Fractals

The multifractal formalism for measures in its original formulation is checked for special classes of measures, such as, doubling, self-similar, and Gibbs-like ones. Out of these classes, suitable conditions should be taken into account to prove the validity of the multifractal formalism. In this work, a large class of measures satisfying a weak condition known as quasi-Ahlfors is considered in the framework of mixed multifractal analysis. A joint multifractal analysis of finitely many quasi-Ahlfors probability measures is developed. Mixed variants of multifractal generalizations of Hausdorff, and packing measures, and corresponding dimensions are introduced. By applying convexity arguments, some properties of these measures, and dimensions are established. Finally, an associated multifractal formalism is introduced, and proved to hold for the class of quasi-Ahlfors measures. Besides, some eventual applications, and motivations, especially, in AI are discussed.

In this paper, we investigate the existence and uniqueness of solutions for Riemann–Liouville fractional integro-differential equations equipped with fractional nonlocal multi-point and strip boundary conditions in the weighted space. The methods of our study include the well-known tools of the fixed point theory, which are commonly applied to establish the existence theory for the initial and boundary value problems after converting them into the fixed point problems. We also discuss the case when the nonlinearity depends on the Riemann–Liouville fractional integrals of the unknown function. Numerical examples illustrating the main results are presented.

In this work, some density estimations associated to vector-valued quasi-Ahlfors measures are developed within the mixed multifractal analysis framework. The principle idea reposes on the fact that being quasi-Ahlfors is sufficient to conduct a mixed multifractal analysis for vector-valued measures. In this work, we introduced a multifractal density for finitely many measures, and showed that such density may be estimated well by means of the mixed multifractal measures. Such estimation induces an exact computation of multifractal spectrum of the vector-valued quasi-Ahlfors measure.

In this paper, some Jensen- and Hardy-type inequalities for convex functions are extended by using Riemann–Liouville delta fractional integrals. Further, some Pólya–Knopp-type inequalities and Hardy–Hilbert-type inequality for convex functions are also proved. Moreover, some related inequalities are proved by using special kernels. Particular cases of resulting inequalities provide the results on fractional calculus, time scales calculus, quantum fractional calculus and discrete fractional calculus.

In this paper, a new set of differential and integral operators has recently been proposed by Abdon et al. by merging the fractional derivative and the fractal derivative, taking into account nonlocality, memory and fractal effects. These operators have demonstrated the complex behavior of many physical, which generally does not predict in ordinary operators or sometimes in fractional operators also. In this paper, we investigate the proposed model by replacing the classic derivative by fractal–fractional derivatives in which fractional derivative is taken in Atangana–Baleanu Caputo sense to study the complex behavior due to nonlocality, memory and fractal effects. Through Schauder’s fixed point theorem, we establish existence theory to ensure that the model posseses at least one solution. Also, Banach fixed theorem guarantees the uniqueness of solution of the proposed model. By means of nonlinear functional analysis, we prove that the proposed model is Ulam–Hyers stable under the new fractal–fractional derivative. We establish the numerical results of the considered model through Lagrangian piece-wise interpolation. For the different values of fractional order and fractal dimension, we study the chaos behavior of the proposed model via simulation at 2D and 3D phase. We show the effect of fractal dimension on integer and fractional order through simulations.

In this paper, the Timoshenko system with distributed delay term, fractional operator in the memory and spatial fractional thermal effect is considered, we will prove under some assumptions the global existence of a weak solution. Furthermore, we show some results about the stability of system by the semigroup method.

In this paper, the mathematical modeling of five different classes for coronavirus disease-19 (COVID-19) is considered using the fractional arbitrary order derivative in Atangana–Baleanu sense. We use nonlinear analysis for the existence theory of the solution for the suggested model. Additionally, the modified Adam–Bashforth method is used for the numerical approximation of the assumed model. Finally, we simulate the results for 100 days with the help of data from the literature to display the excellency of the suggested model.

The most notable inequality pertaining convex functions is Jensen’s inequality which has tremendous applications in several fields. Mercer introduced an important variant of Jensen’s inequality called as Jensen–Mercer’s inequality. Fractal sets are useful tools for describing the accuracy of inequalities in convex functions. The purpose of this paper is to establish a generalized Jensen–Mercer inequality for a generalized convex function on a real linear fractal set ${ℝ}^{\alpha }$ ($0<\alpha \le 1)$. Further, we also demonstrate some generalized Jensen–Mercer-type inequalities by employing local fractional calculus. Lastly, some applications related to Jensen–Mercer inequality and $\alpha$-type special means are given. The present approach is efficient, reliable, and may motivate further research in this area.

The main purpose of the present investigation is to find the solution of fractional coupled equations describing the romantic relationships using $q$-homotopy analysis transform method ($q$-HATM). The considered scheme is a unification of $q$-homotopy analysis technique with Laplace transform (LT). More preciously, we scrutinized the behavior of the obtained solution for the considered model with fractional-order, in order to elucidate the effectiveness of the proposed algorithm. Further, for the different fractional-order and parameters offered by the considered method, the physical natures have been apprehended. The obtained consequences evidence that the proposed method is very effective and highly methodical to study and examine the nature and its corresponding consequences of the system of fractional order differential equations describing the real word problems.

This study highlights the combined effect of heat and mass transfer on MHD Maxwell fluid under time-dependent generalized boundary conditions for velocity, temperature, and concentration. The classical calculus due to the fact that it is assumed as the instant rate of change of the output when the input level changes. Therefore, it is not able to include the previous state of the system called the memory effect. But in the fractional calculus (FC), the rate of change is affected by all points of the considered interval, so it can incorporate the previous history/memory effects of any system. Due to this reason, we applied the modern definition of fractional derivatives (local and nonlocals kernels). Here, the order of fractional derivative will be treated as an index of memory. The exact and semi-analytical solutions are obtained using the integral transform and inversion algorithm. Several important properties of different parameters are analyzed by graphs. Interesting results are revealed by this investigation due to their vast applications in engineering and applied sciences.

In this paper, we study a new class of Hardy-type inequalities involving fractional calculus operators. We derive the Hardy-type inequalities for the variant of Riemann–Liouville fractional calculus operators and $k$-Hilfer fractional derivative operator. The obtained inequalities involving fractional operators are more general as compared to some existing results in the literature.

In this paper, the q-homotopy analysis transform technique is implemented to analyze the solution of fractional-order Burgers and diffusion equations with the help of Caputo operator. The results of the proposed method are shown and analyzed with the help of figures. This approach is used to determine the solution in a convergent sequence and illustrate the q-homotopy analysis transform technique solutions convergence to the exact result. Several examples showed the reliability and simplicity of the technique and highlighted the significance of this work. Therefore, the proposed method is successful in investigating other fractional-order linear and nonlinear partial differential equations.

