Fractal Fract., Volume 6, Issue 6 (June 2022) – 62 articles

In this article, a variable step size strategy is adopted in formulating a new variable step hybrid block method (VSHBM) for the solution of the Kepler problem, which is known to be a rigid and stiff differential equation. To derive the VSHBM, the step size ratio r is left the same, halved, or doubled in order to optimize the total number of steps, minimize the number of formulae stored in the code, and ensure that the method is zero-stable. The method is formulated by integrating the Lagrange polynomial with limits of integration selected at special points. The article further analyzed the stability, order, consistency, and convergence properties of the VSHBM. The stability regions of the VSHBM at different values of the step size ratios were also plotted and plots showed that the method is fit for solving the Kepler problem. The results generated were then compared with some existing methods, including the MATLAB inbuilt stiff solver (ode 15 s), with respect to total number of failure steps, total number of steps, total function calls, maximum error, and computation time. Full article

Viscoelasticity and variable mass are common phenomena in Micro-Electro-Mechanical Systems (MEMS), and could be described by a fractional derivative damping and a stochastic process, respectively. To study the dynamic influence cased by the viscoelasticity and variable mass, stationary response of a kind of nonlinear stochastic systems with stochastic variable-mass and fractional derivative, damping is investigated in this paper. Firstly, an approximately equivalent system of the studied nonlinear stochastic system is presented according to the Taylor expansion technique. Then, based on stochastic averaging of energy envelope, the corresponding Fokker–Plank–Kolmogorov (FPK) equation is deduced, which gives an approximated analytical solution of stationary response. Finally, a nonlinear oscillator with variable mass and fractional derivative damping is proposed in numerical simulations. The approximated analytical solution is compared with Monte Carlo numerical solution, which could verify the effectiveness of the obtained results. Full article

In order to simulate the grinding surface more accurately, a novel modeling method is proposed based on the ubiquitiform theory. Combined with the power spectral density (PSD) analysis of the measured surface, the anisotropic characteristics of the grinding surface are verified. Based on the isotropic fractal Weierstrass–Mandbrot (W-M) function, the expression of the anisotropic fractal surface is derived. Then, the lower bound of scale invariance δ_min is introduced into the anisotropic fractal, and an anisotropic W-M function with ubiquitiformal properties is constructed. After that, the influence law of the δ_min on the roughness parameters is discussed, and the δ_min for modeling the grinding surface is determined to be 10⁻⁸ m. When δ_min = 10⁻⁸ m, the maximum relative errors of Sa, Sq, Ssk, and Sku of the four surfaces are 5.98%, 6.06%, 5.77%, and 4.53%, respectively. In addition, the relative errors of roughness parameters under the fractal method and the ubiquitiformal method are compared. The comparison results show that the relative errors of Sa, Sq, Ssk, and Sku under the ubiquitiformal modeling method are 5.36%, 6.06%, 5.84%, and 4.53%, while the maximum relative errors under the fractal modeling method are 23.21%, 7.03%, 83.10%, and 7.25%. The comparison results verified the accuracy of the modeling method in this paper. Full article

This work considers the three-dimensional incompressible rotating magnetohydrodynamics equation spaces with fractional dissipation ${(-\Delta )}^{\wp }$ for $\frac{1}{2}<\wp \le 1$. Furthermore, we use the Littlewood–Paley decomposition and frequency localization techniques to establish the global well-posedness of fractional rotating magnetohydrodynamics equations in a more generalized Besov spaces characterized by the time evolution semigroup related to the generalized linear Stokes–Coriolis operator. Full article

Milstein and approximate coupling approaches are compared for the pathwise numerical solutions to stochastic differential equations (SDE) driven by Brownian motion. These methods attain an order one convergence under the nondegeneracy assumption of the diffusion term for the approximate coupling method. We use MATLAB to simulate these methods by applying them to a particular two-dimensional SDE. Then, we analyze the performance of both methods and the amount of time required to obtain the result. This comparison is essential in several areas, such as stochastic analysis, financial mathematics, and some biological applications. Full article

