Fractals

In this paper, we utilize local fractional reduced differential transform (LFRDTM) and local fractional Laplace variational iteration methods (LFLVIM) to obtain approximate solutions for coupled KdV equations. The obtained results by both presented methods (the LFRDTM and the LFLVIM) are compared together. The results clearly show that those suggested algorithms are suitable and effective to handle linear and as well as nonlinear problems in engineering and sciences.

A weakly nonlinear analysis provides a system constituting amplitude equations and its related analysis is capable of predicting parameter regimes with different patterns expected to co-exist in dynamical circumstances that exhibit complex fractional-order system characteristics. The Turing mechanism of pattern formation as a result of diffusion-induced instability of the homogeneous steady state is concerned with unpredictable conditions. The Turing instability caused by fractional diffusion in a Human Immunodeficiency Virus model has been addressed in this study. It is important that the effect of the Human Immunodeficiency Virus to the immune system can be modeled by the interaction of uninfected cells, unhealthy cells, virus particles and antigen-specific. Initially, all potential equilibrium points are defined and the stability of the interior equilibrium point is then evaluated using the Routh–Hurwitz criteria. The conditions for Turing instability are obtained by local equilibrium points with stability analysis. In the neighborhood of the Turing bifurcation point, weakly nonlinear analysis is employed to deduce the amplitude equations. After applying amplitude equations, it is observed that this system has a very rich dynamical behavior. The constraints for the formation of the patterns like a hexagon, spot, mixed and stripe patterns are identified for the amplitude equations by dynamic analysis. Furthermore, by using the numerical simulations, the theoretical results are verified. Within this framework, this study through the dynamical behavior of the complex system perspective and bifurcation point based on the viral death rate can provide the basis for several researchers working on Human Immunodeficiency Virus model through various aspects. Accordingly, the Turing bifurcation point and weakly nonlinear analysis employed within the complex fractional-order dynamics addressed herein are highly relevant experimentally since the related effects can be studied and applied concerning different mathematical, physical, engineering and biological models.

In this paper, we solve the $n$-Generalized KdV equation by local fractional homotopy analysis method (LFHAM). Further, we analyze the approximate solution in the form of non-differentiable generalized functions defined on Cantor sets. Some examples and special cases of the main results are also discussed.

In this paper, we have investigated women’s malignant disease, cervical cancer, by constructing the compartmental model. An extended fractal–fractional model is used to study the disease dynamics. The points of equilibria are computed analytically and verified by numerical simulations. The key role of $R_0$ in describing the stability of the model is presented. The sensitivity analysis of $R_0$ for deciding the role of certain parameters altering the disease dynamics is carried out. The numerical simulations of the proposed numerical technique are demonstrated to test the claimed facts.

This paper presents a study on the existence theory of fractional differential equations involving Atangana–Baleanu (AB) derivative of order $1<\alpha \le 2$, with non-separated and integral type boundary conditions. An existence result for the solutions of given AB-fractional differential equation is proved using Krasnoselskii’s fixed point theorem, while the uniqueness of the solution is obtained using Banach contraction principle. Some conditions are proposed under which the given boundary value problem is Hyers–Ulam stable. Examples are given to validate our results.

This research is related to present a novel fractional Mayer neuro-swarming intelligent solver for the numerical treatment of multi-fractional order doubly singular Lane–Emden (LE) equation using combined investigations of the Mayer wavelet (MW) neural networks (NNs) optimized by the global search effectiveness of particle swarm optimization (PSO) and interior-point (IP) method, i.e. MW-NN-PSOIP. The design of novel fractional Mayer neuro-swarming intelligent solver for multi-fractional order doubly singular LE equation is derived from the standard LE model and the shape factors; fractional order terms along with singular points are examined. The modeling based on the MW-NN strength is implemented to signify the multi-fractional order doubly singular LE model using the ability of mean squared error in terms of the merit function and the networks are optimized with the integrated capability of PSOIP scheme. The perfection, verification and validation of the fractional Mayer neuro-swarming intelligent solver for three different cases of the multi-fractional order doubly singular LE equation are recognized through comparative investigations from the reference results on different measures based on the convergence, robustness, stability and accuracy. Furthermore, the statics interpretations further validate the performance of the proposed fractional Mayer neuro-swarming intelligent solvers.