This work is devoted to derive some existence and uniqueness (EU) conditions for the solution to a class of nonlinear delay non-autonomous integro-differential Cauchy evolution problems (CEPs) under Caputo derivative of fractional order. The required results are derived via topological degree method (TDM). TDM is a powerful tool which relaxes strong compact conditions by some weaker ones. Hence, we establish the EU under the situation that the nonlinear function satisfies some appropriate local growth condition as well as of non-compactness measure condition. Furthermore, some results are established for Hyers–Ulam (HU) and generalized HU (GHU) stability. Our results generalize some previous results. At the end, by a pertinent example, the results are verified.

This paper investigates the dynamical semi-analysis of the delayed food chain model under the considered fractional order. The food chain model is composed of three compartments, namely, population of the prey, intermediate predator and a top predator. By using the fixed point theorem approach, we exploit some conditions for existence results and stability for the considered system via Atangana–Baleanu–Caputo derivative with fractional order. Also, using the well-known Adam–Bashforth technique for numerics, we simulate the concerning results for the interference between the prey and intermediate predator. Graphical results are discussed for different fractional-order values for the considered model.

In the present paper, the optimal auxiliary function method (OAFM) has been extended for the first time to fractional-order partial differential equations (FPDEs) with convergence analysis. To find the accuracy of the OAFM, we consider the fractional-order KDV-Burger and fifth-order Sawada–Kotera equations as a test example. The proposed technique has auxiliary functions and convergence control parameters, which accelerate the convergence of the method. The other advantage of this method is that there is no need for a small or large parameter assumption, and it gives an approximate solution after only one iteration. Further, the obtained results have been compared with the exact solution through different graphs and tables, which shows that the proposed method is very effective and easy to implement for different FPDEs.

Fractional differential and integral equations are focus of the researchers owing to their tremendous applications in different field of science and technology, such as physics, chemistry, mathematical biology, dynamical system and engineering. In this work, a power series approach called Residual Power Series Method (RPSM) is applied for the solution of fractional (non-integer) order integro-differential equations (FIDEs). The Caputo sense is used for calculating fractional derivatives. Comparison of the obtained solution is made with the Trigonometric Transform Method (TTM) and Optimal Homotopy Asymptotic Method (OHAM). There is no restrictive condition on the proposed solution. The presented technique is simple in applicability and easily computable.

The scaling exponent of a hierarchy of cities used to be regarded as a fractional. This paper investigates a newly constructed system of equation for Hepatitis B disease in sense of Atanganaa–Baleanu Caputo (ABC) fractional order derivative. The proposed approach has five distinctive quantities, namely, susceptible, acute infections, chronic infection, immunized and vaccinated populace. By applying some well-known results of fixed point theory, we find the Ulam–Hyers type stability and qualitative analysis of the candidate solution. The deterministic stability for the proposed system is also computed. We apply well-known transform due to Laplace and decomposition techniques (LADM) and Adomian polynomial for nonlinear terms for computing the series solution for the proposed model. Graphical results show that LADM is an efficient and robust method for solving nonlinear problems.

In this research paper, we implement the sine-Gordon expansion method to two governing models which are the (2+1)-dimensional Nizhnik–Novikov–Veselov equation and the Caudrey–Dodd–Gibbon–Sawada–Kotera equation. We use conformable derivative to transform these nonlinear partial differential models to ordinary differential equations. We find some wave solutions having trigonometric function, hyperbolic function. Under the strain conditions of these solutions obtained in this paper, various simulations are plotted.

The applications and the fields that use the anomalous sub-diffusion equations cannot be easily listed due to their wide area. Sure, one of the main physical reasons for using and researching fractional order diffusion equations is to explain anomalous diffusion that occurs in transport processes through complex and/or disordered structures, such as fractal media. One of the important applications is their use in chemical reactions, where a single material continues to shift from a high concentration area to a low concentration area until the concentration across the space is equal. The mathematical model that describes these physical-chemical phenomena is called the reaction sub-diffusion equation. In our study, we try to solve the 2D variable order version of these equations (2DVORSE) (linear and nonlinear) by using an accurate numerical technique which is the variable weighted average finite difference method (WAFDM). We will analyze the stability of the resulting scheme by using a modified suitable version of the John Von Neumann procedure. Specific stability conditions that occur or some parameters in the resulting schemes are derived and checked. At the end of the study, numerical examples are simulated to check the stability and the accuracy of the proposed technique.

Anomalous Reaction-Sub-diffusion equations play an important role transferred in a lot of our daily applications in our life, especially in applied chemistry. In the presented work, a modified type of these models is considered which is the Reaction-Sub-diffusion equation of variable order, the linear and nonlinear models and we will refer to it by VORSDE. An accurate technique depends on a mix of the finite difference methods (FDM) together with Hermite formula is introduced to study these important types of anomalous equations. Regarding the analysis of the stability for the mentioned, it is done using the variable Von-Neumann technique; also the convergent analysis is introduced. As a result of the previous steps, we derived a stability condition which will be held for many discretization schemes of the variable order derivative and some other parameters and we checked it numerically.

In this paper, we study a newly updated nonlinear fractional Kawahara equation (KE) using Atangana–Baleanu fractional operator in the sense of Caputo (ABC). To find the approximate solution, one of the famous techniques of the Laplace Adomian decomposition method (LADM) is used along with a time-fractional derivative. For evaluation, the required quantity is decomposing into small particles along with the application of Adomian polynomial to the nonlinear term. By the addition of the first few evaluating terms, the required convergent quantity is obtained. To explain the authenticity and the manageability of the procedure, few examples are present at different fractional orders both in three and two dimensions. Further, to compare the obtained results between fractional derivative and integer derivative, some graphical presentations are given. So, the newly updated version of the KE equation is analyzed in fraction operator providing the whole density of the total dynamics at any fractional value between two different integers.

In this paper, we develop the theory of fractional order hybrid differential equations involving Riemann–Liouville differential operators of order $\ell \in (0,1)$. We study the existence theory to a class of boundary value problems for fractional order hybrid differential equations. The sum of three operators is used to prove the key results for a couple of hybrid fixed point theorems. We obtain sufficient conditions for the existence and uniqueness of positive solutions. Moreover, examples are also presented to show the significance of the results.

In our research work, we develop the analysis of a noninteger-order model for hepatitis B (HBV) under singular type Caputo fractional-order derivative. We investigated our proposed system for an approximate or semi-analytical solution using Laplace transform along with decomposition techniques by Adomian polynomial of nonlinear terms and some perturbation techniques of Homotopy (HPM). The obtained solutions have been compared with each other against some real data by simulation via MATLAB. The graphical simulation in fractional form shows a better general result as compared to integer-order simulation.