In this paper, a (4+1)-dimensional nonlinear integrable Fokas equation is studied. It is rarely studied because the order of the highest-order derivative term of this equation is higher than the common generalized (4+1)-dimensional Fokas equation. Firstly, the (4+1)-dimensional time-fractional Fokas equation with the Riemann–Liouville fractional derivative is derived by the semi-inverse method and variational method. Further, the symmetry of the time-fractional equation is obtained by the fractional Lie symmetry analysis method. Based on the symmetry, the conservation laws of the time fractional equation are constructed by the new conservation theorem. Then, the $\left(\frac{G^{\prime }}{G}\right)$-expansion method is used here to solve the equation and obtain the exact traveling wave solutions. Finally, the spectral method in the spatial direction and the Gr$\ddot{u}$nwald–Letnikov method in the time direction are considered to obtain the numerical solutions of the time-fractional equation. The numerical solutions are compared with the exact solutions, and the error results confirm the effectiveness of the proposed numerical method. Full article

Magnesium phosphate cement (MPC) paste is hardened by the acid–base reaction between magnesium oxide and phosphate. This work collects and evaluates the thermodynamic data at 25 °C and a pressure of 0.1 MPa and establishes the hydration reaction model of MPC pastes. The influence of the magnesium–phosphorus molar (M/P) ratio and water-to-binder (W/B) ratio on the hydration product is explored by the thermodynamic simulation. Following this, the initial and ultimate states of the hydration state of MPC pastes are visualized, and the porosity of different pastes as well as fractal analysis are presented. The result shows that a small M/P ratio is beneficial for the formation of main hydration products. The boric acid acts as a retarder, has a significant effect on lowering the pH of the paste, and slows down the formation of hydration products. After the porosity comparison, it can be concluded that the decreasing of M/P and W/B ratios helps reduce porosity. In addition, the fractal dimension D_f of MPC pastes is positively proportional to the porosity, and small M/P ratios as well as small W/B ratios are beneficial for reducing the D_f of MKPC pastes. Full article

In this paper, to investigate mixed-mode I-II fracture behaviors, three different asymmetric notched semi-circular bending specimens (ANSCB) were designed by adjusting the angle and the distance between supporting rollers to conduct asymmetric three-point bending tests. Several aid technologies, including acoustic emission (AE), digital image correlation (DIC), crack propagation gauge (CPG), and scanning electron microscopy (SEM), was utilized to monitor and assess the fracture characteristic. Meanwhile, the fractal dimension of the fracture surface was assessed based on the reconstructed digital fracture surface. The results show that mixed-mode I-II ANSCB three-point bending fracture is a brittle failure with the characteristics of the main crack being rapidly transfixed and the bearing capacity decreasing sharply. Based on the DIC method, the whole fracture process consists of a nonlinear elastic stage, fracture process zone, crack initiation stage and crack propagation stage. The crack initiation is mainly caused by the tension-shear strain concentration at the pre-existing crack tip. At the microscale, the crack propagation path is always along the grain boundary where the resultant stress is weakest. According to the monitoring of the AE, it can be found that micro-tensile cracks are mainly responsible for the asymmetric three-point bending fracture. The data obtained by CPG suggest that the subcritical crack growth rate is positively correlated to the ultimate load. In addition, asymmetric loading leads to a coarser fracture surface, and thus a higher fractal dimension of the fracture surface. The current study can provide a better understanding of the mixed-mode I-II fracture behaviors of rock. Full article

The crack resistance of face slab concretes to various shrinkages is crucial for the structural integrity and the normal operation of concrete-faced rockfill dams (CFRDs). In this work, the effects of fly ash with four dosages (i.e., 10%, 20%, 30% and 40%) on the drying shrinkage, autogenous shrinkage and the cracking resistance of face slab concrete were studied. Besides, the difference in shrinkage behavior due to fly ash addition was revealed from the viewpoint of the pore structure and fractal dimension of the pore surface (D_s). The findings demonstrate that (1) the incorporation of 10–40% fly ash could slightly reduce the drying shrinkage by about 2.2–13.5% before 14 days of hydration, and it could reduce the drying shrinkage at 180 days by about 5.1–23.2%. By contrast, the fly ash addition could markedly reduce the autogenous shrinkage at early, middle and long-term ages. (2) Increasing fly ash dosage from 0 to 40% considerably improves the crack resistance of concrete to plastic shrinkage. Nevertheless, the increase in fly ash dosage increases the drying-induced cracking risk under restrained conditions. (3) The pore structures of face slab concrete at 3 and 28 days become coarser with the increase in fly ash dosage up to 40%. At 180 days, the pore structures become more refined as the fly ash dosage increases to 30%; however, this refinement effect is not as appreciable as the fly ash dosage increases from 30% to 40%. (4) The D_s of face slab concrete is closely related with the concrete pore structures. The D_s of face slab concrete at a. late age increases from 2.902 to 2.946 with increasing of the fly ash dosage. The pore structure and D_s are closely correlated with the shrinkage of face slab concrete. (5) The fly ash dosage around 30% is optimal for face slab concretes in terms of lowering shrinkage and refining the pore structures, without compromising much mechanical property. However, the face slab concretes with a large fly ash dosage should be well cured under restrained and evaporation conditions at an initial hydration age. Full article

