Fractal and Fractional

10 pages, 284 KiB

Open AccessArticle

Some Local Fractional Hilbert-Type Inequalities

by Predrag Vuković

Fractal Fract. 2023, 7(2), 205; https://doi.org/10.3390/fractalfract7020205 - 19 Feb 2023

Cited by 1 | Viewed by 1100

The main purpose of this paper is to prove some new local fractional Hilbert-type inequalities. Our general results are applicable to homogeneous kernels. Furthermore, the best possible constants in terms of local fractional hypergeometric function are obtained. The obtained results prove that the [...] Read more.

The main purpose of this paper is to prove some new local fractional Hilbert-type inequalities. Our general results are applicable to homogeneous kernels. Furthermore, the best possible constants in terms of local fractional hypergeometric function are obtained. The obtained results prove that the employed method is very simple and effective for treating various kinds of local fractional Hilbert-type inequalities. Full article

(This article belongs to the Special Issue Discrete Fractional Calculus, Local Fractional Inequalities, and Applications)

23 pages, 847 KiB

Open AccessArticle

Solving and Numerical Simulations of Fractional-Order Governing Equation for Micro-Beams

by Aimin Yang, Qunwei Zhang, Jingguo Qu, Yuhuan Cui and Yiming Chen

Fractal Fract. 2023, 7(2), 204; https://doi.org/10.3390/fractalfract7020204 - 18 Feb 2023

Cited by 4 | Viewed by 1730

Abstract

This paper applies a recently proposed numerical algorithm to discuss the deflection of viscoelastic micro-beams in the time domain with direct access. A nonlinear-fractional order model for viscoelastic micro-beams is constructed. Before solving the governing equations, the operator matrices of shifted Chebyshev polynomials [...] Read more.

This paper applies a recently proposed numerical algorithm to discuss the deflection of viscoelastic micro-beams in the time domain with direct access. A nonlinear-fractional order model for viscoelastic micro-beams is constructed. Before solving the governing equations, the operator matrices of shifted Chebyshev polynomials are derived first. Shifted Chebyshev polynomials are used to approximate the deflection function, and the nonlinear fractional order governing equation is expressed in the form of operator matrices. Next, the collocation method is used to discretize the equations into the form of algebraic equations for solution. It simplifies the calculation. MATLAB software was used to program this algorithm to simulate the numerical solution of the deflection. The effectiveness and accuracy of the algorithm are verified by the numerical example. Finally, numerical simulations are carried out on the viscoelastic micro-beams. It is found that the viscous damping coefficient is inversely proportional to the deflection, and the length scale parameter of the micro-beam is also inversely proportional to the deflection. In addition, the stress and strain of micro-beam, the change of deflection under different simple harmonic loads, and potential energy of micro-beam are discussed. The results of the study fully demonstrated that the shifted Chebyshev polynomial algorithm is effective for the numerical simulations of viscoelastic micro-beams. Full article

(This article belongs to the Special Issue Application of Fractional Calculus as an Interdisciplinary Modeling Framework)

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18 pages, 1263 KiB

Open AccessArticle

Forecasting Cryptocurrency Prices Using LSTM, GRU, and Bi-Directional LSTM: A Deep Learning Approach

by Phumudzo Lloyd Seabe, Claude Rodrigue Bambe Moutsinga and Edson Pindza

Fractal Fract. 2023, 7(2), 203; https://doi.org/10.3390/fractalfract7020203 - 18 Feb 2023

Cited by 45 | Viewed by 17320

Abstract

Highly accurate cryptocurrency price predictions are of paramount interest to investors and researchers. However, owing to the nonlinearity of the cryptocurrency market, it is difficult to assess the distinct nature of time-series data, resulting in challenges in generating appropriate price predictions. Numerous studies [...] Read more.

Highly accurate cryptocurrency price predictions are of paramount interest to investors and researchers. However, owing to the nonlinearity of the cryptocurrency market, it is difficult to assess the distinct nature of time-series data, resulting in challenges in generating appropriate price predictions. Numerous studies have been conducted on cryptocurrency price prediction using different Deep Learning (DL) based algorithms. This study proposes three types of Recurrent Neural Networks (RNNs): namely, Long Short-Term Memory (LSTM), Gated Recurrent Unit (GRU), and Bi-Directional LSTM (Bi-LSTM) for exchange rate predictions of three major cryptocurrencies in the world, as measured by their market capitalization—Bitcoin (BTC), Ethereum (ETH), and Litecoin (LTC). The experimental results on the three major cryptocurrencies using both Root Mean Squared Error (RMSE) and the Mean Absolute Percentage Error (MAPE) show that the Bi-LSTM performed better in prediction than LSTM and GRU. Therefore, it can be considered the best algorithm. Bi-LSTM presented the most accurate prediction compared to GRU and LSTM, with MAPE values of 0.036, 0.041, and 0.124 for BTC, LTC, and ETH, respectively. The paper suggests that the prediction models presented in it are accurate in predicting cryptocurrency prices and can be beneficial for investors and traders. Additionally, future research should focus on exploring other factors that may influence cryptocurrency prices, such as social media and trading volumes. Full article

(This article belongs to the Special Issue Recent Advances in Fractional-Order Neural Networks: Theory and Application)

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23 pages, 12535 KiB

Open AccessArticle

Lateral Fractal Formation by Crystallographic Silicon Micromachining

by Lucas Johannes Kooijman, Yasser Pordeli, Johan Willem Berenschot and Niels Roelof Tas

Fractal Fract. 2023, 7(2), 202; https://doi.org/10.3390/fractalfract7020202 - 18 Feb 2023

Cited by 2 | Viewed by 1558

Abstract

A novel wafer-scale silicon fractal fabrication method is presented here for forming pyramids only in the lateral direction using the crystal orientation of silicon. Fractals are fabricated in silicon by masking only the corners (corner lithography) of a cavity in silicon with silicon [...] Read more.

