to validate the correctness and practicability of the obtained results. Full article

In this paper, circuit implementation and anti-synchronization are studied in coupled nonidentical fractional-order chaotic systems where a fractance module is introduced to approximate the fractional derivative. Based on the open-plus-closed-loop control, a nonlinear coupling strategy is designed to realize the anti-synchronization in the fractional-order Rucklidge chaotic systems and proved by the stability theory of fractional-order differential equations. In addition, using the frequency-domain approximation and circuit theory in the Laplace domain, the corresponding electronic circuit experiments are performed for both uncoupled and coupled fractional-order Rucklidge systems. Finally, our circuit implementation including the fractance module may provide an effective method for generating chaotic encrypted signals, which could be applied to secure communication and data encryption. Full article

The authors of this paper systematically studied the mechanical properties and durability of concrete prepared with copper slag instead of natural aggregates. An analysis index was used to assess compressive strength, and a statistical method was used to establish a mix proportion design theory of copper slag aggregate concrete. The analysis was used to quantify the effect of copper slag aggregate concrete on resistance to chloride ion migration. Combined with the morphological analysis of SEM images and fractal calculations, the tests were used to explain the improvement mechanism of copper slag as a fine aggregate on concrete’s mechanics and durability from the microscopic mechanism perspective. The results showed that replacing a natural sand fine aggregate with copper slag improved the compressive strength of concrete, and the optimum replacement rate was found to be 40%. The influence of the water–cement ratio on the strength of copper slag aggregate concrete was exceptionally conspicuous—the more significant the water–cement proportion was the lower the compressive strength of the concrete. The optimum dosage of the water-reducing agent was found to be 3.8 kg/m³. A rapid chloride ion migration test and potential corrosion analysis showed that copper slag aggregate concrete’s initial density and corrosion resistance were higher than those of natural aggregate concrete. Electrochemical impedance spectroscopy analysis results showed that the structural concrete comprising copper slag aggregate instead of natural sand had a better anticorrosion effect on embedded steel bars. SEM morphology and fractal dimension analyses showed that the incorporation of steel slag aggregate decreased the initial damage to the concrete internal section. Full article

In this paper, we implement computational methods, namely the local fractional natural homotopy analysis method (LFNHAM) and local fractional natural decomposition method (LFNDM), to examine the solution for the local fractional Lighthill–Whitham–Richards (LFLWR) model occurring in a fractal vehicular traffic flow. The LWR approach preferably models the traffic flow and represents the traffic patterns via the supposition of speed–density equilibrium relationship and continuity equation. This model is mostly preferred for modeling of traffic flow because of its simple approach and interpretive ability to examine the qualitative patterns of traffic flow. The methods applied here incorporate the local fractional natural transform (LFNT) and derive the solutions for the LFLWR model in a closed form. Two examples are provided to demonstrate the accuracy and efficiency of the suggested methods. Furthermore, the numerical simulations have also been presented for each of the examples in the fractal domain. Additionally, the explored solutions for both examples have also been compared and are in good match with already existing solutions in literature. The methods applied in this work make the computational process easier as compared to other iterative methods and still provide precise solutions. Full article

The celebrated Korteweg–de Vries and Kadomtsev–Petviashvili (KP) equations are prototypical examples of integrable evolution equations in one and two spatial dimensions, respectively. The question of constructing integrable evolution equations in three-spatial dimensions has been one of the most important open problems in the history of integrability. Here, we study an integrable extension of the KP equation in three-spatial dimensions, which can be derived using a specific reduction of the integrable generalization of the KP equation in four-spatial and two-temporal dimensions derived in (Phys. Rev. Lett. 96, (2006) 190201). For this new integrable extension of the KP equation, we construct smooth multi-solitons, high-order breathers, and high-order rational solutions, by using Hirota’s bilinear method. Full article

This manuscript mainly discusses the approximate controllability for certain fractional delay evolution equations in Banach spaces. We introduce a suitable complete space to deal with the disturbance due to the time delay. Compared with many related papers on this issue, the major tool we use is a set of differentiable properties based on resolvent operators, rather than the theory of $C_0$-semigroup and the properties of some associated characteristic solution operators. By implementing an iterative method, some new controllability results of the considered system are derived. In addition, the system with non-local conditions and a parameter is also discussed as an extension of the original system. An instance is proposed to support the theoretical results. Full article

Cyclic loading always results in great damage to the pore structure and fractal characteristics of soft soil. Scanning electron microscope (SEM) can help collect data to describe the microstructure of soft soil. This paper conducted a series of SEM tests to interpret the effect of consolidation confining pressure, circulating dynamic stress ratios and overconsolidation ratio on soil’s micro-pore structure and fractal characteristics. The results demonstrate that fractal dimension can well represent the complex characteristics of the microstructure of the soil; the larger the consolidation confining pressure, the greater the cyclic dynamic stress ratio, and the greater the overconsolidation ratio, the smaller the fractal dimension number of soil samples. Finally, an empirical fitting formula for cumulative strain considering microstructure parameters is established through data fitting. Full article

A compact triple-band operation ultra-wideband (UWB) antenna with dual-notch band characteristics is presented in this paper. By inserting three metamaterial (MTM) square split-ring resonators (MTM-SSRRs) and a triangular slot on the radiating patch, the antenna develops measured dual-band rejection at 4.17–5.33 GHz and 6.5–8.9 GHz in the UWB frequency range (3–12 GHz). The proposed antenna offers three frequency bands of operation in the UWB range, which are between 3–4.17 GHz (~1.2 GHz bandwidth), 5.33–6.5 GHz (~1.17 GHz bandwidth), and 8.9–12 GHz (~3.1 GHz bandwidth), respectively. The higher resonating frequency band can be tuned/controlled by varying the width of the triangle slot, while the medium operational band can be controlled by adjusting the width of the SSRR slot. Initially, the simulated S-parameter response, 2D and 3D radiation patterns, gain, and surface current distribution of the proposed UWB inverted triangular antenna has been studied using epoxy glass FR4 substrate having parameters ε_r = 4.3, h = 1.6 mm, and tan δ = 0.025, respectively. In order to validate the simulation results, the proposed UWB antenna with dual-notch band characteristics is finally fabricated and measured. The fabricated antenna’s return-loss and far-field measurements show good agreement with the simulated results. The proposed antenna achieved the measured gain of 2.3 dBi, 4.9 dBi, and 5.2 dBi at 3.5 GHz, 6.1 GHz, and 9.25 GHz, respectively. Additionally, an in-depth comparative study is performed to analyze the performance of the proposed antenna with existing designs available in the literature. The results show that the proposed antenna is an excellent candidate for fifth-generation (5G) mobile base-stations, next-generation WiFi-6E indoor distributed antenna systems (IDAS), as well as C-band and X-band applications. Full article

The aim of this research is to identify an efficient model to describe the fluctuations around the trend of the soil temperatures monitored in the volcanic caldera of the Campi Flegrei area in Naples (Italy). This study focuses on the data concerning the temperatures in the mentioned area through a seven-year period. The research is initially finalized to identify the deterministic component of the model given by the seasonal trend of the temperatures, which is obtained through an adapted regression method on the time series. Subsequently, the stochastic component from the time series is tested to represent a fractional Brownian motion (fBm). An estimation based on the periodogram of the data is used to estabilish that the data series follows an fBm motion rather than fractional Gaussian noise. An estimation of the Hurst exponent H of the process is also obtained. Finally, an inference test based on the detrended moving average of the data is adopted in order to assess the hypothesis that the time series follows a suitably estimated fBm. Full article