This study aims to identify soliton structures as an inherent fractional discrete nonlinear electrical transmission lattice. Here, the analysis is founded on the idea that the electrical properties of a capacitor typically contain a non-integer-order time derivative in a realistic system. We construct a non-integer order nonlinear partial differential equation of such voltage dynamics using Kirchhoff’s principles for the model under study. It was discovered that the behavior for newly generated soliton solutions is impacted by both the non-integer-order time derivative and connected parameters. Regardless of structure, the fractional-order alters the propagation velocity of such a voltage wave, thus bringing up a localized framework under low coupling coefficient values. The generalized auxiliary equation method drove us to these solitary structures while employing the modified Riemann–Liouville derivatives and the non-integer order complex transform. As well as addressing sensitivity testing, we also investigate how our model’s altered dynamical framework shows quasi-periodic properties. Some randomly selected solutions are shown graphically for physical interpretation, and conclusions are held at the end. Full article

This work presents the numerical performances of the fractional kind of food supply (FKFS) model. The fractional kinds of the derivatives have been used to acquire the accurate and realistic solutions of the FKFS model. The FKFSM system contains three types, special kind of the predator L(x), top-predator M(x) and prey populations N(x). The numerical solutions of three different cases of the FKFS model are provided through the stochastic procedures of the scaled conjugate gradient neural networks (SCGNNs). The data selection for the FKFS model is chosen as 82%, for training and 9% for both testing and authorization. The precision of the designed SCGNNs is provided through the achieved and Adam solutions. To rationality, competence, constancy, and correctness is approved by using the stochastic SCGNNs along with the simulations of the regression actions, mean square error, correlation performances, error histograms values and state transition measures. Full article

The time scaling exponent for the analytical expression of capillary rise $\ell \sim t^{\delta }$ for several theoretical fractal curves is derived. It is established that the actual distance of fluid travel in self-avoiding fractals at the first stage of imbibition is in the Washburn regime, whereas at the second stage it is associated with the Hausdorff dimension $d_{\mathcal{H}}$. Mapping is converted from the Euclidean metric into the geodesic metric for linear fractals $\mathcal{F}$ governed by the geodesic dimension $d_g=d_{\mathcal{H}}/d_{\ell }$, where $d_{\ell }$ is the chemical dimension of $\mathcal{F}$. The imbibition measured by the chemical distance ${\ell }_g$ is introduced. Approximate spatiotemporal maps of capillary rise activity are obtained. The standard differential equations proposed for the von Koch fractals are solved. Illustrative examples to discuss some physical implications are presented. Full article

We deal with backward stochastic differential equations driven by a pure jump Markov process and an independent Brownian motion (BSDEJs for short). We start by proving the existence and uniqueness of the solutions for this type of equation and present a comparison of the solutions in the case of Lipschitz conditions in the generator. With these tools in hand, we study the existence of a (minimal) solution for BSDE where the coefficient is continuous and satisfies the linear growth condition. An existence result for BSDE with a left-continuous, increasing and bounded generator is also discussed. Finally, the general result is applied to solve one kind of quadratic BSDEJ. Full article

Navier–Stokes (NS) equation, in fluid mechanics, is a partial differential equation that describes the flow of incompressible fluids. We study the fractional derivative by using fractional differential equation by using a mild solution. In this work, anomaly diffusion in fractal media is simulated using the Navier–Stokes equations (NSEs) with time-fractional derivatives of order $\beta \in (0,1)$. In ${\mathbf{H}}^{\gamma ,\wp }$, we prove the existence and uniqueness of local and global mild solutions by using fuzzy techniques. Meanwhile, we provide a local moderate solution in Banach space. We further show that classical solutions to such equations exist and are regular in Banach space. Full article