This paper aims to investigate the notion of $p$-convex functions on fractal sets ${ℝ}^{\widehat{\alpha }}(0<\widehat{\alpha }\le 1).$ Based on these novel ideas, we derived an auxiliary result depend on a three-step quadratic kernel by employing generalized $p$-convexity. Take into account the local fractal identity, we established novel Newton’s type variants for the local differentiable functions. Several special cases are apprehended in the light of generalized convex functions and generalized harmonically convex functions. This novel strategy captures several existing results in the relative literature. Application is obtained in cumulative distribution function and generalized special weighted means to confirm the relevance and computational effectiveness of the considered method. Finally, we supposed that the consequences of this paper can stimulate those who are interested in fractal analysis.

This work addresses several novel classes of convex function involving arbitrary non-negative function, which is known as approximately generalized $(\psi ,\hslash )$-convex and approximately $\psi$-quasiconvex function, with respect to Raina’s function, which are elaborated in Hilbert space. To ensure the feasibility of the proposed concept and with the discussion of special cases, it is presented that these classes generate other classes of generalized $(\psi ,\hslash )$-convex functions such as higher-order strongly (HOS) generalized $(\psi ,\hslash )$-convex functions and HOS generalized $\psi$-quasiconvex function. The core of the proposed method is a newly developed Simpson’s type of identity in the settings of Riemann–Liouville fractional integral operator. Based on the HOS generalized $(\psi ,\hslash )$-convex function representation, we established several theorems and related novel consequences. The presented results demonstrate better performance for HOS generalized $\psi$-quasiconvex functions where we can generate several other novel classes for convex functions that exist in the relative literature. Accordingly, the assortment in this study aims at presenting a direction in the related fields.

The meminductor is a new type of memory circuit element which is defined based on the memristor. To explore the application of the meminductor in the nonlinear circuits, a mathematical model of meminductor is proposed and applied to nonlinear circuits. In this work, a simple meminductor-based chaotic system is designed. The equilibrium point of the system is controlled by the externally excited sinusoidal signal in the circuit. No matter what the value of the externally excited signal is, the chaotic attractor generated by the proposed system is hidden. The dynamic characteristics of the system are analyzed by theoretical analysis and numerical simulation. The results show that the dynamic behaviors of the system are affected by the circuit parameters and the circuit running time. The proposed system shows some novel nonlinear phenomena, such as transient chaos and state transitions. In addition, the existence of coexisting attractors, such as chaotic, periodic and quasi-periodic attractors, is analyzed in different initial states.

We consider the following abstract version of the Moore–Gibson–Thompson (MGT) equation:

This research work is to design a neural-swarming heuristic procedure for numerical investigations of Singular Multi-Pantograph Delay Differential (SMP-DD) equation by applying the function approximation aptitude of Artificial Neural Networks (ANNs) optimized efficient swarming mechanism based on Particle Swarm Optimization (PSO) integrated with convex optimization with Active Set (AS) algorithm for rapid refinements, named as ANN-PSO-AS. A merit function (MF) on mean squared error sense is designed by using the differential ANN models and boundary condition. The optimization of this MF is executed with the global PSO and local search AS approaches. The planned ANN-PSO-AS approach is instigated for three different SMP-DD model-based equations. The assessment with available standard results relieved the effectiveness, robustness and precision that is further authenticated through statistical investigations of Variance Account For, Root Mean Squared Error, Semi-Interquartile Range and Theil’s inequality coefficient performances.