A novel wafer-scale silicon fractal fabrication method is presented here for forming pyramids only in the lateral direction using the crystal orientation of silicon. Fractals are fabricated in silicon by masking only the corners (corner lithography) of a cavity in silicon with silicon nitride, where the shape is determined by the crystal {111} planes of the silicon. The octahedral cavity shaped by the {111} planes was previously only used for forming octahedral fractals in all directions, but by using a planar silicon dioxide hard-mask on a silicon (100) wafer, the silicon octahedral cavity is “cut in half”. This creates a pyramid with sharper edges and vertices at its base than those determined by just the {111} planes. This allows selective corner lithography patterning at the vertices of the base while leaving the apex unpatterned, leading to lateral growing of pyramidal fractals. This selective patterning is shown mathematically and then demonstrated by creating a fractal of four generations, with the initial pyramid being 8 µm and the two final generations being of submicron size. Full article

(This article belongs to the Special Issue The Materials Structure and Fractal Nature)

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33 pages, 6636 KiB

Open AccessArticle

Bifurcations and the Exact Solutions of the Time-Space Fractional Complex Ginzburg-Landau Equation with Parabolic Law Nonlinearity

by Wenjing Zhu, Zijie Ling, Yonghui Xia and Min Gao

Fractal Fract. 2023, 7(2), 201; https://doi.org/10.3390/fractalfract7020201 - 18 Feb 2023

Cited by 8 | Viewed by 1564

Abstract

This paper studies the bifurcations of the exact solutions for the time–space fractional complex Ginzburg–Landau equation with parabolic law nonlinearity. Interestingly, for different parameters, there are different kinds of first integrals for the corresponding traveling wave systems. Using the method of dynamical systems, [...] Read more.

This paper studies the bifurcations of the exact solutions for the time–space fractional complex Ginzburg–Landau equation with parabolic law nonlinearity. Interestingly, for different parameters, there are different kinds of first integrals for the corresponding traveling wave systems. Using the method of dynamical systems, which is different from the previous works, we obtain the phase portraits of the the corresponding traveling wave systems. In addition, we derive the exact parametric representations of solitary wave solutions, periodic wave solutions, kink and anti-kink wave solutions, peakon solutions, periodic peakon solutions and compacton solutions under different parameter conditions. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications)

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13 pages, 3565 KiB

Open AccessArticle

Dynamics and Stability of a Fractional-Order Tumor–Immune Interaction Model with B-D Functional Response and Immunotherapy

by Xiaozhou Feng, Mengyan Liu, Yaolin Jiang and Dongping Li

Fractal Fract. 2023, 7(2), 200; https://doi.org/10.3390/fractalfract7020200 - 17 Feb 2023

Cited by 1 | Viewed by 1452

Abstract

In this paper, we investigate a fractional-order tumor–immune interaction model with B-D function item and immunotherapy. First, the existence, uniqueness and nonnegativity of the solutions of the model are established. Second, the local and global asymptotic stability of some tumor-free equilibrium points and [...] Read more.

In this paper, we investigate a fractional-order tumor–immune interaction model with B-D function item and immunotherapy. First, the existence, uniqueness and nonnegativity of the solutions of the model are established. Second, the local and global asymptotic stability of some tumor-free equilibrium points and a unique positive equilibrium point are obtained. Finally, we use numerical simulation method to visualize and verify the theoretical conclusions. It is known that the fractional-order parameter

β

has a stabilization effect, and the tumor cells can be destroyed or controlled by using immunotherapy. Full article

(This article belongs to the Special Issue Fractional Differential Equations in Anomalous Diffusion)

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16 pages, 329 KiB

Open AccessArticle

Differential Subordination and Superordination Results for q-Analogue of Multiplier Transformation

by Alina Alb Lupaş and Adriana Cătaş

Fractal Fract. 2023, 7(2), 199; https://doi.org/10.3390/fractalfract7020199 - 17 Feb 2023

Cited by 6 | Viewed by 1359

Abstract

The results obtained by the authors in the present paper refer to quantum calculus applications regarding the theories of differential subordination and superordination. These results are established by means of an operator defined as the q-analogue of the multiplier transformation. Interesting differential [...] Read more.

