Fractal and Fractional

In this paper, the nonlinear version of the Henry–Gronwall integral inequality with the tempered $\Psi$-Hilfer fractional integral is proved. The particular cases, including the linear one and the nonlinear integral inequality of this type with multiple tempered $\Psi$-Hilfer fractional integrals, are presented as corollaries. To illustrate the results, the problem of the nonexistence of blowing-up solutions of initial value problems for fractional differential equations with tempered $\Psi$-Caputo fractional derivative of order $0<\alpha <1$, where the right side may depend on time, the solution, or its tempered $\Psi$-Caputo fractional derivative of lower order, is investigated. As another application of the integral inequalities, sufficient conditions for the $\Psi$-exponential stability of trivial solutions are proven for these kinds of differential equations. Full article

An eco-epidemiological model involving competition regarding the predator and quarantine on infected prey is studied. The prey is divided into three compartments, namely susceptible, infected, and quarantine prey, while the predator only attacks the infected prey due to its weak condition caused by disease. To include the memory effect, the Caputo fractional derivative is employed. The model is validated by showing the existence, uniqueness, non-negativity, and boundedness of the solution. Three equilibrium points are obtained, namely predator-disease-free, predator-free-endemic, and predator-endemic points, which, respectively, represent the extinction of both predator and disease, the extinction of predator only, and the existence of all compartments. The local and global stability properties are investigated using the Matignon condition and the Lyapunov direct method. The numerical simulations using a predictor–corrector scheme are provided not only to confirm the analytical findings but also to explore more the dynamical behaviors, such as the impact of intraspecific competition, memory effect, and the occurrence of bifurcations. Full article

The main idea in this research is introducing another approximate method to calculate solutions of the fractional Torsion equation, which is one of the applied equations in civil engineering. Since the fractional order is closed to an integer, we convert the fractional Torsion equation to a perturbed ordinary differential equation involving a small parameter epsilon. Then we can find the exact solutions and approximate symmetries for the alternative approximation equation. Also, with help of the definition of conserved vector and the concept of nonlinear self-adjointness, approximate conservation laws(ACL) are obtained without approximate Lagrangians by using their approximate symmetries. In order to apply the presented theory, we apply the Lie symmetry analysis (LSA) and concept of nonlinear self-adjoint Torsion equation, which are very important in mathematics and engineering sciences, especially civil engineering. Full article

This paper presents an investigation into the dynamics of the emerging three-compartment financial bubble problem using a new non-singular kernel Atangana–Baleanu derivative operator. The problem is tested for at least one solution, and a unique root is determined using an iterative Newton approximation method, providing a globally stable fractional analysis technique. Curve sketches of the globalized model are provided, considering integers and other conformable orders. Sensitivities of the fractional order and other model parameters are examined, offering insights into their impact on the system dynamics. This research contributes to understanding financial bubbles and lays the groundwork for future studies in this field. Full article

This paper is devoted to studying the $\varrho$-Hilfer fractional snap dynamic system under the $\varrho$-Riemann–Liouville fractional integral conditions on unbounded domains $[a,\infty ),a\ge 0$, for the first time. The results concerning the existence and uniqueness, along with the Ulam–Hyers, Ulam–Hyers–Rassias, and semi-Ulam–Hyers–Rassias stabilities, are established in an appropriate special Banach space according to fractional calculus, fixed point theory, and nonlinear analysis. At the end, a numerical example is presented for the interpretation of the main results. Full article

Low-heat Portland (LHP) cement is a kind of high-belite cement, which has the characteristic of low hydration heat. Currently, it is extensively used in the temperature control of mass concrete. Based on the thermodynamic database of OPC-based materials, the thermodynamic software GEM-Selektor (noted as GEMS) is used for simulating the hydration products of the LHP cement paste. Then, according to the GEMS thermodynamic simulation results, MATLAB is used to visualize the initial and ultimate stages of LHP cement pastes; the effects of curing temperature and water to cement (w/c) ratio on hydration products are addressed; and the porosity, fractal dimension, and tortuosity of different pastes are calculated. It is found that an appropriately high curing temperature is important for reducing porosity, especially in the early hydration stage. Hydration time also has a significant impact on the hydration of LHP cement paste; long hydration time may reduce the impact of temperature on hydration products. The w/c ratio is another important consideration regarding the hydration degree and porosity of LHP paste, and under different curing temperatures, hydration times, and w/c ratios, the porosity varies from 5.91–32.91%. The fractal dimension of this work agrees with the previous findings. From tortuosity analysis, it can be concluded that the high curing temperature may cause significant tortuosity, further affecting the effective diffusivity of LHP cement paste. For cement pastes with low w/c ratio, this high curing temperature effect is mainly reflected in the early hydration stage, for ones with high w/c ratio, it is in turn evident under long-term curing. Full article