The study of nonlinear phenomena associated with physical phenomena is a hot topic in the present era. The fundamental aim of this paper is to find the iterative solution for generalized quintic complex Ginzburg–Landau (GCGL) equation using fractional natural decomposition method (FNDM) within the frame of fractional calculus. We consider the projected equations by incorporating the Caputo fractional operator and investigate two examples for different initial values to present the efficiency and applicability of the FNDM. We presented the nature of the obtained results defined in three distinct cases and illustrated with the help of surfaces and contour plots for the particular value with respect to fractional order. Moreover, to present the accuracy and capture the nature of the obtained results, we present plots with different fractional order, and these plots show the essence of incorporating the fractional concept into the system exemplifying nonlinear complex phenomena. The present investigation confirms the efficiency and applicability of the considered method and fractional operators while analyzing phenomena in science and technology.

Discrete fractional calculus (DFC) is suggested to interpret neural schemes with memory impacts. This study seeks to formulate some discrete fractional nonlinear inequalities with $\widehat{\xi }$ fractional sum operators that are used with some conventional and forthright inequalities. Taking into consideration, we recreate the explicit bounds of Gronwall-type inequalities by observing the principle of DFC for unknown functions here. Such inequalities are of new version relative to the current literature findings so far and can be used as a helpful method to evaluate the numerical solutions of discrete fractional differential equations. We show a few uses of the rewarded inequalities to mirror the advantages of our work. Regarding applications, we can apply the acquainted results to discuss boundedness, uniqueness, and continuous dependency on the initial value problem for the solutions of certain underlying worth problems of fractional difference equations. The leading consequences may be proven by the usage of the analysis process and the methodology of the mean value theorem. These variations can be used as an advantageous device in the subjective examination solutions of discrete fractional difference equations.

Our main concern in this paper, is to prove the stability combined well-posedness results of solutions for a linear Timoshenko’s beam laminated with thermoplastic and past history together with distributed delay terms for the both cases nonequal and equal speeds of wave propagation. By using of some of the usual basic theorems and well-known methods in functional analysis.

The purpose of this paper is to provide novel estimates of Ostrowski-type inequalities in a much simpler and shorter way of some recent significant results in the context of a fractal set ${ℝ}^{\tilde{\alpha }}.$ By using our new approach, we established an auxiliary result that correlates with generalized convex (${𝒢}{𝒞}$) and concave functions for absolutely continuous functions with second-order local differentiable mappings. Moreover, we derived some companions of Ostrowski-type inequalities belonging to ${{𝒱}}^{(2\tilde{\alpha })}\in L_{\infty }[s_1,s_2],{{𝒱}}^{(2\tilde{\alpha })}\in L_p[s_1,s_2]$ and ${{𝒱}}^{(2\tilde{\alpha })}\in L_1[s_1,s_2]$ in local fractional sense. Our results generalize and offer better bounds than many known results in the existing literature associated with trapezoidal and midpoint formula. As an application perspective, we derived several estimation-type outcomes by the use of generalized $\tilde{\alpha }$-type special means formula provided here to illustrate the usability of the obtained results. Our study contributes to a better understanding of fractal analysis and proves beneficial in exploring real-world phenomena.

A user-friendly approach depending on nonlocal kernel has been constituted in this study to model nonlocal behaviors of fractional differential and difference equations, which is known as a generalized proportional fractional operator in the Hilfer sense. It is deemed, for differentiable functions, by a fractional integral operator applied to the derivative of a function having an exponential function in the kernel. This operator generalizes a novel version of Čebyšev-type inequality in two and three variables sense and furthers the result of existing literature as a particular case of the Čebyšev inequality is discussed. Some novel special cases are also apprehended and compared with existing results. The outcome obtained by this study is very broad in nature and fits in terms of yielding an enormous number of relating results simply by practicing the proportionality indices included therein. Furthermore, the outcome of our study demonstrates that the proposed plans are of significant importance and computationally appealing to deal with comparable sorts of differential equations. Taken together, the results can serve as efficient and robust means for the purpose of investigating specific classes of integrodifferential equations.

In this paper, we establish several results involving the minimum and maximum principles and the comparison principles for elliptic equations and parabolic equations on finite graphs. The results are then used to prove the monotonicity and asymptotic properties of solutions for parabolic equations whose initial values are given by the equation $\mathrm{\Delta }\psi +f(\psi )=0$ with Dirichlet boundary conditions. Finally, an illustration with numerical experiments is provided to demonstrate our main results.