On the grounds of few perceived Gronwall inequalities, we inspect some new relevant nonlinear dynamic inequalities on time scales related to one independent variable. To guarantee the evenness of information division and with the conversation of peculiar cases, it is indicated that these inequalities catch dynamic, delay, Volterra–Fredholm, discrete, fractional, etc. The inequalities suggested right here to other specific bounds on unknown parameters may be handled as powerful equipment for determining the features of various dynamic equations on time scales. It is trusted that this research plan will open new possibilities in in-depth examination of time scale scheme structure of varying nature.

This paper is devoted to investigating or computing the solution to one-dimensional partial fuzzy fractional order heat equation. In particular, one-dimensional fuzzy partial heat equation is hybridized into two equations by hybrid techniques along with the concept of parametric fuzzy number. For this investigation, a hybrid method of decomposition due to Adomian and Laplace transform is used. The considered techniques are presented for the computation of series of solutions of partial fractional order heat equation. The applied techniques have also provided the accuracy, simplicity and efficiency as compared to other existing methods. Finally, some illustrations are solved for the justification of our theoretical solution.

In this paper, some attempts have been devoted to investigating the dynamic features of discrete fractional calculus (DFC). To date, discrete fractional systems with complex dynamics have attracted the most consideration. By considering discrete $\hslash$-proportional fractional operator with nonlocal kernel, this study contributes to the major consequences of the certain novel versions of reverse Minkowski and related Hölder-type inequalities via discrete $\hslash$-proportional fractional sums, as presented. The proposed system has an intriguing feature not investigated in the literature so far, it is characterized by the nabla $\hslash$ fractional sums. Novel special cases are reported with the intention of assessing the dynamics of the system, as well as to highlighting the several existing outcomes. In terms of applications, we can employ the derived consequences to investigate the existence and uniqueness of fractional difference equations underlying worth problems. Finally, the projected method is efficient in analyzing the complexity of the system.

This work deals with the proof of local existence theorem of solutions for coupled nonlocal singular viscoelastic system with respect to the nonlinearity of source terms by using the Faedo–Galerkin method together with energy methods. This work makes a new contribution, since most of the previous works did not address the proof of the theorem of the local existence of solutions. It is also a completed study of Boulaaras et al. [Adv. Differ. Equ. 2020 (2020) 310].

In this paper, we solve the local fractional generalized Fokker–Planck equation. To solve the problem, local fractional Mohand transform with Adomian decomposition method is introduced due to its simple approach and less computational work. Furthermore, for the applicability of the technique, we illustrate some examples and their exact or approximate solutions with their graphical representations.

The basic idea of this paper is to investigate the approximate solution to a well-known Fornberg–Whitham equation of arbitrary order. We consider the stated problem under $\mathrm{\text{ABC}}$ fractional order derivative. The proposed derivative is non-local and contains non-singular kernel of Mittag-Leffler type. With the help of Modified Homotopy Perturbation Method (MHPM), we find approximate solution to the aforesaid equations. The required solution is computed in the form of infinite series. The method needs no discretization or collocation and easy to implement to compute the approximate solution that we intend. We also compare our results with that of the exact solution for the initial four terms approximate solution as well as with that computed by the Laplace decomposition method. We also plot the approximate solution of considered model through surface plots. For numerical illustration, we use Matlab throughout this work.

In this paper, we develop a numerical method for the solution of nonlinear fractional integral equations (NFIEs) based on Haar wavelet collocation technique (HWCT). Under certain conditions, we also prove the uniqueness and existence as well as Hyers–Ulam (HU) stability of the solution. With the help of the mentioned technique, the considered problem is transformed to a system of algebraic equations which is then solved for the required results by using Broyden algorithm. To check the validation and convergence of the proposed technique, some examples are given. For different number of collocation points (CPs), maximum absolute and mean square root errors are computed. The results show that for solving these equations, the HWCT is effective. The convergence rate is also measured for different CPs, which is nearly equal to $2.$

The purpose of this research is to explore the spread dynamics of a novel coronavirus outbreak, or 2019-nCOV via a fractional approach of type fractal-fractional (FF) derivative. We considered the FF approach in sense of the Atangana–Baleanu derivative for the system 2019-nCOV. In the FF operator, when we choose fractional-order one, we achieve the fractal model and when choosing fractal order one then we obtain a fractional model and while considering both the operators together we obtain the fractal-fractional model. The obtained results show via graphics for the different collections of fractal and fractional orders. The graphical results show the new operator impacts on a practical situation in a more visual way.

The Ebola virus infection (EVI), generally known as Ebola hemorrhagic fever, is a major health concern. The occasional outbreaks of virus occur primarily in certain parts of Africa. Many researches have been devoted to the study of the Ebola virus disease. In this paper, we have taken susceptible-infected-recovered-deceased-environment (SIRDP) system to investigate the dynamics of Ebola virus infection. We adopted fractional operators for a better illustration of model dynamics and memory effects. Initially, the Ebola disease model is modified with Caputo–Fabrizio arbitrary operator in Caputo sense (CFC) and we employed the fixed-point results for the existence and uniqueness of the solution of the fractional system. Further, we adopted the arbitrary fractional conformable and $\beta$-conformable derivatives to the alternative representation of the model. For the numerical approximation of the system, we show a numerical technique based on the fundamental theorem of fractional calculus for CFC derivative and a numerical scheme called the Adams–Moulton for conformable derivatives. Finally, for the validation of theoretical results, the numerical simulations are displayed.

In this paper, we present a class of stochastic system of fractional space diffusion equations forced by additive noise. Our goal here is to approximate the solutions of this system via a system of ordinary differential equations. Moreover, we study the influence of the same degenerate additive noise on the stability of the solutions of the stochastic system of fractional diffusion equations. We are interested in the systems that have nonlinear polynomial and give applications as Lotka–Volterra system from biology and the Brusselator model for the Belousov–Zhabotinsky chemical reaction from chemistry to illustrate our results.

The social habit of smoking has affected the whole world in a social manner. It is the main cause of diseases like cancers, asthma, bad breath, etc., and a source of spreading of infectious diseases like COVID-19. This work is related to an existing smoking model with relapse habit converted in fractional order. First, formulation of fractional-order smoking model is presented and then the dynamics of proposed problem is analyzed. Fixed-point theory via Banach contraction and Schauder theorems is used to derive the existence and uniqueness of the model. At last, the adaptive predictor–corrector algorithm and Runge–Kutta fourth-order (RK4) strategy are used to perform simulation. To bolster the validity of the theoretical results, a set of numerical simulations are performed. A good agreement between hypothetical and numerical results is demonstrated via numerical simulations using MATLAB software.

In this paper, we develop a numerical method for the solutions of mixed type Volterra–Fredholm fractional integral equations (FIEs). The proposed algorithm is based on Haar wavelet collocation technique (HWCT). Under certain conditions, we prove the existence and uniqueness of the solution. Also, some stability results are given of Hyers–Ulam (H–U) type. With the help of the HWCT, the considered problem is transformed into a system of algebraic equations which is then solved for the required results by using Gauss elimination algorithm. Some numerical examples for convergence of the proposed technique are taken from the literature. Maximum absolute and root mean square errors are calculated for different collocation points (CPs). The results show that the HWCT is an effective method for solving FIEs. The convergence rate for different CPS is also calculated, which is nearly equal to $2$.