The concrete–rock interfacial transition zone (ITZ) is generally considered the weak layer in hydraulic engineering, for it is more permeable than the intact concrete or rocks. The water permeability of the ITZ is a critical parameter concerned with structural safety and durability. However, the permeability and pore structure of the ITZ has not been investigated previously, and the mathematical model of ITZ permeability has not been established. This study performed multi-scale experiments on the concrete–rock ITZ with various rock types (limestone, granite, and sandstone). A series of quantitative and qualitative analysis techniques, including NMR, SEM-EDS, and XRD, characterize the ITZ pore structures. The controlled constant flow method was used to determine the permeability of the concrete, rock, and ITZ. The mathematical model of ITZ permeability was proposed using the fractal theory. The consistency between the experimental data and the proposed model indicates the reliability of this study. The results of the experiment show that ITZ permeability is between 4.08 × 10⁻¹⁸ m² and 5.74 × 10⁻¹⁸ m². The results of the experiment and the proposed model could determine ITZ permeability in hydraulic structure safety and durability analysis. Full article

We study some fractal properties of the hereditary $\alpha \omega$-dynamo model in the two-mode approximation. The phase variables of the model describe the temporal dynamics of the toroidal and poloidal components of the magnetic field. The hereditary operator of the quenching the $\alpha$-effect by field helicity in numerical simulation is determined using the Riemann–Liouville fractional differentiation operator. The model also includes a stochastic term. The structure of this term corresponds to the effect of coherent structures from small-scale magnetic field and velocity modes. A difference scheme and a program code for numerical simulation have been developed and verified. A series of computational experiments with the model has been carried out. The Hausdorff dimension of the polarity scale in the model and the distribution of polarity intervals are calculated. It is shown that the Hausdorff dimension of the polarity scale is less than 1, i.e., this scale is a fractal. The numerical value of the dimension for some values of the control parameters is 0.87, which is consistent with the dimension of the real geomagnetic polarity scale. The distribution histogram of polarity intervals in the model has a pronounced power-law tail, which also agrees with the properties of real polarity scales. Full article

It is well known that multicomponent integrable systems provide a method for analyzing phenomena with numerous interactions, due to the interactions between their different components. In this paper, we derive the multicomponent higher-order Chen–Lee–Liu (mHOCLL) system through the zero-curvature equation and recursive operators. Then, we apply the trace identity to obtain the bi-Hamiltonian structure of mHOCLL system, which certifies that the constructed system is integrable. Considering the spectral problem of the Lax pair, a related Riemann–Hilbert (RH) problem of this integrable system is naturally constructed with zero background, and the symmetry of this spectral problem is given. On the one hand, the explicit expression for the mHOCLL solution is not available when the RH problem is regular. However, according to the formal solution obtained using the Plemelj formula, the long-time asymptotic state of the mHOCLL solution can be obtained. On the other hand, the N-soliton solutions can be explicitly gained when the scattering problem is reflectionless, and its long-time behavior can still be discussed. Finally, the determinant form of the N-soliton solution is given, and one-, two-, and three-soliton solutions as specific examples are shown via the figures. Full article

In this paper, we study the existence of solutions for a multiplied system of fractional differential equations with nonlocal integro multi-point boundary conditions by using the p-Laplacian operator and the $\phi$-Hilfer derivatives. The presented results are obtained by the fixed point theorems of Krasnoselskii. An illustrative example is presented at the end to show the applicability of the obtained results. To the best of our knowledge, this is the first time where such a problem is considered. Full article

Conventionally, crack width is used to assess the corrosion level, whereas other important characteristics such as the variation in crack width at different locations on the surface are disregarded. These important characteristics of surface crack can be described comprehensively using the fractal theory to facilitate the assessment of the corrosion level. In this study, the relationship between steel corrosion and the fractal characterization of concrete surface cracking is investigated. Reinforced concrete prisms with steel bars of different diameters and with different corrosion rates were evaluated. High-resolution images of cracks on the surfaces of these specimens were captured and processed to obtain their fractal dimensions. Finally, a relationship between the fractal dimension, steel bar diameter, and the corrosion rate is established. The results show that the fractal dimension is associated closely with the corrosion rate and steel bar diameter. This study provides new ideas for evaluating corroded reinforced concrete structures. Full article