The results obtained by the authors in the present paper refer to quantum calculus applications regarding the theories of differential subordination and superordination. These results are established by means of an operator defined as the q-analogue of the multiplier transformation. Interesting differential subordination and superordination results are derived by the authors involving the functions belonging to a new class of normalized analytic functions in the open unit disc U, which is defined and investigated here by using this q-operator. Full article

(This article belongs to the Special Issue Fractional Operators and Their Applications)

20 pages, 373 KiB

Open AccessArticle

Stability of p(·)-Integrable Solutions for Fractional Boundary Value Problem via Piecewise Constant Functions

by Mohammed Said Souid, Ahmed Refice and Kanokwan Sitthithakerngkiet

Fractal Fract. 2023, 7(2), 198; https://doi.org/10.3390/fractalfract7020198 - 17 Feb 2023

Cited by 3 | Viewed by 1309

Abstract

The goal of this work is to study a multi-term boundary value problem (BVP) for fractional differential equations in the variable exponent Lebesgue space (

L^{p (\cdot)}

). Both the existence, uniqueness, and the stability in the sense of Ulam–Hyers [...] Read more.

The goal of this work is to study a multi-term boundary value problem (BVP) for fractional differential equations in the variable exponent Lebesgue space (

L^{p (\cdot)}

). Both the existence, uniqueness, and the stability in the sense of Ulam–Hyers are established. Our results are obtained using two fixed-point theorems, then illustrating the results with a comprehensive example. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 2nd Edition)

26 pages, 13945 KiB

Open AccessArticle

Design, Hardware Implementation on FPGA and Performance Analysis of Three Chaos-Based Stream Ciphers

by Fethi Dridi, Safwan El Assad, Wajih El Hadj Youssef and Mohsen Machhout

Fractal Fract. 2023, 7(2), 197; https://doi.org/10.3390/fractalfract7020197 - 17 Feb 2023

Cited by 9 | Viewed by 2311

Abstract

In this paper, we come up with three secure chaos-based stream ciphers, implemented on an FPGA board, for data confidentiality and integrity. To do so, first, we performed the statistical security and hardware metrics of certain discrete chaotic map models, such as the [...] Read more.

In this paper, we come up with three secure chaos-based stream ciphers, implemented on an FPGA board, for data confidentiality and integrity. To do so, first, we performed the statistical security and hardware metrics of certain discrete chaotic map models, such as the Logistic, Skew-Tent, PWLCM, 3D-Chebyshev map, and 32-bit LFSR, which are the main components of the proposed chaotic generators. Based on the performance analysis collected from the discrete chaotic maps, we then designed, implemented, and analyzed the performance of three proposed robust pseudo-random number generators of chaotic sequences (PRNGs-CS) and their corresponding stream ciphers. The proposed PRNGs-CS are based on the predefined coupling matrix M. The latter achieves a weak mixing of the chaotic maps and a chaotic multiplexing technique or XOR operator for the output function. Therefore, the randomness of the sequences generated is expanded as well as their lengths, and divide-and-conquer attacks on chaotic systems are avoided. In addition, the proposed PRNGs-CS contain polynomial mappings of at least degree 2 or 3 to make algebraic attacks very difficult. Various experimental results obtained and analysis of performance in opposition to different kinds of numerical and cryptographic attacks determine the high level of security and good hardware metrics achieved by the proposed chaos system. The proposed system outperformed the state-of-the-art works in terms of high-security level and a high throughput which can be considered an alternative to the standard methods. Full article

(This article belongs to the Special Issue Advances in Fractional-Order Embedded Systems)

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14 pages, 3212 KiB

Open AccessArticle

A Computational Technique for Solving Three-Dimensional Mixed Volterra–Fredholm Integral Equations

by Amr M. S. Mahdy, Abbas S. Nagdy, Khaled M. Hashem and Doaa Sh. Mohamed

Fractal Fract. 2023, 7(2), 196; https://doi.org/10.3390/fractalfract7020196 - 16 Feb 2023

Cited by 18 | Viewed by 1953

Abstract

In this article, a novel and efficient approach based on Lucas polynomials is introduced for solving three-dimensional mixed Volterra–Fredholm integral equations for the two types (3D-MVFIEK2). This method transforms the 3D-MVFIEK2 into a system of linear algebraic equations. The error evaluation for the [...] Read more.

In this article, a novel and efficient approach based on Lucas polynomials is introduced for solving three-dimensional mixed Volterra–Fredholm integral equations for the two types (3D-MVFIEK2). This method transforms the 3D-MVFIEK2 into a system of linear algebraic equations. The error evaluation for the suggested scheme is discussed. This technique is implemented in four examples to illustrate the efficiency and fulfillment of the approach. Examples of numerical solutions to both linear and nonlinear integral equations were used. The Lucas polynomial method and other approaches were contrasted. A collection of tables and figures is used to present the numerical results. We observe that the exact solution differs from the numerical solution if the exact solution is an exponential or trigonometric function, while the numerical solution is the same when the exact solution is a polynomial. The Maple 18 program produced all of the results. Full article

(This article belongs to the Topic Evolutionary Differential Equations, Dynamic Systems, Computation and Optimization)

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12 pages, 309 KiB

Open AccessArticle

Certain Sharp Coefficient Results on a Subclass of Starlike Functions Defined by the Quotient of Analytic Functions

by Lei Shi and Muhammad Arif

Fractal Fract. 2023, 7(2), 195; https://doi.org/10.3390/fractalfract7020195 - 15 Feb 2023

Cited by 5 | Viewed by 1386

Abstract

In the present paper, we consider a subclass of starlike functions

G_{3 / 2}

defined by the ratio of analytic representations of convex and starlike functions. The main aim is to determine the bounds of Fekete–Szegö-type inequalities and Hankel determinants for functions [...] Read more.