In this paper, we consider a fractional mathematical model describing the co-infection of HBV and HCV under the non-singular Mittag-Leffler derivative. We also investigate the qualitative analysis for at least one solution and a unique solution by applying the approach fixed point theory. For an approximate solution, the technique of the iterative fractional order Adams–Bashforth scheme has been implemented. The simulation for the proposed scheme has been drawn at various fractional order values lying between (0,1) and integer-order of 1 via using Matlab. All the compartments have shown convergence and stability with time. A detailed comparative result has been given by the different fractional orders, which showed that the stability was achieved more rapidly at low orders.

We investigate the appropriate and sufficient conditions for the existence and uniqueness of a solution for a coupled system of Atangana–Baleanu fractional equations with a p-Laplacian operator. We also study the HU-stability of the solution by using the Atangana–Baleanu–Caputo (ABC) derivative. To achieve these goals, we convert the coupled system of Atangana–Baleanu fractional equations into an integral equation form with the help of Green functions. The existence of the solution is proven by using topological degree theory and Banach’s fixed point theorem, with which we analyze the solution’s continuity, equicontinuity and boundedness. Then, we use Arzela–Ascolli theory to ensure that the solution is completely continuous. Uniqueness is established using the Banach contraction principle. We also investigate several adequate conditions for HU-stability and generalized HU-stability of the solution. An illustrative example is presented to verify our results.

A new differentiation technique, fractional pseudospectral shifted Legendre differentiation matrices (FSL D-matrices), was introduced. It depends on shifted Legendre polynomials (SLPs) as a base function. We take into consideration its extreme points and inner product. The technique was used to solve fractional ordinary differential equations (FODEs). Moreover, it extended to approximate fractional integro-differential equations (FIDEs) and fractional optimal control problems (FOCPs). The novel FSL D-matrices transformed these fractional differential problems (FDPs) into an algebraic system of equations. Also, an error and a convergence analysis for that technique were investigated. Finally, the correctness and efficiency of this technique were examined with test functions and several examples. All the results were compared with the results of other methods to ensure the investigated error analysis.

Boson intensity correlations were examined to explore the fluid characterization produced in the collisions at the largest colliders and were perceived to have an astonishing curtailment during the computation of three-pion quantum interferences. Such analogous suppression can be probed to investigate the characteristics of particle production sources created during the collisions of heavy nuclei at unprecedented energies. We have demonstrated and analyzed the particle emissions from radiated sources with Bose–Einstein coherence that induces three-particle interferences to investigate the peculiarity of the particle emitted fluids. We are perspicacious that the bosons resemble the pertinent aspirant of coherence, and the normalized three-particle correlations evaluate the occurrence of such conglomerate fluid phases of chaotic and coherence significantly. Moreover, we also explored and analyzed the cumulant and the normalized correlations to examine the specific features of particle emission sources during the smashing of heavy nuclei. With such particular and pioneering approach, we have observed a consequential difference in the three-particle correlations and the normalized correlator at small temperature regimes.

We propose a new model to investigate the recent coronavirus (COVID-19) epidemic. The fundamental reproductive number is determined using the next-generation matrix, and the model’s soundness is tested using stability theory. The local asymptotic stability conditions for the disease-free equilibrium are determined. The model is extended to the fractional domain, and its properties are investigated using the Caputo–Fabrizio (CFC) robust nonsingular and nonlocal fractional operator. We focused on numerous exciting elements of the expanded model, such as solution positivity. In numerical aspects, different values of numerous parameters are employed to view certain physical behavior of the model.

Currently, a study has come out with a novel class of differential operators using fractional-order and variable-order fractal Atangana–Baleanu derivative, which in turn, became the source of inspiration for new class of differential equations. The aim of this paper is to apply the operation matrix to get numerical solutions to this new class of differential equations and help us to simplify the problem and transform it into a system of an algebraic equation. This method is applied to solve two types, linear and nonlinear of fractal differential equations. Some numerical examples are given to display the simplicity and accuracy of the proposed technique and compare it with the predictor–corrector and mixture two-step Lagrange polynomial and the fundamental theorem of fractional calculus methods.

In this paper, we introduce and investigate a new class of coupled fractional $q$-integro-difference equations involving Riemann–Liouville fractional $q$-derivatives and $q$-integrals of different orders, equipped with $q$-integral-coupled boundary conditions. The given problem is converted into an equivalent fixed-point problem by introducing an operator whose fixed-points coincide with solutions of the problem at hand. The existence and uniqueness results for the given problem are, respectively, derived by applying Leray–Schauder nonlinear alternative and Banach contraction mapping principle. Illustrative examples for the obtained results are constructed. This paper concludes with some interesting observations and special cases dealing with uncoupled boundary conditions, and non-integral and integral types nonlinearities.

In this paper, cubic polynomial and nonpolynomial splines are developed to solve solutions of 10th- and 12th-order nonlinear boundary value problems (BVPs). Such types of BVPs occur when a consistent magnetized force field is applied crosswise the fluid in the substance of gravitational force. We will amend our problem into such a form that converts the system of $10$th- $\&$ $12$th-order BVPs into a new system of $2$nd-order BVPs. The appropriate outcomes by using CP Spline and CNP Spline are compared with the exact root. To show the efficiency of our results, absolute errors calculated by using CP Spline and CNP Spline have been compared with other methods like differential transform method, Adomian decomposition method, variational iteration method, cubic B-spline, homotopy perturbation method, $5$th- and $6$th-order B-spline and our results are very encouraging. Graphs and tables are also presented in the numerical section of this paper.

The enhancement of the working ability of the industrial fluid is the need of the present era; nanofluid is an emerging field in science and technology. In this study, the Brinkman-type fluid model is used and is generalized using the Fourier’s and Fick’s laws. The graphene oxide nanoparticles are dispersed in the base fluid water. The fractional partial differential equations are then solved via the Laplace and Fourier transform method. The obtained solutions for velocity, heat transfer, and mass transfer are plotted in graphs. The results show that velocity profile decreases for Brinkman-type fluid parameter and volume fraction of the nanoparticles. The plot for the fractional parameter shows that different plots can be drawn for a fixed time and other physical parameters, which is the memory effect.