This paper demonstrates an effective and powerful technique, namely fractional He–Laplace method (FHe-LM), to study a nonlinear coupled system of equations with time fractional derivative. The FHe-LM is designed on the basis of Laplace transform to elucidate the solution of nonlinear fractional Hirota–Satsuma coupled KdV and coupled mKdV system but the series coefficients are evaluated in an iterative process with the help of homotopy perturbation method manipulating He’s polynomials. The fractional derivatives are considered in the Caputo sense. The obtained results confirm the suggested approach is extremely convenient and applicable to provide the solution of nonlinear models in the form of a convergent series, without any restriction. Also, graphical representation and the error estimate when compared with the exact solution are presented.

In this study, the scaling characteristics of root-mean-squared roughness ($Rq$) was investigated for both fractal and non-fractal profiles by using roughness scaling extraction (RSE) method proposed in our previous work. The artificial profiles generated through Weierstrass–Mandelbrot (W–M) function and the actual profiles, including surface contours of silver thin films and electroencephalography signals, were analyzed. Based on the relationship curves between $Rq$ and scale, it was found that there was a continuous variation of the dimension value calculated with RSE method ($D_{\mathrm{\text{RSE}}}$) across the fractal and non-fractal profiles. In the range of fractal region, $D_{\mathrm{\text{RSE}}}$ could accurately match with the ideal fractal dimension ($FD$) input for W–M function. In the non-fractal region, $D_{\mathrm{\text{RSE}}}$ values could characterize the complexity of the profiles, similar to the functionality of $FD$ value for fractal profiles, thus enabling the detection of certain incidents in signals such as an epileptic seizure. Moreover, the traditional methods (Box-Counting and Higuchi) of $FD$ calculation failed to reflect the complexity variation of non-fractal profiles, because their $FD$ was generally 1. The feasibility of abnormal implementation of W–M function and the capability of RSE method were discussed according to the analysis on the properties of W–M function, which would be promising to make more understandings of the nonlinear behaviors of both theoretical and practical features.

The present work presents a simplified mathematical model to calculate the flowrate of the combined electroosmotic flow (EOF) and pressure driven laminar flow (PDLF) in the symmetric tree-like microchannel network under the assumptions of small zeta potential and thin electrical double layer. A numerical analysis of the combined EOF and PDLF in symmetric Y-shaped microchannel is also carried out to validate the mathematical model. The analytical results and numerical results are found to be in good agreement with each other. Using the mathematical model, the present work further investigates the effect of diameter ratio of the tree-like network on the flowrate of the combined EOF and PDLF to recognize a possible conclusion being similar to the Murray’s law. Based on the present work, it is found that the symmetric tree-like network has an optimal diameter ratio to achieve the maximum flowrate for the combined EOF and PDLF when the total microchannel volume is constant; however, this optimal diameter ratio for the combined flow disobeys the generalized Murray’s law in a simple form of power function of the branching number $N$, and it is not only related on the branching number, but also depends on the branching level and channel length ratio of the tree-like network. Furthermore, the optimal diameter ratio shows a monotonous transition from $N^{-1/3}$ for the pure PDLF to $N^{-1/2}$ for the pure EOF with the increasing ratio of the driven voltage and driven pressure. The present work discusses the effects of these parameters on the optimal diameter ratio for the combined EOF and PDLF.

Let $K$ be a self-similar set in $ℝ$. Generally, if the iterated function system (IFS) of $K$ has some overlaps, then some points in $K$ may have multiple codings. If an $x\in K$ has a unique coding, then we call $x$ a univoque point. We denote by ${𝒰}$ (univoque set) the set of points in $K$ having unique codings. In this paper, we shall consider the following natural question: if two self-similar sets are bi-Lipschitz equivalent, then are their associated univoque sets also bi-Lipschitz equivalent. We give a class of self-similar sets with overlaps, and answer the above question affirmatively.