In the present paper, we consider a subclass of starlike functions

G_{3 / 2}

defined by the ratio of analytic representations of convex and starlike functions. The main aim is to determine the bounds of Fekete–Szegö-type inequalities and Hankel determinants for functions in this class. It is proved that

max \{|H_{3, 1} (f)| : f \in G_{3 / 2}\}

is equal to

\frac{1}{81}

. The bounds for

f \in G_{3 / 2}

are sharp. Full article

(This article belongs to the Special Issue Nonlinear Functional Analysis and Applications)

40 pages, 9182 KiB

Open AccessReview

Generalized Beta Models and Population Growth: So Many Routes to Chaos

by M. Fátima Brilhante, M. Ivette Gomes, Sandra Mendonça, Dinis Pestana and Pedro Pestana

Fractal Fract. 2023, 7(2), 194; https://doi.org/10.3390/fractalfract7020194 - 15 Feb 2023

Cited by 2 | Viewed by 1797

Abstract

Logistic and Gompertz growth equations are the usual choice to model sustainable growth and immoderate growth causing depletion of resources, respectively. Observing that the logistic distribution is geo-max-stable and the Gompertz function is proportional to the Gumbel max-stable distribution, we investigate other models [...] Read more.

Logistic and Gompertz growth equations are the usual choice to model sustainable growth and immoderate growth causing depletion of resources, respectively. Observing that the logistic distribution is geo-max-stable and the Gompertz function is proportional to the Gumbel max-stable distribution, we investigate other models proportional to either geo-max-stable distributions (log-logistic and backward log-logistic) or to other max-stable distributions (Fréchet or max-Weibull). We show that the former arise when in the hyper-logistic Blumberg equation, connected to the Beta

(p, q)

function, we use fractional exponents

p - 1 = 1 \mp 1 / α

and

q - 1 = 1 \pm 1 / α

, and the latter when in the hyper-Gompertz-Turner equation, the exponents of the logarithmic factor are real and eventually fractional. The use of a BetaBoop function establishes interesting connections to Probability Theory, Riemann–Liouville’s fractional integrals, higher-order monotonicity and convexity and generalized unimodality, and the logistic map paradigm inspires the investigation of the dynamics of the hyper-logistic and hyper-Gompertz maps. Full article

(This article belongs to the Special Issue Feature Papers in Fractal and Fractional 2022–2023)

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12 pages, 516 KiB

Open AccessArticle

Structured Doubling Algorithm for a Class of Large-Scale Discrete-Time Algebraic Riccati Equations with High-Ranked Constant Term

by Bo Yu, Chengxu Jiang and Ning Dong

Fractal Fract. 2023, 7(2), 193; https://doi.org/10.3390/fractalfract7020193 - 14 Feb 2023

Viewed by 1337

Abstract

Consider the computation of the solution for a class of discrete-time algebraic Riccati equations (DAREs) with the low-ranked coefficient matrix G and the high-ranked constant matrix H. A structured doubling algorithm is proposed for large-scale problems when A is of lowrank. Compared [...] Read more.

Consider the computation of the solution for a class of discrete-time algebraic Riccati equations (DAREs) with the low-ranked coefficient matrix G and the high-ranked constant matrix H. A structured doubling algorithm is proposed for large-scale problems when A is of lowrank. Compared to the existing doubling algorithm of

O (2^{k} n)

flops at the k-th iteration, the newly developed version merely needs

O (n)

flops for preprocessing and

O ({(k + 1)}^{3} m^{3})

flopsfor iterations and is more proper for large-scale computations when

m ≪ n

. The convergence and complexity of the algorithm are subsequently analyzed. Illustrative numerical experiments indicate that the presented algorithm, which consists of a dominant time-consuming preprocessing step and a trivially iterative step, is capable of computing the solution efficiently for large-scale DAREs. Full article

(This article belongs to the Special Issue Applications of Iterative Methods in Solving Nonlinear Equations)

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26 pages, 1514 KiB

Open AccessArticle

Deterministic and Fractional-Order Co-Infection Model of Omicron and Delta Variants of Asymptomatic SARS-CoV-2 Carriers

by Waqas Ali Faridi, Muhammad Imran Asjad, Shabir Ahmad, Adrian Iftene, Magda Abd El-Rahman and Mohammed Sallah

Fractal Fract. 2023, 7(2), 192; https://doi.org/10.3390/fractalfract7020192 - 14 Feb 2023

Viewed by 1956

Abstract

The Delta and Omicron variants’ system was used in this research study to replicate the complex process of the SARS-CoV-2 outbreak. The generalised fractional system was designed and rigorously analysed in order to gain a comprehensive understanding of the transmission dynamics of both [...] Read more.