In this paper, we explore the nonlinear dynamics of two dipole-coupled-trapped ions in the Lamb–Dicke regime. The dynamics of the generated two-trapped-ions correlations under intrinsic decoherence is quantified by Bell function, the uncertainty-induced nonlocality, and the concurrence. We investigate the effects of the Lamb–Dicke nonlinearity, the intrinsic decoherence, and the two-trapped-ions coupling. It is found that the two-trapped-ions state has different nonlocal correlations. These correlations can be controlled. In the absence of decoherence, the nonlocal correlations can be enhanced by the Lamb–Dicke nonlinearity and the two-trapped-ions coupling. The stable value and the sudden death-birth phenomenon of the entanglement can be apperceived due to the increase of the Lamb–Dicke nonlinearity. The intrinsic decoherence reduces and stabilizes the nonlocal correlations. They are more pronounced with large Lamb–Dicke values. The decoherence effects are reduced by the strong Lamb–Dicke nonlinearity.

This paper considers an arbitrary-order mathematical model that analyzes the syndrome type Middle Eastern Coronavirus (MERS-CoV) under the nonsingular Mittag-Leffler derivative. Such types of viruses were transferred from camels to the population of humans in the Arabian deserts. We investigate the said problem for theoretical results and determine the existence of a solution by using the fixed point theory concept. For the approximate solution, the well-known method of iteration arbitrary-order Adams–Bashforth (AB) technique has been used. A numerical scheme is developed for the analyzed problem in the continuation of simulations at different noninteger and natural order for the interval ($0,1$]. The graphical representation shows that all classes show convergence and achieve a stable position with growing time. A good comparative result has been given by the analysis of various noninteger orders with natural orders and achieves stability quickly at fewer non-integer orders.

Aim: Tuberculosis is an infectious disease caused by Mycobacterium tuberculosis bacteria. This study plans to build a novel deep learning-based model for the accurate recognition of tuberculosis. Methods: We propose a novel model — rotation angle vector grid-based fractional Fourier entropy and deep stacked sparse autoencoder (RAVG-FrFE–DSSAE) — which uses RAVG-FrFE as a feature extractor and harnesses DSSAE as the classifier. Moreover, an 18-way MDA is introduced on the training set to avoid overfitting. Results: Experimental results of 10 runs of 10-fold CV showcase that this proposed RAVG-FrFE–DSSAE algorithm yields a reasonable performance including of 93.68±1.11% sensitivity, 94.38±1.11% specificity, 94.35±1.04% precision, 94.03±0.69% accuracy, 94.01±0.70% $F_1$-score, 88.07±1.38% MCC, 94.01±0.70% FMI, and 0.9725 AUC, respectively. Conclusions: Our result outperforms the eight state-of-the-art approaches. Besides, the result shows the effectiveness of the 18-way MDA.

Chemical processes are constantly occurring in all existing creatures, and most of them contain proteins that are enzymes and perform as catalysts. To understand the dynamics of such phenomena, mathematical modeling is a powerful tool of study. This study is carried out for the dynamics of cooperative phenomenon based on chemical kinetics. Observations indicate that fractional models are more practical to describe complex systems’ dynamics, such as recording the memory in partial and full domains of particular operations. Therefore, this model is modeled in terms of classical-order-coupled nonlinear ODEs. Then the classical model is generalized with two different fractional operators of Caputo and Atangana–Baleanu in a Caputo sense. Some fundamental theoretical analysis for both the fractional models is also made. Reaction speeds for the extreme cases of positive/negative and no cooperation are also calculated. The graphical solutions are achieved via numerical schemes, and the simulations for both the models are carried out through the computational software MATLAB. It is observed that both the fractional models of Caputo and Atangana–Baleanu give identical results for integer order, i.e. $\alpha =\beta =1$. By decreasing the fractional parameters, the concentration profile of the substrate $S$ takes more time to vanish. Moreover, binding of first substrate increases the reaction rate at another binding site in the case of extreme positive cooperation, while the opposite effect is noticed for the case of negative cooperativity. Furthermore, the effects of other parameters on concentration profiles of different species are shown graphically and discussed physically.

The usage of Lévy processes involving big moves or jumps over a short period of time has proven to be a successful strategy in financial analysis to capture such rare or extreme events of stock price dynamics. Models that follow the Lévy process are FMLS, Kobol, and CGMY models. Such simulations steadily raise the attention of researchers in science because of the certain best options they produce. Thus, the issue of resolving these three separate styles has gained more interest. In the new paper, we introduce the computational method of such models. At first, the left and right tempered fractional derivative with arbitrary order is approximated by using the basis function of the shifted Chebyshev polynomials of the third kind (SCPTK). In the second point, by implementing finite difference approximation, we get the semi-discrete structure to solve the tempered fractional B–S model (TFBSM). We show that this system is stable and ${𝒪}(\delta \tau )$ is the convergence order. In practice, the processing time and the calculation time per iteration will be reduced by a quickly stabilized system. Then we use SCPTK to approximate the spatial fractional derivative to get the full design. Finally, two numerical examples are provided to illustrate the established system’s reliability and effectiveness.

The aim of this paper is to model corona-virus (COVID-19) taking into account random perturbations. The suggested model is composed of four different classes i.e. the susceptible population, the smart lockdown class, the infectious population, and the recovered population. We investigate the proposed problem for the derivation of at least one unique solution in the positive feasible region of nonlocal solution. For one stationary ergodic distribution, the necessary result of existence is developed by applying the Lyapunov function and the condition for the extinction of the disease is also established. The obtained results show that the effect of Brownian motion and noise terms on the transmission of the epidemic is very high. If the noise is large the infection may decrease or vanish. For validation of our obtained scheme, the results for all the classes of the problem have been simulated numerically.

In this paper, we consider the trapping problem on the level-3 Sierpinski gasket $({\mathrm{\text{SG}}}_3)$ when both nearest-neighbor (NN) and non-nearest-neighbor (NNN) jumps are included. Based on the topological structure of network and the method of probability generation function, we get the analytical expression of the average trapping time (ATT). Therefore, compared with the case where only NN jumps are allowed, we modify the specific value of ATT. Our results prove the exponent of the scaling expression has nothing to do with the NNN jump probability $q$, i.e. the scaling expression of ATT still scales superlinearly with the large network size. According to the analytical expression, we do numerical simulations of ATT with respect to parameters $q$ and $n$ (the number of generations of the network). The results show that the ATT will decrease with the increase of $q$. These also indicate that NNN jump is helpful to improve the efficiency of random walk on network ${\mathrm{\text{SG}}}_3$.

To minimize the thermal resistance of the fractal treelike heat conduction network and develop an optimization principle applicable for the network with an arbitrary cross-sectional shape, this paper first establishes a theoretical model regarding the total thermal resistance of the symmetric treelike network with arbitrary cross-sectional shapes and then studies the effects of the geometric and structural parameters of the network on its total thermal resistance. The numerical simulations are also performed to analyze the influences of the geometric and structural parameters of symmetric treelike networks with circular, rectangular and triangular cross-sectional shapes on the total thermal resistance. Both the theoretical model and the numerical simulation show that the total thermal resistance of the network with an arbitrary cross-sectional shape first decreases and then increases with increasing cross-sectional area ratio but always increases with increasing length ratio of branches at two successive branching levels when the total branch volume is constant. When the cross-sectional area ratio is equal to the reciprocal of the branching number, the treelike network has the minimum total thermal resistance. This scaling law is applicable for the treelike network with an arbitrary cross-sectional shape to achieve the minimum total thermal resistance.