This paper introduces a new type of polynomials generated through the convolution of generalized multivariable Hermite polynomials and Appell polynomials. The paper explores several properties of these polynomials, including recurrence relations, explicit formulas using shift operators, and differential equations. Further, integrodifferential and partial differential equations for these polynomials are also derived. Additionally, the study showcases the practical applications of these findings by applying them to well-known polynomials, such as generalized multivariable Hermite-based Bernoulli and Euler polynomials. Thus, this research contributes to advancing the understanding and utilization of these hybrid polynomials in various mathematical contexts. Full article

In this paper, we present an efficient, new, and simple programmable method for finding approximate solutions to fractional differential equations based on Bernoulli wavelet approximations. Bernoulli Wavelet functions involve advantages such as orthogonality, simplicity, and ease of usage, in addition to the fact that fractional Bernoulli wavelets have exact operational matrices that improve the accuracy of the applied approach. A fractional differential equation was simplified to a system of algebraic equations using the fractional order integration operational matrices of Bernoulli wavelets. Examples are used to demonstrate the technique’s precision. Full article

Recent developments in electrical power grids have witnessed high utilization levels of renewable energy sources (RESs) and increased trends that benefit the batteries of electric vehicles (EVs). However, modern electrical power grids cause increased concerns due to their continuously reduced inertia resulting from RES characteristics. Therefore, this paper proposes an improved fractional-order frequency controller with a design optimization methodology. The proposed controller is represented by two cascaded control loops using the one-plus-proportional derivative (1 + PD) in the outer loop and a fractional-order proportional integral derivative (FOPID) in the inner loop, which form the proposed improved 1 + PD/FOPID. The main superior performance characteristics of the proposed 1 + PD/FOPID fractional-order frequency controller over existing methods include a faster response time with minimized overshoot/undershoot peaks, an ability for mitigating both high- and low-frequency disturbances, and coordination of EV participation in regulating electrical power grid frequency. Moreover, simultaneous determination of the proposed fractional-order frequency controller parameters is proposed using the recent manta ray foraging optimization (MRFO) algorithm. Performance comparisons of the proposed 1 + PD/FOPID fractional-order frequency controller with existing PID, FOPID, and PD/FOPID controllers are presented in the paper. The results show an improved response, and the disturbance mitigation is also obtained using the proposed MRFO-based 1 + PD/FOPID control and design optimization methodology. Full article

Recent years have seen an increase in scientific interest in the El Nio/La Nia Southern Oscillation (ENSO), a quasiperiodic climate phenomenon that takes place throughout the tropical Pacific Ocean over five years and causes significant harm. It is associated with the warm oceanic stage known as El Nio and the cold oceanic stage known as La Nia. In this research, the ENSO model is considered under a fractional operator, which is defined via a nonsingular and nonlocal kernel. Some theoretical features, such as equilibrium points and their stability, bifurcation maps, the existence of a unique solution via the Picard–Lindelof approach, and the stability of the solution via the Ulam–Hyres stability approach, are deliberated for the proposed ENSO model. The Adams–Bashforth numerical method, associated with Lagrangian interpolation, is used to obtain a numerical solution for the considered ENSO model. The complex dynamics of the ENSO model are displayed for a few fractional orders via MATLAB-18. Full article

A mathematical model consisting of weakly coupled time fractional one parabolic PDE and one ODE equations describing dynamical processes in porous media is our physical motivation. As is often performed, by solving analytically the ODE equation, such a system is reduced to an integro-parabolic equation. We focus on the numerical reconstruction of a diffusion coefficient at finite number space-points measurements. The well-posedness of the direct problem is investigated and energy estimates of their solutions are derived. The second order in time and space finite difference approximation of the direct problem is analyzed. The approach of Lagrangian multiplier adjoint equations is utilized to compute the Fréchet derivative of the least-square cost functional. A numerical solution based on the conjugate gradient method (CGM) of the inverse problem is studied. A number of computational examples are discussed. Full article

White noise is fundamentally linked to many processes; it has a flat power spectral density and a delta-correlated autocorrelation. Operators acting on white noise can result in coloured noise, whether they operate in the time domain, like fractional calculus, or in the frequency domain, like spectral processing. We investigate whether any of the white noise properties remain in the coloured noises produced by the action of an operator. For a coloured noise, which drives a physical system, we provide evidence to pinpoint the mother process from which it came. We demonstrate the existence of two indices, that is, kurtosis and codifference, whose values can categorise coloured noises according to their mother process. Four different mother processes are used in this study: Gaussian, Laplace, Cauchy, and Uniform white noise distributions. The mother process determines the kurtosis value of the coloured noises that are produced. It maintains its value for Gaussian, never converges for Cauchy, and takes values for Laplace and Uniform that are within a range of its white noise value. In addition, the codifference function maintains its value for zero lag-time essentially constant around the value of the corresponding white noise. Full article