The Delta and Omicron variants’ system was used in this research study to replicate the complex process of the SARS-CoV-2 outbreak. The generalised fractional system was designed and rigorously analysed in order to gain a comprehensive understanding of the transmission dynamics of both variants. The proposed dynamical system has heredity and memory effects, which greatly improved our ability to perceive the disease propagation dynamics. The non-singular Atangana–Baleanu fractional operator was used to forecast the current pandemic in order to meet this challenge. The Picard recursions approach can be used to ensure that the designed fractional system has at least one solution occupying the growth condition and memory function regardless of the initial conditions. The Hyers–Ulam–Rassias stability criteria were used to carry out the stability analysis of the fractional governing system of equations, and the fixed-point theory ensured the uniqueness of the solution. Additionally, the model exhibited global asymptotically stable behaviour in some conditions. The approximate behaviour of the fatal virus was investigated using an efficient and reliable fractional numerical Adams–Bashforth approach. The outcome demonstrated that there will be a significant decline in the population of those infected with the Omicron and Delta SARS-CoV-2 variants if the vaccination rate is increased (in both the symptomatic and symptomatic stages). Full article

(This article belongs to the Special Issue Fractional Order Systems: Deterministic and Stochastic Analysis II)

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25 pages, 1516 KiB

Open AccessArticle

The Propagating Exact Solitary Waves Formation of Generalized Calogero–Bogoyavlenskii–Schiff Equation with Robust Computational Approaches

by Basem Al Alwan, Muhammad Abu Bakar, Waqas Ali Faridi, Antoniu-Claudiu Turcu, Ali Akgül and Mohammed Sallah

Fractal Fract. 2023, 7(2), 191; https://doi.org/10.3390/fractalfract7020191 - 14 Feb 2023

Cited by 22 | Viewed by 1863

Abstract

The generalized Calogero–Bogoyavlenskii–Schiff equation (GCBSE) is examined and analyzed in this paper. It has several applications in plasma physics and soliton theory, where it forecasts the soliton wave propagation profiles. In order to obtain the analytically exact solitons, the model under consideration is [...] Read more.

The generalized Calogero–Bogoyavlenskii–Schiff equation (GCBSE) is examined and analyzed in this paper. It has several applications in plasma physics and soliton theory, where it forecasts the soliton wave propagation profiles. In order to obtain the analytically exact solitons, the model under consideration is a nonlinear partial differential equation that is turned into an ordinary differential equation by using the next traveling wave transformation. The new extended direct algebraic technique and the modified auxiliary equation method are applied to the generalized Calogero–Bogoyavlenskii–Schiff equation to get new solitary wave profiles. As a result, novel and generalized analytical wave solutions are acquired in which singular solutions, mixed singular solutions, mixed complex solitary shock solutions, mixed shock singular solutions, mixed periodic solutions, mixed trigonometric solutions, mixed hyperbolic solutions, and periodic solutions are included with numerous soliton families. The propagation of the acquired soliton solution is graphically presented in contour, two- and three-dimensional visualization by selecting appropriate parametric values. It is graphically demonstrated how wave number impacts the obtained traveling wave structures. Full article

(This article belongs to the Special Issue Numerical Simulations and Advanced Techniques for Nonlinear Fractional Evolution Models)

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14 pages, 4114 KiB

Open AccessArticle

A Daisyworld Ecological Parable Including the Revenge of Gaia and Greenhouse Effect

by Marcelo A. Savi and Flavio M. Viola

Fractal Fract. 2023, 7(2), 190; https://doi.org/10.3390/fractalfract7020190 - 14 Feb 2023

Viewed by 2043

Abstract

The Daisyworld model illustrates the concept of biological homeostasis in the global environment by establishing a connection between the biota and environment, resulting in a single intertwined system known as Gaia. In essence, the Daisyworld model represents life by daisy populations whereas temperature [...] Read more.

The Daisyworld model illustrates the concept of biological homeostasis in the global environment by establishing a connection between the biota and environment, resulting in a single intertwined system known as Gaia. In essence, the Daisyworld model represents life by daisy populations whereas temperature represents the environment, establishing a population dynamics model to represent life–environment ecological interactions. The recent occurrence of extreme weather events due to climate change and the critical crises brought on by the COVID-19 pandemic are strengthening the arguments for the revenge of Gaia, a term used to describe the protective response of the global biota-environment system. This paper presents a novel Daisyworld parable to describe ecological life–environment interactions including the revenge of Gaia and the greenhouse effect. The revenge of Gaia refers to a change in the interplay between life and environment, characterized by the Gaia state that establishes the life-environment state of balance and harmony. This results in reaction effects that impact the planet’s fertile regions. On the other hand, the greenhouse effect is incorporated through the description of the interactions of greenhouse gases with the planet, altering its albedo. Numerical simulations are performed using a nonlinear dynamics perspective, showing different ecological scenarios. An investigation of the system reversibility is carried out together with critical life–environment interactions. This parable provides a qualitative description that can be useful to evaluate ecological scenarios. Full article

(This article belongs to the Special Issue Advances in Nonlinear Dynamics: Theory, Methods and Applications)

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27 pages, 3368 KiB

Open AccessArticle

Fractal-Fractional Caputo Maize Streak Virus Disease Model

by Joseph Ackora-Prah, Baba Seidu, Eric Okyere and Joshua K. K. Asamoah

Fractal Fract. 2023, 7(2), 189; https://doi.org/10.3390/fractalfract7020189 - 13 Feb 2023

Cited by 18 | Viewed by 1897

Abstract

Maize is one of the most extensively produced cereals in the world. The maize streak virus primarily infects maize but can also infect over 80 other grass species. Leafhoppers are the primary vectors of the maize streak virus. When feeding on plants, susceptible [...] Read more.