The effective thermal conductivity of porous media is of steady interest in the design of new materials. In this study, an analytical model for dimensionless effective thermal conductivity of media embedded with fracture networks of the power law length distributions is proposed. It is found that the proposed dimensionless effective thermal conductivity is a function of micro-structural parameters of media, such as the porosity $(\varphi )$ and fracture orientation (dip $𝜃$ and azimuth $\phi$). The present results show that the dimensionless effective thermal conductivity of the media increases with the increase in the ratio $(k_s/k_f)$ and the power law exponent $\alpha$ at $k_s/k_f<1$. Inversely, when $k_s/k_f>1$, it decreases with the increase in the power law exponent $\alpha$. In addition, the dimensionless effective thermal conductivity is gradually independent of the orientation as the ratio $k_s/k_f>1$. The present model may provide a significant insight into the mechanism of heat transfer in the media embedded with fracture networks of power law length distributions.

Permeability is a critical parameter for characterizing the migration behavior of a fluid in porous media, thus being of great importance in guiding the geotechnical engineering practice. In this study, an improved permeability model was developed on the basis of the fractal features of the porous geomaterials. It further takes into account the periodic variation of the cross-section of the pore path in the flow direction and the roughness of the pore surfaces. This model indicates that the permeability is a function of the porosity, maximum pore radius, proportion of minimum pore radius to the maximum pore radius, roughness of the capillary tube, and variation in the cross-sectional characteristics of the tube. It provides a better insight into the transport mechanism of fluids in porous geomaterials. The permeability of the model is related to the variable cross-sectional characteristics of the pore channel, i.e. the belly-to-throat ratio and the roughness, as a quadratic function. These two parameters are the key to affecting the level of permeability. The model was verified over a wide range of permeability variations by changing from high-permeability media (10² mD) to low-permeability media (10⁻³ mD). Accurate permeability prediction of high–low-permeability porous media can be achieved by considering the variation of parameters.

The study of strange nonchaotic attractors (SNAs) has been mainly restricted to quasiperiodically forced systems. At present, SNAs have also been uncovered in several periodically forced smooth systems with noise. In this work, we consider a periodically forced nonsmooth system and find that SNAs are created by a small amount of noise. SNAs can be generated in different periodic windows with weak noise perturbation. If the parameter is varied further from the chaotic range, a larger noise intensity is required to induce SNAs. Besides, noise-induced SNAs can be generated by the periodic attractors near the boundary crisis. In addition, with the increasing noise intensity, the intermittency between SNAs and periodic attractors can be induced by transient chaos. The characteristics of SNAs are analyzed by the Lyapunov exponent, power spectrum, singular continuous spectrum, spectral distribution functions, and finite time Lyapunov exponent.

It is an open question whether Wada basin exists in control systems. We investigate coexisting attractors and their Wada basins for a given set of parameters in an uncontrolled and controlled vibration isolation system. Wada basins are verified by the Nusse–Yorke method. All coexisting chaotic attractors can be converted into periodic attractors but the basin structure can still possess Wada property by two control methods (an external drive control and a piecewise damping control) with the appropriate control parameters. The Wada basins of controlled systems imply a particular kind of unpredictability, and we should avoid selecting such control parameters although chaotic attractors are effectively controlled.

Ultraslow diffusion describes the long-time diffusion of particles whose mean square displacement (MSD) grows logarithmically in time. We prove the well-posedness of a Caputo–Hadamard time-fractional diffusion model in multiple space dimensions, in which the MSD in time grows logarithmically and thus provides adequate descriptions for the ultraslow diffusion processes, as well as the smoothing properties of the solutions.

Seepage flow in low permeability reservoirs is strongly influenced by the boundary layer, which is characterized by a low-velocity non-Darcy flow phenomenon, and a nonlinear relationship between velocity and pressure gradient. The viscosity is not uniformly distributed in the pore throat and has a great effect on the nonlinear flow behavior. In this paper, a low-velocity non-Darcy flow model is proposed. By quantitatively characterizing the boundary layer thickness and considering the non-uniform distribution of viscosity for the first time, a modified Hagen–Poiseuille equation combining the boundary layer effect and viscosity distribution in a single capillary is derived. Then, the analytical expression of the nonlinear relationship between the flow rate and pressure gradient in the fractal porous media is established, assuming that the pores in the tight formation are fractal distribution. Finally, the validity of the model is verified by experimental data. The relative error between the model and experimental data, with a permeability smaller than 3mD, is 3.27%. And the parameters affecting low-velocity non-Darcy flow are also analyzed. The results showed that the influence of boundary layer is one of the causes of low-velocity non-Darcy flow in tight reservoir. The study has also revealed that the non-uniform distribution of viscosity in the pore throat has a great influence on the nonlinear flow, which is often neglected in previous works.

The paper aims to obtain the cushion curves of a polymer foam commonly used in the protective packaging of fragile products and its fractal cushioning performance. First, the impact curves are obtained to evaluate the dropping damage of a critical component considering a fractal nonlinear dynamic packaging system model, more specifically a fractal cubic-quintic-heptic Duffing’s equation. Then, the two-scale fractal derivative transform is introduced to determine the packaging material fractal behavior due to the material molecular structure. The findings explain the dynamic response behavior of the porous cushion material, and the proposed fractal dynamic model sheds new light on the design and analysis of cushioning packaging systems.

The unsmooth boundary has a great influence on the solitary wave form of a nonlinear wave equation. It this work, we for the first time ever propose the fractal regularized long-wave equation which can describe the shallow water waves under the unsmooth boundary (such as the fractal seabed). The fractal variational principle is established and is proved to have a strong minimum condition by the He–Weierstrass theorem. Then, the solitary wave solution is obtained by the fractal variational method which can reduce the order of differential equation and obtain the optimal solution by the stationary condition. Finally, the impact of the unsmooth boundary on the solitary wave is presented. It shows that the fractal order can affect the wave morphology, but cannot affect its peak value. The finding in this paper is important for the coast protection and expected to bring a light to the study of the fractal theoretical basis in the geosciences.

This paper argues that in the microgravity space, the fractal nonlinear oscillators’ models with discontinuities are established by the fractal calculus, and their fractal variational principles are obtained via using the fractal semi-inverse method. Finally, a novel technology is proposed to find the frequency of the fractal nonlinear oscillators’ model by using the Fourier series. The results illustrate that the proposed method is efficient and accurate.