In this research, we present a qualitative analysis for studying a new modification of a nonlinear hyperbolic fractional integro-differential equation (NHFIDEq) in dual Banach space $C_E\left(E, J\right)$. Under some suitable conditions, the existence and uniqueness of a solution are demonstrated with the use of fixed-point theorems. The verification of the offered method has been conducted by applying the Lerch matrix collocation (LMC) method as a numerical treatment. The major motivation for selecting the LMC approach is that it reduces the solution of the given NHFIDEq to a matrix representation form corresponding to a linear system of algebraic equations; additionally, to demonstrate that the proposed strategy has better precision than alternative numerical methods, we study the error and the convergence analysis. Finally, we introduce numerical examples illustrating comparisons between the exact solutions and numerical solutions for different values of the Lerch parameters $\lambda$ and time $t$ as well as how the absolute error in each example is calculated. Full article

Diabetic retinopathy (DR), which is seen in approximately one-third of diabetes patients worldwide, leads to irreversible vision loss and even blindness if not diagnosed and treated in time. It is vital to limit the progression of DR disease in order to prevent the loss of vision in diabetic patients. It is therefore essential that DR disease is diagnosed at an early phase. Thanks to retinal screening at least twice a year, DR disease can be diagnosed in its early phases. However, due to the variations and complexity of DR, it is really difficult to determine the phase of DR disease in current clinical diagnoses. This paper presents a robust artificial intelligence (AI)-based model that can overcome nonlinear dynamics with low computational complexity and high classification accuracy using fundus images to determine the phase of DR disease. The proposed model consists of four stages, excluding the preprocessing stage. In the preprocessing stage, fractal analysis is performed to reveal the presence of chaos in the dataset consisting of 12,500 color fundus images. In the first stage, two-dimensional stationary wavelet transform (2D-SWT) is applied to the dataset consisting of color fundus images in order to prevent information loss in the images and to reveal their characteristic features. In the second stage, 96 features are extracted by applying statistical- and entropy-based feature functions to approximate, horizontal, vertical, and diagonal matrices of 2D-SWT. In the third stage, the features that keep the classifier performance high are selected by a chaotic-based wrapper approach consisting of the k-nearest neighbor (kNN) and chaotic particle swarm optimization algorithms (CPSO) to cope with both chaoticity and computational complexity in the fundus images. At the last stage, an AI-based classification model is created with the recurrent neural network-long short-term memory (RNN-LSTM) architecture by selecting the lowest number of feature sets that can keep the classification performance high. The performance of the DR disease classification model was tested on 2500 color fundus image data, which included five classes: no DR, mild non-proliferative DR (NPDR), moderate NPDR, severe NPDR, and proliferative DR (PDR). The robustness of the DR disease classification model was confirmed by the 10-fold cross-validation. In addition, the classification performance of the proposed model is compared with the support vector machine (SVM), which is one of the machine learning techniques. The results obtained show that the proposed model can overcome nonlinear dynamics in color fundus images with low computational complexity and is very effective and successful in precisely diagnosing all phases of DR disease. Full article

The key issues in fractal geometry concern scale invariance (self-similarity or self-affinity) and the notion of a fractal dimension $D$ which exceeds the topological dimension $d$. In this regard, we point out that the constitutive inequality $D>d$ can have either a geometric or topological origin, or both. The main topological features of fractals are their connectedness, connectivity, ramification, and loopiness. We argue that these features can be specified by six basic dimension numbers which are generally independent from each other. However, for many kinds of fractals, the number of independent dimensions may be reduced due to the peculiarities of specific kinds of fractals. Accordingly, we survey the paradigmatic fractals from a topological perspective. Some challenging points are outlined. Full article