The fuzzy order relation $\left(\succcurlyeq \right)$ and fuzzy inclusion relation $\left(\supseteq \right)$ are two different relations in fuzzy-interval calculus. Due to the importance of $p$-convexity, in this article we consider the introduced class of nonconvex fuzzy-interval-valued mappings known as $p$-convex fuzzy-interval-valued mappings ($p$-convex f-i-v-ms) through fuzzy order relation. With the support of a fuzzy generalized fractional operator, we establish a relationship between $p$-convex f-i-v-ms and Hermite–Hadamard (ℋ–ℋ) inequalities. Moreover, some related ℋ–ℋ inequalities are also derived by using fuzzy generalized fractional operators. Furthermore, we show that our conclusions cover a broad range of new and well-known inequalities for $p$-convex f-i-v-ms, as well as their variant forms as special instances. The theory proposed in this research is shown, with practical examples that demonstrate its usefulness. These findings and alternative methodologies may pave the way for future research in fuzzy optimization, modeling, and interval-valued mappings (i-v-m). Full article

This paper investigates an environmental protection expenses model, which considers the relations between the visitors to the protected areas V, the quality of the environmental resource E, and the capital stock K. In this model, the total tourism income is used partly to increase the capital stock or as the environmental protection expenses. Two time delays are introduced into the number of visitors, since the visitors need time to respond the changes of the environment, and the environment will take time to respond to the input of money. Stability crossing curves in the plane of delays (${\tau }_1,$${\tau }_2$) are used to obtain the stable region of equilibrium. Numerical simulations represent the mutual transformation of the supercritical bifurcation and the subcritical bifurcation. Our model shows that under some parameter conditions, the share of tourism income $\eta$ is related closely to the delay ${\tau }_1$, while the capital stock and the environmental quality can be maintained persistently if the delay ${\tau }_1$ is not too large. Full article

The magnetic field intensity will be nondeterminacy with the flow of charged particles thrown out by solar activities, the overlap of adjacent magnetic islands or non-axisymmetric magnetic interference in tokamaks and so on. The model of a generalized Oldroyd-B fluid with fractional derivative under oscillating pressure gradient and magnetic field with some disturbance will be considered in this paper. The disturbance is regarded as the background noise of the system, and the model is described by a fractional stochastic differential equation. Time and space are discretized by L1, L2 schemes based on piecewise linear interpolation and the central difference quotient method. We demonstrate the effects of the amplitude and period of the oscillating pressure gradient, magnetic parameter, fractional parameters and noise on the velocity field, and two special cases are given. Full article

In this work, the restricted three-body system is studied in the framework of the continuation fractional potential with its application on the Earth–Moon system. With the help of a numerical technique, we obtained thirteen equilibrium points, such that nine of them are collinear while the remaining four are non-collinear points. We found that the collinear points near the smaller primary were shifted outward from the Moon, whereas the points near the bigger primary were shifted towards the Earth as the value of the continuation fractional parameter increased. We analyzed the zero-velocity curves and discussed the perturbation of the continuation fractional potential effect on the possible regions of the motion. We also discussed the linear stability of all the equilibrium points and found that out of thirteen only two were stable. Due to such a prevalence, the continuation fractional potential is a source of significant perturbation, which embodies the lack of sphericity of the body in the restricted three-body problem Full article

Plant leaf surfaces can contain interesting, reproducible spatial patterns that can be used for several industrial purposes. In this paper, the main goal was to analyze the surface microtexture of Amazon Anacardium occidentale L. using multifractal theory. AFM images were used to evaluate the multifractal spatial surface patterns of the adaxial and abaxial sides of the leaf. The 3D maps revealed that the abaxial side is dominated by stomach cells, while striated structures were observed on the adaxial side. The surface of the abaxial side is rougher than the adaxial side. The autocorrelation function calculations showed that the abaxial side has an isotropic surface compared to the adaxial side. Despite this, Minkowski functionals demonstrated that the morphological spatial patterns have robust statistical similarity. Both sides exhibit multifractal behavior, which was verified by the trend observed in the mass exponent and generalized dimension. However, the adaxial side exhibits stronger multifractality and increased vertical complexity compared to the abaxial side. Our findings show that the multifractal spatial patterns of the leaf surface depend on the rough dynamics of the topographic profile. The identification of the multifractal patterns of the structures present on the surface of plant leaves is useful for the fabrication of leaf-architecture-based materials. Full article