Gas diffusion coefficient is an important parameter in characterizing gas transport in porous media. Based on the fractal characteristics of microscopic capillaries, the effect of capillary surface roughness on gas diffusion is considered by the Monte Carlo technique. We proposed a probability model of the effective gas diffusion coefficient of the porous media with rough surfaces. This model is expressed as a function of the geometrical parameters of porous media, such as the porosity, the pore fractal dimension, the tortuosity fractal dimension, and the relative roughness. The simulated effective gas diffusion coefficient is compared with the available experimental data and a good agreement between them is obtained. The effect of the structural parameters of porous media on gas diffusion is analyzed in detail.

This paper mainly investigated the influence of the fractal dimension of mixed mode I/II crack on its near-tip plastic zones and propagation angles. The near-tip distortional strain energy density for mixed mode I/II crack is modified by considering its fractal dimension. Using the Von Mises yield condition, the plastic zones at the fractal crack tip are evaluated. The effects of the fractal dimension of crack on the near-tip plastic zones are analyzed qualitatively. A modified criterion minimizes the fractal distortional strain energy density to predict the propagation direction of mixed mode I/II crack. Based on this, the effects of fractal dimension on the mixed mode I/II crack propagation direction at different inclination angles are discussed quantitatively. In addition, the modified criterion is compared to the experimental data in the literature, as well as the classical maximum tangential stress criterion and minimum strain energy density criterion. The results indicate that the fractal dimension of crack is an important factor in evaluating the crack-tip plastic zones and predicting the crack propagation angle, and the modified fractal criterion is more accurate and effective in predicting the crack propagation angles.

Due to the alarming rise in types of crime committed and the number of criminal activities across the world, there is a great need to amend the existing policies and models adopted by jurisdictional institutes. The majority of the mathematical models have not included the history of the crime committed by the individual, which is vital to control crime transmission in stipulated time. Further, due to various external factors and policies, a considerable number of criminals have not been imprisoned. To address the aforementioned issues prevailing in society, this research proposes a fractional-order crime transmission model by categorizing the existing population into four clusters. These clusters include law-abiding citizens, criminally active individuals who have not been imprisoned, prisoners, and prisoners who completed the prison tenure. The well-posedness and stability of the proposed fractional model are discussed in this work. Furthermore, the proposed model is extended to the delayed model by introducing the time-delay coefficient as time lag occurs between the individual’s offense and the judgment. The endemic equilibrium of the delayed model is locally asymptotically stable up to a certain extent, after which bifurcation occurs.

In theoretical setting, associated with a fractional $\psi$-Fueter operator that depends on an additional vector of complex parameters with fractional real parts, this paper establishes a fractional analog of Borel–Pompeiu formula as a first step to develop a fractional $\psi$-hyperholomorphic function theory and the related operator calculus.

This paper studies the efficiency of Brazilian activity sectors. For that, we apply the Macroeconophysics Indicator of Economic Efficiency (MIEE) for each sector’s index of the daily closing price in the stock market. The MIEE quantifies efficiency considering permutation entropy and Fisher Information. We divide the indices time series into two periods: before COVID-19 and during COVID-19. The overall results indicate that efficiency has decreased for the majority of stock market indices, suggesting that the recent crisis has had a deleterious effect on stock efficiency.

In this paper, the sufficient conditions for the stability of the equilibrium to the fractional-order differential dynamical system with Caputo–Hadamard derivative are investigated. The existence of Hopf bifurcation in two-dimensional fractional dynamical system with a positive parameter is shown. Then, some numerical simulations are presented for a better understanding.

The vibration of carbon nanotubes embedded in elastic medium is a hot topic and many research findings are available. But when the medium is a fractal medium such as chips, ceramics, etc., its vibration characteristics are difficult to explain with the existing theories. So in this paper, we mainly study the nonlinear vibration of carbon nanotube embedded in a fractal medium. The fractal variational principle of the problem is successfully established, and is proved to have a strong minimum condition according to the He–Weierstrass Theorem. Then the Hamiltonian theory combined with the residual equation is implemented to find the solution of the fractal nonlinear vibration. Finally, the impacts of different fractal orders on the vibration characteristics are also presented.

Based on local fractional calculus theory, the Hölder double local fractional integral inequality with Weighted is proved. By using the methods of weight function and some analysis techniques on the fractal real set, a local fractional integral inequality with the best constant is given, which is generalization of Yang–Hardy–Hilbert’s inequality on the fractal set ${ℝ}_{+}^{\alpha }$ of fractal dimension $\alpha (0<\alpha \le 1)$ and its equivalent form is considered.

We give general method of constructing invisible sets on square Sierpiński carpets which can be asymmetrical. As an application, we obtain upper bounds of the conformal dimensions of square Sierpiński carpets. The bounds are less than the Hausdorff dimensions of square Sierpiński carpets.

We study the graphs associated with Vicsek sets in higher dimensional settings. First, we study the eigenvalues of the Laplacians on the approximating graphs of the Vicsek sets, finding a general spectral decimation function. This is an extension of earlier results on two-dimensional Vicsek sets. Second, we study the Vicsek set lattices, which are natural analogues to the Sierpinski lattices. We have a criterion when two different Vicsek set lattices are isomorphic.

Large-scale simulations and analysis of the original powers of 2 geometric source sum algorithm for simulating $1/f$ noise confirm that it provides a reasonably accurate approximation to an exact $1/f$ spectral density with a Gaussian amplitude distribution over any arbitrarily large frequency range. This incremental algorithm is computationally efficient with a computation time, after initialization, that varies linearly with the number of samples generated. A new variation allows non-integer and random ratios for the geometric sequence that reduces variations about the exact power-law spectral density and, by varying individual source amplitudes, produces generalized $1/f^{\beta }$ noises.

In this paper, a power-law strain-dependent variable order is first incorporated into the fractional constitutive model and employed to describe mechanical behaviors of aluminum foams under quasi-static compression and tension. Comparative results illustrate that power-law strain-dependent variable order is capable of better describing stress–strain responses compared with the traditional linear one. The evolution of fractional order along with the porosities or relative densities can be well qualitatively interpreted by its physical meaning. Furthermore, the model is also extended to characterize the impact behaviors under large constant strain rates. It is observed that fractional model with sinusoidal variable order agrees well with the experimental data of aluminum foams with impact and non-impact surfaces.

On analogy of the regular continued fractions, for any fixed positive integer $N\in \mathbb{N}$, every $x\in [0,1)$ can be expanded into an $N$-continued fraction, denoted by $x={[a_1(x),a_2(x),\dots ]}_N$, where $a_n(x)$ are called the partial quotients. In this paper, we concern the metric theory of the partial quotients. More precisely, let $\varphi :\mathbb{N}\to (0,\infty )$, the Borel–Bernstein theorem and Hausdorff dimension of the set $\{x\in [0,1):a_n(x)\ge \varphi (n)$ for infinitely many $n\in \mathbb{N}\}$ are determined. This generalizes the results of regular continued fractions.