Maize is one of the most extensively produced cereals in the world. The maize streak virus primarily infects maize but can also infect over 80 other grass species. Leafhoppers are the primary vectors of the maize streak virus. When feeding on plants, susceptible vectors can acquire the virus from infected plants, and infected vectors can transmit the virus to susceptible plants. However, because maize is normally patchy and leafhoppers are mobile, leafhoppers will always be foraging for food. Therefore, we want to look at how leafhoppers interact on maize farms using Holling’s Type III functional response in a Caputo fractal-fractional derivative sense. We show that the proposed model has unique positive solutions within a feasible region. We employed the Newton polynomial scheme to numerically simulate the proposed model to illustrate the qualitative results obtained. We also studied the relationship between the state variables and some epidemiological factors captured as model parameters. We observed that the integer-order versions of the model exaggerate the impact of the disease. We also observe that the increase in the leafhopper infestation on maize fields has a devastating effect on the health of maize plants and the subsequent yield. Furthermore, we noticed that varying the conversion rate of the infected leafhopper leads to a crossover effect in the number of healthy maize after 82 days. We also show the dynamics of varying the maize streak virus transmission rates. It indicates that when preventive measures are taken to reduce the transmission rates, it will reduce the low-yielding effect of maize due to the maize streak virus disease. Full article

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16 pages, 4456 KiB

Open AccessArticle

Numerical Solutions of the (2+1)-Dimensional Nonlinear and Linear Time-Dependent Schrödinger Equations Using Three Efficient Approximate Schemes

by Neveen G. A. Farag, Ahmed H. Eltanboly, Magdi S. El-Azab and Salah S. A. Obayya

Fractal Fract. 2023, 7(2), 188; https://doi.org/10.3390/fractalfract7020188 - 13 Feb 2023

Cited by 1 | Viewed by 2234

Abstract

In this paper, the (2+1)-dimensional nonlinear Schrödinger equation (2D NLSE) abreast of the (2+1)-dimensional linear time-dependent Schrödinger equation (2D TDSE) are thoroughly investigated. For the first time, these two notable 2D equations are attempted to be solved using three compelling pseudo-spectral/finite difference approaches, [...] Read more.

In this paper, the (2+1)-dimensional nonlinear Schrödinger equation (2D NLSE) abreast of the (2+1)-dimensional linear time-dependent Schrödinger equation (2D TDSE) are thoroughly investigated. For the first time, these two notable 2D equations are attempted to be solved using three compelling pseudo-spectral/finite difference approaches, namely the split-step Fourier transform (SSFT), Fourier pseudo-spectral method (FPSM), and the hopscotch method (HSM). A bright 1-soliton solution is considered for the 2D NLSE, whereas a Gaussian wave solution is determined for the 2D TDSE. Although the analytical solutions of these partial differential equations can sometimes be reached, they are either limited to a specific set of initial conditions or even perplexing to find. Therefore, our suggested approximate solutions are of tremendous significance, not only for our proposed equations, but also to apply to other equations. Finally, systematic comparisons of the three suggested approaches are conducted to corroborate the accuracy and reliability of these numerical techniques. In addition, each scheme’s error and convergence analysis is numerically exhibited. Based on the MATLAB findings, the novelty of this work is that the SSFT has proven to be an invaluable tool for the presented 2D simulations from the speed, accuracy, and convergence perspectives, especially when compared to the other suggested schemes. Full article

(This article belongs to the Special Issue Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics)

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16 pages, 344 KiB

Open AccessArticle

On Caputo Fractional Derivatives and Caputo–Fabrizio Integral Operators via (s, m)-Convex Functions

by Ammara Nosheen, Maria Tariq, Khuram Ali Khan, Nehad Ali Shah and Jae Dong Chung

Fractal Fract. 2023, 7(2), 187; https://doi.org/10.3390/fractalfract7020187 - 13 Feb 2023

Cited by 12 | Viewed by 2790

Abstract

This paper contains a variety of new integral inequalities for

(s, m)

-convex functions using Caputo fractional derivatives and Caputo–Fabrizio integral operators. Various generalizations of Hermite–Hadamard-type inequalities containing Caputo–Fabrizio integral operators are derived for those functions whose derivatives are [...] Read more.