We used multifractals to analyze the Lebanese topography focusing on Mount Lebanon. The elevation data were obtained from NASA STRM Global Digital Elevation of Earth Land, spaced at 80m in the East-West direction, and at 90m in the North-South direction. After transforming the grid to be perpendicular and parallel to the range, we found anisotropic scaling from 500m to 10,000m, and it reflected the fact that the Lebanese topography was more correlated in the direction perpendicular to the mountain range, probably due to occurrence of valleys and ridges in that direction. We estimated the parameters of the Universal Multifractal (UM) model and found $\alpha =1.45$ and $c1=0.05$, consistent with values reported for topography. The UM parameter $H$ was found to be 0.72 across the range and 0.57 along the range, the latter value agrees with prior observations. However, the larger value across the range is consistent with the higher spatial correlation in that direction. We introduced a new expression for the 2D power spectral density, and we showed that it can decently capture the anisotropic scaling. We also generated multiple realizations and we showed that the generation of anisotropic scaling did not alter the underlying parameter values $\alpha$ and $c1$ of the UM model. We also proposed an approximate method for generating conditional simulations, and we showed that through a judicious selection of values, one may reproduce approximately the observed field values at the desired locations. We believe such an approach could be used to generate realistic simulations of fields that are time-invariants, such as topography and soil properties.

In the current study, we propose a bifractal stratified characterization method to evaluate the surface profile evolution of a plateau honing cylinder liner during the wear process. This method is suitable for the characterization of two-process and worn surface profiles with bifractal and stratified characteristics. We develop a bifractal truncated separation structure function method to calculate the characterization parameters of the proposed method. In particular, the upper- and lower-strata surface profiles are truncated and separated in order to calculate their fractal parameters. Experimental results demonstrate the ability of the proposed method to outperform the traditional fractal method in characterizing the evolution of the plateau honing cylinder liner surface profile during the wearing process.

The effective thermal conductivity (ETC) of roughened porous media (RPM) is of interest in a number of applications of heat transfer. In this work, a fractal analytical model for the ETC of RPM with microscale effect is proposed. The proposed fractal model is expressed in terms of relative roughness, the molecular mean free path, porosity, fractal dimensions (pore area fractal dimension and tortuosity fractal dimension), maximum pore diameter, capillary straight length, and the thermal conductivity of the solid matrix and gas. It is observed that the dimensionless ETC of RPM decreases with increasing relative roughness, pore area fractal dimension and tortuosity fractal dimension. Besides, it is found that the dimensionless ETC of RPM increases with thermal conductivity ratio of gas phase over solid phase. In addition, it is found that the dimensionless ETC of RPM is slightly dependent on the relative roughness and tortuosity fractal dimension when $k_g/k_s<1$. The determined dimensionless ETC of RPM is in good agreement with experimental data and existing models reported in the literature. With the proposed fractal model, the physical mechanisms of heat transport through RPM with microscale effect are better elucidated. Every parameter in the fractal analytical model has clear physical meaning, with no empirical constant.

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.

In this work, the nonlinear fractal $K(p,q)$ model is defined via the fractal derivative. The variational principle of the fractal model is constructed by using the fractal semi-inverse method, and it is of great significance to study the structure and properties of its soliton solution. Based on the established variational principle, we successfully proposed a new method to solve the fractal model. The efficiency and accuracy of the proposed approach are illustrated by an example.

A concrete beam is a basic element for many applications in architectural engineering, and its vibration property plays an important role in safety and life-span. Previous vibration models were based on continuum beam theories, which could not take into account the porous property of the concrete beam. This paper adopts a two-scale fractal vibration model, and its low frequency property is revealed. This new finding can explain the vibration attenuation and vibration absorption of the porous concrete, and the present theory sheds a new light on the concrete design.

In this paper, considering the structure of the shape factor and the rough element, combined with fractal geometry theory, the shape factor, the relative increase in the heat transfer between the rough surface and the smooth surface in a fractal unit, and the fractal analytical expression of the dimensionless equivalent thermal conductivity caused by convection thermal conductivity are derived. In addition, the relationship between the microstructure parameters of the rough surface and the shape factor, the relative increase in convective heat transfer and the dimensionless equivalent thermal conductivity are studied. It is found that the relative increment of the convective heat transfer and the shape factor are proportional to the ratio of the minimum height and maximum height of the rough element and the fractal dimension, and the dimensionless equivalent thermal conductivity model deduced in this paper is compared with the existing experimental data to verify the validity of the model.