This paper studies the capability of digital architecture to mimic fractal behavior. As chaotic attractors realized digitally had opened many tracks, digital designs mimicking fractals may ultimately achieve the same. This study is based on a complex single-dimensional discrete chaotic system known as the generalized positive logistic map. The fractals realized from this system are linked to the results of the mathematical analysis to understand the fractal behavior with different variations. A digital hardware architecture manifesting the fractal behavior is achieved on FPGA, showing a fractal entity experimentally. With this digital realization, it is hoped that fractals can follow the example of chaotic attractors digital applications.

In recent years, a new idea of the fractal fractional derivative has been introduced. However, it is not used for the free convection flow drilling nanoliquid with clay nanoparticles. In this paper, we have deliberated this new approach of the fractal fractional derivative with a power-law kernel for heat transfer in a drilling nanofluid with clay nanoparticle in a vertical channel. Water is taken as the base fluid. The flow of the fluid is between two fixed vertical parallel plates such that one of them is constantly heated. The resulting problem is modeled with a fractal fractional derivative operator with a power-law kernel. As the exact solution to this problem is not possible, therefore, the Crank–Nicolson Finite Difference Scheme (CNFDS) is used for numerical solutions. This idea for the fractal fractional fluid problem is used here for the first time in the literature. Results are portrayed in the various graphs using Maple-15 software. The importance of both fractal and fractional parameters is discussed in detail. Results showed that the fractal fractional parameter gives a more general solution of the velocity and temperature for the fixed fractional operators. Therefore, a mutual approach of fractal fractional clarifies the memory of the function better than fractional only.

In this paper, we investigate a class of self-similar networks modeled on a self-similar fractal tree, and use the self-similar measure and the method of finite pattern to obtain the asymptotic formula of node-weighted average Fermat distances on fractal tree networks.

This paper aims at establishing two different types of wave models with unsmooth boundaries by the fractal calculus, and their fractal variational principles are successfully designed by employing the fractal semi-inverse transform method. A new approximate technology is proposed to solve the two fractal models based on the variational principle and fractal two-scale transform method. Finally, two numerical examples show that the proposed method is efficient and accurate, which can be extended to solve different types of fractal models.

In this paper, the convergence of the Nash-Q-Learning algorithm will be studied mainly. In the previous proof of convergence, each stage of the game must have a global optimal point or a saddle point. Obviously, the assumption is so strict that there are not many application scenarios for the algorithm. At the same time, the algorithm can also get a convergent result in the two Grid-World Games, which do not meet the above assumptions. Thus, previous researchers proposed that the assumptions may be appropriately relaxed. However, a rigorous theoretical proof is not given. The convergence point is a fractal attractor from the view of Fractals, general proof of convergence of the Nash-Q-Learning algorithm will be shown by the mathematical method. Meanwhile, some discussions on the efficiency and scalability of the algorithm are also described in detail.

The seepage in tree-like bifurcating networks is a very common phenomenon in nature. The research on the transport characteristics of tree-like bifurcating networks has always been a hot topic. In this paper, a novel permeability model for fluid flow in damaged tree-like bifurcating networks is proposed. In the proposed model, the influence of roughness on permeability is considered by means of the fractal method. It is found that the permeability is not only related to the structural parameters of the network but also related to the damaged position and the number of damaged tubes at the damaged position. The effects of these parameters and damaged structure on permeability are discussed separately. The results show that the permeability reduces along with an increase in the roughness level, the length ratio, the number of damaged tubes, and the number of total bifurcating levels. Another major finding is that the permeability increases with an increase in the diameter ratio. Besides, we found that the damaged position and the number of damaged tubes at the damaged position have an important effect on the permeability. Increasing the number of damaged tubes and bringing the damaged position close to the front end of the network will reduce the permeability. Compared with the undamaged network, the permeability of damaged network has a significant decline. The proposed model may provide potential applications for the analysis of fluid flow in damaged tree-like bifurcating network.

In view of the fractal evolution phenomenon in Internet information dissemination, this paper constructs a generalized fractal network by analyzing the characteristics of user dissemination behavior and according to the dissemination trend of actual information. On this basis, it reveals the fractal evolution mechanism in Internet information dissemination and its dissemination dynamic characteristics and rules. First, based on node similarity, a fractal propagation model of Internet information is constructed. Second, based on the scale-free network and according to the user characteristics, the information dissemination process is simulated and a generalized fractal network is evolved. Then, the key factors influencing the formation of fractal networks, the characteristics, and the rules of propagation dynamics are analyzed. Finally, the rationality and validity of the model are verified by crawling the real user forwarded data in the micro-blog.

Lazy random walks have been used in many scientific fields such as image segmentation and optimal transport; however, related theoretical results are much less for this dynamical process. In this paper, we study lazy random walks in a pseudofractal scale-free web, where the self-loop jumps on graph vertexes are considered. For a special random walks with one trap fixed at a hub node, also known as the trapping problem, we derive the exact analytic formulas of the average trapping time (ATT), an indicator measuring the efficiency of the trapping process, by using two different methods. The results obtained by the two methods are consistent. Analyzing and comparing the obtained solutions, we find that the ATT is related to the walking rule with the self-loop jumps. Specifically, adding the self-loop to change the walking rule can affect the coefficient of the ATT formula, but it cannot change the leading scaling of the trapping efficiency. We hope that these results in this paper can help us better understand the biased random walk process in complex systems.

The focus of this paper is to study unbounded variation functions from the perspective of Hölder conditions. Three special unbounded variation functions have been constructed. The first is a continuous function of unbounded variation that satisfies the Hölder condition of a given order and the second is a continuous function of unbounded variation that does not satisfy the Hölder condition of any order. The third is a continuous function of unbounded variation defined on any sub-interval of the interval $I$. Then, specific fractal dimension analysis of the above functions and relevant conclusions have been investigated. Finally, combining functional analysis and reinforcement learning, the convergence of reinforcement learning algorithms can be proved in unified framework.

In this paper, we defined a generalized weighted Caputo-type differential operator. Then, it can express as a convergent series through the Riemann–Liouville integral. At the same time, by solving related linear differential equation, we can construct the generalized weighted Caputo-type integral operator. Lastly, some theorems and properties of these considered operators were also studied.

In this paper, we analyzed the variations in brain activation between different activities. Since Electroencephalogram (EEG) signals as an indicator of brain activation contain information and have complex structures, we employed complexity and information-based analysis. Specifically, we used fractal theory and Shannon entropy for our analysis. Eight subjects performed three different activities (standing, walking, and walking with a brain–computer interface) while their EEG signals were recorded. Based on the results, the complexity and information content of EEG signals have the greatest and smallest values in walking and standing, respectively. Complexity and information-based analysis can be applied to analyze the activations of other organs in different conditions.