This paper contains a variety of new integral inequalities for

(s, m)

-convex functions using Caputo fractional derivatives and Caputo–Fabrizio integral operators. Various generalizations of Hermite–Hadamard-type inequalities containing Caputo–Fabrizio integral operators are derived for those functions whose derivatives are

(s, m)

-convex. Inequalities involving the digamma function and special means are deduced as applications. Full article

(This article belongs to the Special Issue Numerical Simulations and Advanced Techniques for Nonlinear Fractional Evolution Models)

19 pages, 5218 KiB

Open AccessArticle

A Compact Scheme Combining the Fast Time Stepping Method for Solving 2D Fractional Subdiffusion Equations

by Yibin Xu, Yanqin Liu, Xiuling Yin, Libo Feng and Zihua Wang

Fractal Fract. 2023, 7(2), 186; https://doi.org/10.3390/fractalfract7020186 - 13 Feb 2023

Cited by 1 | Viewed by 1760

Abstract

In this paper, in order to improve the calculation accuracy and efficiency of α-order Caputo fractional derivative (0 < α ≤ 1), we developed a compact scheme combining the fast time stepping method for solving 2D fractional nonlinear subdiffusion equations. In the [...] Read more.

In this paper, in order to improve the calculation accuracy and efficiency of α-order Caputo fractional derivative (0 < α ≤ 1), we developed a compact scheme combining the fast time stepping method for solving 2D fractional nonlinear subdiffusion equations. In the temporal direction, a time stepping method was applied. It can reach second-order accuracy. In the spatial direction, we utilized the compact difference scheme, which can reach fourth-order accuracy. Some properties of coefficients are given, which are essential for the theoretical analysis. Meanwhile, we rigorously proved the unconditional stability of the proposed scheme and gave the sharp error estimate. To overcome the intensive computation caused by the fractional operators, we combined a fast algorithm, which can reduce the computational complexity from O(N²) to O(Nlog(N)), where N represents the number of time steps. Considering that the solution of the subdiffusion equation is weakly regular in most cases, we added correction terms to ensure that the solution can achieve the optimal convergence accuracy. Full article

(This article belongs to the Special Issue Fractional Differential Equations in Anomalous Diffusion)

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23 pages, 698 KiB

Open AccessArticle

Order of Convergence and Dynamics of Newton–Gauss-Type Methods

by Ramya Sadananda, Santhosh George, Ioannis K. Argyros and Jidesh Padikkal

Fractal Fract. 2023, 7(2), 185; https://doi.org/10.3390/fractalfract7020185 - 13 Feb 2023

Cited by 2 | Viewed by 1333

Abstract

On the basis of the new iterative technique designed by Zhongli Liu in 2016 with convergence orders of three and five, an extension to order six can be found in this paper. The study of high-convergence-order iterative methods under weak conditions is of [...] Read more.

On the basis of the new iterative technique designed by Zhongli Liu in 2016 with convergence orders of three and five, an extension to order six can be found in this paper. The study of high-convergence-order iterative methods under weak conditions is of extreme importance, because higher order means that fewer iterations are carried out to achieve a predetermined error tolerance. In order to enhance the practicality of these methods by Zhongli Liu, the convergence analysis is carried out without the application of Taylor expansion and requires the operator to be only two times differentiable, unlike the earlier studies. A semilocal convergence analysis is provided. Furthermore, numerical experiments verifying the convergence criteria, comparative studies and the dynamics are discussed for better interpretation. Full article

(This article belongs to the Section General Mathematics, Analysis)

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14 pages, 535 KiB

Open AccessArticle

Robust Localization for Near- and Far-Field Signals with an Unknown Number of Sources

by Tao Liu, Hao Feng, Tianshuang Qiu, Shengyang Luan and Jiacheng Zhang

Fractal Fract. 2023, 7(2), 184; https://doi.org/10.3390/fractalfract7020184 - 12 Feb 2023

Cited by 1 | Viewed by 1450

Abstract

Source location is a constant issue of importance of both theoretical study and practical engineering. Many pioneers have put out the corresponding solutions for near- or far-field signals, and preferred contributions are suggested. To our best knowledge, there are currently few focused approaches [...] Read more.

Source location is a constant issue of importance of both theoretical study and practical engineering. Many pioneers have put out the corresponding solutions for near- or far-field signals, and preferred contributions are suggested. To our best knowledge, there are currently few focused approaches to the complicated situation where both near- and far-field signals exist with an unknown number of sources. Additionally, the robustness of the method must be taken into account when the additive background noise does not follow Gaussian or super-Gaussian distribution. To solve these problems, a novel method based on phased fractional lower-order moment (PFLOM) is proposed to simultaneously better preserve the signal and suppress the noise. Secondly, the whole procedure of the method containing direction of arrival (DOA) estimation, range estimation, separation of near-and far-field sources, and crucial parameter settings are studied in detail. Finally, comprehensive Monte Carlo experiments are carried out in the simulation to demonstrate the superiority of the proposed method compared to the existing competitive methods. Due to the novel method’s effectiveness with an unknown number of sources and robustness against various noises, it is believed that it could be fully utilized in more fields. Full article

(This article belongs to the Special Issue Fractional-Order Circuits, Systems, and Signal Processing)

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20 pages, 386 KiB

Open AccessArticle

On the 1st-Level General Fractional Derivatives of Arbitrary Order

by Yuri Luchko

Fractal Fract. 2023, 7(2), 183; https://doi.org/10.3390/fractalfract7020183 - 12 Feb 2023

Cited by 7 | Viewed by 1371

Abstract

In this paper, the 1st-level general fractional derivatives of arbitrary order are defined and investigated for the first time. We start with a generalization of the Sonin condition for the kernels of the general fractional integrals and derivatives and then specify a set [...] Read more.