The paleoenvironment influences trace-making behavior significantly, and progressively more evidence shows that this behavior follows scale-invariance properties. In this study, we focused on the definition of behavioral complexity in the trace fossil Zoophycos to explore its paleoenvironmental significance. Firstly, the fractal topography in Zoophycos was established by introducing virtual tertiary lamellae. Next, a fractal dimension ($D$) model was proposed and the spatio-temporal evolution trend of the complexity of the trace-making behavior was established. In application, the correlation of $D$ to the paleo-water depth and substrate consistency of the paleoenvironment was analyzed with the aid of variations of ${\delta }^{13}$C and lamellae deformations in Zoophycos from the Carboniferous-Permian Taiyuan Formation of North China. The results show that $D$ increases with paleo-water depth, although it decreases when the degree of substrate consistency increases. Our investigation indicates that fractal topography not only provides an essential explanation for the complexity of trace-making behavior but also plays a reliable and significant role in paleoenvironmental interpretation. Most importantly, this work may shed some light on the subject of fractal ichnology and provide a mathematical framework toward studies in quantitative ichnology.

There are still mathematical predictions in the fight against epidemics. Speedy expansion, ways and procedures for the pandemic control require early understanding when solutions with better computer-based mathematical modeling and prognosis are developed. Despite high uncertainty in each of these models, one of the important tools for public health management system is epidemiology models. The fractional order is shown to be more effective in modeling epidemic diseases, in relation to the memory effects. Notably, recently founded calculus tools, called fractal-fractional calculus, having a fractional order and fractal dimension, enable us to study the behavior of a real-world problem under both fractal and fractional tools. This paper is about the dynamical behavior of a new mathematical model of novel corona disease (COVID-19) under the fractal-fractional Atangana–Baleanu derivative. The considered model has three compartments, namely, susceptible, infected and recovered or removed (SIR). The existence and uniqueness of the model’s solution will be proved via Krasnoselskii’s and Banach’s fixed point theorems, respectively. The stability of the solution in the sense of Hyers–Ulam (HU) will be built up by nonlinear functional analysis. Moreover, the numerical simulations for different values of isolation parameters corresponding to various fractal-fractional orders are analyzed using fractional Adams–Bashforth (AB) method with two-step Lagrange polynomial. Finally, the obtained simulation results are applied to the real data of disease spread from Pakistan. The graphical interpretations demonstrate that increasing the isolation parameters which is caused by strict precautionary measures will reduce the disease infection transmission in society.

We analytically investigate a nonlinear fractional-order sine-Gordon (sG) equation. The derivatives considered herein, are taken in Caputo’s sense. The Laplace transform together with the Adomian decomposition method (LADM) is applied to attain analytical approximation of the aforesaid equation. The sG equation having Caputo derivative is solved in the order of series solutions and the results are confirmed by considering two examples with appropriate initial conditions. The numerical simulations are accomplished to compare with the analytical approximations, where qualitatively better agreements are achieved.

The well-known coupled Boussinesq–Burger equation can be used to describe the flow of the shallow water of the harbor. But when the boundary is nonsmooth, it becomes powerless. So, the fractal calculus is needed to be applied to it and the fractal coupled Boussinesq–Burger equation (FCBBe) is presented for the first time in this paper. By using the semiinverse method, we have successfully established the fractal variational formulation of the FCBBe, which can not only provide the conservation laws in an energy form in the fractal space but also reveal the possible solution structures of the equation. Then a novel variational approach based on He’s variational method and the two-scale transform are used to seek its periodic wave solutions. The main advantage of variational approach is that it can reduce the order of differential equation and make the equation more simple. Finally, the numerical results have been shown through graphs to discuss the effect of different fractal orders on the wave motion. The obtained results in this work are expected to shed a bright light on the study of fractal nonlinear partial differential equations in the fractal space.