A simplified method of determining lattice Boltzmann boundary conditions based on self-affine microchannels with an inherent roughness in a tight reservoir is presented in this paper to address nonlinear efficiency problems in fluid simulation. This approach effectively combines the influence of rough surfaces in the simulation of the flow field, the description of L-fractal theory applied to rough surfaces, and a generalized lattice Boltzmann method with equivalent composite slip boundary conditions for inherent roughness. The numerical simulations of gas slippage in a two-dimensional plate model and rough surfaces to induce gas vortex reflux flow are also successfully carried out, and the results are in good agreement with the simulation results, which establishes the reliability and flexibility of the proposed simplified method of rough surfaces. The effects of relative average height and fractal dimensions of the rough surfaces under exact boundary conditions and equivalent coarsened ones are investigated from three perspectives, namely those of the average lattice velocity, the lattice velocity at average height position at the outlet, and the coefficient of variation for lattice velocity at average height position. It was found that the roughness effect on gas flow behavior was more obvious when it was associated with the enhanced rarefaction effect. In addition, the area of gas seepage was reduced, and the gas flow resistance was increased. When the fractal dimension of the wall was about 1.20, it has the greatest impact on the fluid flow law. In addition, excessive roughness of the wall surface tends to lead to vortex backflow of the gas in the region adjacent to the wall, which greatly reduces its flow velocity. For gas flow in the nanoscale seepage space, wall roughness hindered gas migration rate by 84.7%. For pores larger than 200 nm, the effects of wall roughness on gas flow are generally negligible. Full article

Cancer is a complex disease, responsible for a significant portion of global deaths. The increasing prioritisation of know-why over know-how approaches in biological research has favoured the rising use of both white- and black-box mathematical techniques for cancer modelling, seeking to better grasp the multi-scale mechanistic workings of its complex phenomena (such as tumour-immune interactions, drug resistance, tumour growth and diffusion, etc.). In light of this wide-ranging use of mathematics in cancer modelling, the unique memory and non-local properties of Fractional Calculus (FC) have been sought after in the last decade to replace ordinary differentiation in the hypothesising of FC’s superior modelling of complex oncological phenomena, which has been shown to possess an accumulated knowledge of its past states. As such, this review aims to present a thorough and structured survey about the main guiding trends and modelling categories in cancer research, emphasising in the field of oncology FC’s increasing employment in mathematical modelling as a whole. The most pivotal research questions, challenges and future perspectives are also outlined. Full article

This paper is concerned with the tracking control problem for the lower-triangular systems with unknown fractional powers and nonparametric uncertainties. A prescribed performance control approach is put forward as a means of resolving this problem. The proposed control law incorporates a set of barrier functions to guarantee error constraints. Unlike the previous works, our approach works for the cases where the fractional powers, the nonlinearities, and their bounding functions or bounds are totally unknown; no restrictive conditions on the powers, such as power order restriction, specific size limitation or homogeneous condition, are made. Moreover, neither the powers and system nonlinearities nor their bounding functions or bounds are needed. It achieves reference tracking with the preassigned tracking accuracy and convergence speed. In addition, our controller is simple, as it does not necessitate parameter identification, function approximation, derivative calculation, or adding a power integrator technique. At the end, a comparative simulation demonstrates the effectiveness and advantage of the proposed approach. Full article

The composition $H\left(z\right)=f(\Phi (z\left)\right)$ is studied, where f is an entire function of a single complex variable and $\Phi$ is an entire function of n complex variables with a vanished gradient. Conditions are presented by the function $\Phi$ providing boundedness of the $\mathbf{L}$-index in joint variables for the function H, if the function f has bounded l-index for some positive continuous function l and $\mathbf{L}\left(z\right)=l(\Phi \left(z\right))(max\{1,|{\Phi }_{z_1}^{\prime }\left(z\right)\left|\right\},\dots ,max\{1,|{\Phi }_{z_n}^{\prime }\left(z\right)\left|\right\}),$$z\in {\mathbb{C}}^n.$ Such a constrained function $\mathbf{L}$ allows us to consider a function $\Phi$ with a nonempty zero set. The obtained results complement earlier published results with $\Phi \left(z\right)\ne 0.$ Also, we study a more general composition $H\left(\mathbf{w}\right)=G(\mathbf{\Phi }(\mathbf{w}\left)\right)$, where $G:{\mathbb{C}}^n\to \mathbb{C}$ is an entire function of the bounded $\mathbf{L}$-index in joint variables, $\mathbf{\Phi }:{\mathbb{C}}^m\to {\mathbb{C}}^n$ is a vector-valued entire function, and $\mathbf{L}:{\mathbb{C}}^n\to {\mathbb{R}}_{+}^n$ is a continuous function. If the $\mathbf{L}$-index of the function G equals zero, then we construct a function $\tilde{\mathbf{L}}:{\mathbb{C}}^m\to {\mathbb{R}}_{+}^m$ such that the function H has bounded $\tilde{\mathbf{L}}$-index in the joint variables $z_1,\dots ,z_n$. The other group of our results concern a sum of entire functions in several variables. As a general case, a sum of functions with bounded index is not of bounded index. The same is also valid for the multidimensional case. We found simple conditions proving that $f_1\left(z_1\right)+f_2\left(z_2\right)$ belongs to the class of functions having bounded index in joint variables $z_1,z_2$. We formulate some open problems based on the deduced results and on the usage of fractional differentiation operators in the theory of functions with bounded index. Full article