Let H be a compact metric space. The metric of H is denoted by d. And let $(H,f_{1,\infty })$ be a non-autonomous discrete system where $f_{1,\infty }={\left\{f_n\right\}}_{n=1}^{\infty }$ is a mapping sequence. This paper discusses infinite sensitivity, m-sensitivity, and m-cofinitely sensitivity of $f_{1,\infty }$. It is proved that, if $f_n(n\in \mathbb{N})$ are feebly open and uniformly converge to $f:H\to H$, $f_i\circ f=f\circ f_i$ for any $i\in \{1,2,\dots \}$, and ${\sum }_{i=1}^{\infty }D(f_i,f)<\infty$, then $(H,f)$ has the above sensitive property if and only if $(H,f_{1,\infty })$ has the same property where $D(·,·)$ is the supremum metric. Full article

In this paper, the authors introduce the notion of neutrosophic double controlled metric spaces as a generalization of neutrosophic metric spaces. For this purpose, two non-comparable functions, ξ and Γ, are used in triangle inequalities. The authors prove several interesting results for contraction mappings with non-trivial examples. At the end of the paper, the authors prove the existence, and the uniqueness, of the integral equation to support the main result. Full article

Fracture energy, as an important characteristic parameter of the fracture properties of materials, has been extensively studied by scholars. However, less research has been carried out on ubiquitiformal fracture energy and the main method used by scholars is the uniaxial tensile test. In this paper, based on previous research, the first Brazilian splitting test was used to study the ubiquitiformal crack extension of slate and granite, and the complexity and ubiquitiformal fracture energy of rock material were obtained. The heterogeneity of the material was then characterized by the Weibull statistical distribution, and the cohesive model is applied to the ABAQUS numerical software to simulate the effect of heterogeneity on the characteristics of ubiquitiformal cracks. The results demonstrate that the ubiquitiformal complexity of slate ranges from 1.54 to 1.60, and that of granite ranges from 1.58 to 1.62. The mean squared deviations of the slate and granite ubiquitiformal fracture energy are the smallest compared with the other fracture energies, which are 0.038 and 0.037, respectively. When the homogeneity of the heterogeneous model is less than 1.5, its heterogeneity has a greater influence on the Brazilian splitting strength, and the heterogeneity of the rock is obvious. However, when the homogeneity is greater than five, the effect on the Brazilian splitting strength is much less, and the Brazilian splitting strength tends to be the average strength. Therefore, it is particularly important to study the fracture problem of cracks from the nature of the material structure by combining the macroscopic and mesoscopic views through the ubiquitiform theory. Full article

In this paper, we familiarize a class of multivalent functions with respect to symmetric points related to the differential operator and discuss the impact of Janowski functions on conic regions. Inclusion results, the subordination property, and coefficient inequalities are obtained. Further, the applications of our results that are extensions of those given in earlier works are presented as corollaries. Full article

In this paper, we focus on the existence of positive solutions for a boundary value problem of the changing-sign differential equation on time scales. By constructing a translation transformation and combining with the properties of the solution of the nonhomogeneous boundary value problem, we transfer the changing-sign problem to a positone problem, then by means of the known fixed-point theorem, several sufficient conditions for the existence of positive solutions are established for the case in which the nonlinear term of the equation may change sign. Full article

In this paper, based on the modified block-by-block method, we propose a higher-order numerical scheme for two-dimensional nonlinear fractional Volterra integral equations with uniform accuracy. This approach involves discretizing the domain into a large number of subdomains and using biquadratic Lagrangian interpolation on each subdomain. The convergence of the high-order numerical scheme is rigorously established. We prove that the numerical solution converges to the exact solution with the optimal convergence order $O(h_x^{4-\alpha }+h_y^{4-\beta })$ for $0<\alpha ,\beta <1$. Finally, experiments with four numerical examples are shown, to support the theoretical findings and to illustrate the efficiency of our proposed method. Full article