Fractal coding has been widely used as an image compression technique in many image processing problems in the past few decades. On the other hand side, most of the natural images have the characteristic of nonlocal self-similarity that motivates low-rank representations of them. We would employ both the fractal image coding and the nonlocal self-similarity priors to achieve image compression in image denoising problems. Specifically, we propose a new image denoising model consisting of three terms: a patch-based nonlocal low-rank prior, a data-fidelity term describing the closeness of the underlying image to the given noisy image, and a quadratic term measuring the closeness of the underlying image to a fractal image. Numerical results demonstrate the superior performance of the proposed model in terms of peak-signal-to-noise ratio, structural similarity index and mean absolute error.

The purpose of this work is to investigate some inequalities for generalized $s$-convexity on fractal sets ${ℝ}^{\alpha }$, which are associated with Simpson-like inequalities. To this end, an improved version of Hölder inequality and a Simpson-like identity on fractal sets are established, in view of which we give several estimation-type results involving Simpson-like inequalities for the first-order differentiable mappings. Moreover, we provide five examples to illustrate our results. As applications with respect to local fractional integrals, we derive two inequalities according to $\alpha$-type special means and generalized probability density functions.

In this paper, we develop the multifractal detrending weighted average algorithm of historical volatility (MF-DHV) for one-dimensional multifractal measure based on the classical multifractal detrended fluctuation analysis (MF-DFA). In the calculation process of getting a local trend for MF-DHV, historical volatility is taken to develop an moving average algorithm, which is different from the simple moving average function in multifractal detrended moving average (MF-DMA). We assess the performance of three methods such as MF-DFA, MF-DMA, and MF-DHV based on the p-model multiplicative cascading constructed time series. The computational results show that all the estimated generalized Hurst exponent $H(q)$, the scaling exponent $\tau (q)$, and the singularity spectrum $f(\alpha )$ of MF-DHV are in good agreement with the theoretical values. In addition, we also calculate the standard deviations of $H_{\mathrm{\text{err}}}$ and ${\tau }_{\mathrm{\text{err}}}$ for three methods, and the lowest errors in MF-DHV provides the most accurate estimates. To avoid the accidental selection of parameters, we change the total length of the generated multifractal simulation data and p-value, respectively. It is found that in all the cases, the MF-DHV outperforms the other two methods.

Analysis of the variations of heart activity during different human activities is an important area of research in sport sciences. Therefore, in this paper, we evaluated the variations of heart activity for 23 subjects while sitting, hand biking, walking, and running. Since the obtained R-R time series (as the indicator of heart rate variability (HRV)) has a complex structure that contains information, we employed fractal dimension, sample entropy, and Shannon entropy for our analysis. According to the results, doing a harder activity causes a more significant alteration in the complexity and information content of HRV. The results of statistical analyses also verified the obtained results. Similar investigations can be conducted in case of other activities to evaluate the variations in heart activity in different conditions.

Attention deficit hyperactivity disorder (ADHD) is a mental health disorder that is very common among children and may last into their adulthood. It is known that ADHD affects the attention of patients due to problems with short-term memory. Therefore, analysis of attention and memory of these patients should come into consideration. In this study, the complexity and memory of Electroencephalogram (EEG) signals are analyzed to investigate the reduction in attention and memory of patients with ADHD compared to normal subjects while playing a serious game. To achieve this, the fractal dimension and sample entropy of EEG signals are analyzed to evaluate the alterations in the complexity of EEG signals. Moreover, the Hurst exponent of EEG signals for ADHD and non-ADHD subjects is calculated to discuss the memory of EEG signals. The results showed a smaller fractal dimension and sample entropy of EEG signals for patients with ADHD that reflects their lower attention. Besides, the Hurst exponent of EEG signals for these patients reflects their lower memory than normal subjects. Therefore, it can be concluded that the reductions of attention and memory in ADHD subjects are mapped on the reduction of complexity and memory of their EEG signals.