In this paper, the 1st-level general fractional derivatives of arbitrary order are defined and investigated for the first time. We start with a generalization of the Sonin condition for the kernels of the general fractional integrals and derivatives and then specify a set of the kernels that satisfy this condition and possess an integrable singularity of the power law type at the origin. The 1st-level general fractional derivatives of arbitrary order are integro-differential operators of convolution type with the kernels from this set. They contain both the general fractional derivatives of arbitrary order of the Riemann–Liouville type and the regularized general fractional derivatives of arbitrary order considered in the literature so far. For the 1st-level general fractional derivatives of arbitrary order, some important properties, including the 1st and the 2nd fundamental theorems of fractional calculus, are formulated and proved. Full article

(This article belongs to the Special Issue New Trends on Generalized Fractional Calculus)

32 pages, 929 KiB

Open AccessArticle

Existence Results for Caputo Tripled Fractional Differential Inclusions with Integral and Multi-Point Boundary Conditions

by Muath Awadalla and Murugesan Manigandan

Fractal Fract. 2023, 7(2), 182; https://doi.org/10.3390/fractalfract7020182 - 12 Feb 2023

Cited by 3 | Viewed by 1135

Abstract

In this study, based on Coitz and Nadler’s fixed point theorem and the non-linear alternative for Kakutani maps, existence results for a tripled system of sequential fractional differential inclusions (SFDIs) with integral and multi-point boundary conditions (BCs) in investigated. A practical examples are [...] Read more.

In this study, based on Coitz and Nadler’s fixed point theorem and the non-linear alternative for Kakutani maps, existence results for a tripled system of sequential fractional differential inclusions (SFDIs) with integral and multi-point boundary conditions (BCs) in investigated. A practical examples are given to illustrate the obtained the theoretical results. Full article

(This article belongs to the Special Issue New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus Volume II)

23 pages, 390 KiB

Open AccessArticle

On a System of Sequential Caputo Fractional Differential Equations with Nonlocal Boundary Conditions

by Alexandru Tudorache and Rodica Luca

Fractal Fract. 2023, 7(2), 181; https://doi.org/10.3390/fractalfract7020181 - 12 Feb 2023

Cited by 6 | Viewed by 1270

Abstract

We obtain existence and uniqueness results for the solutions of a system of Caputo fractional differential equations which contain sequential derivatives, integral terms, and two positive parameters, supplemented with general coupled Riemann–Stieltjes integral boundary conditions. The proofs of our results are based on [...] Read more.

We obtain existence and uniqueness results for the solutions of a system of Caputo fractional differential equations which contain sequential derivatives, integral terms, and two positive parameters, supplemented with general coupled Riemann–Stieltjes integral boundary conditions. The proofs of our results are based on the Banach fixed point theorem and the Leray–Schauder alternative. Full article

(This article belongs to the Special Issue Advances in Boundary Value Problems for Fractional Differential Equations, 2nd Edition)

16 pages, 374 KiB

Open AccessArticle

Globally Existing Solutions to the Problem of Dirichlet for the Fractional 3D Poisson Equation

by Toshko Boev and Georgi Georgiev

Fractal Fract. 2023, 7(2), 180; https://doi.org/10.3390/fractalfract7020180 - 11 Feb 2023

Viewed by 1124

Abstract

A general approach to solving the Dirichlet problem, both for bounded

3 D

domains and for their unbounded complements, in terms of the fractional (

3 D

) Poisson equation, is presented. Lauren Schwartz class solutions are sought for tempered distributions. The solutions [...] Read more.

A general approach to solving the Dirichlet problem, both for bounded

3 D

domains and for their unbounded complements, in terms of the fractional (

3 D

) Poisson equation, is presented. Lauren Schwartz class solutions are sought for tempered distributions. The solutions found are represented by a formula that contains the volume Riesz potential and the one-layer potential, the latter depending on the boundary data. Infinite regularity of fractional harmonic functions, analogous to the infinite smoothness of the classical harmonic functions, is also proved in the respective domain, no matter what the boundary conditions are. Other properties of the solutions, that are presumably of interest to mathematical physics, are also investigated. In particular, an intrinsic decay property, valid far from the common boundary, is shown. Full article

(This article belongs to the Special Issue Newly Developments in Fractional Laplacian: Numerical Methods and Inverse Problems)

12 pages, 4108 KiB

Open AccessArticle

Intelligent Measurement of Void Fractions in Homogeneous Regime of Two Phase Flows Independent of the Liquid Phase Density Changes

by Abdullah M. Iliyasu, Farhad Fouladinia, Ahmed S. Salama, Gholam Hossein Roshani and Kaoru Hirota

Fractal Fract. 2023, 7(2), 179; https://doi.org/10.3390/fractalfract7020179 - 10 Feb 2023

Cited by 15 | Viewed by 1719

Abstract

Determining the amount of void fraction of multiphase flows in pipelines of the oil, chemical and petrochemical industries is one of the most important challenges. Performance of capacitance based two phase flow meters highly depends on the fluid properties. Fluctuation of the liquid [...] Read more.

Determining the amount of void fraction of multiphase flows in pipelines of the oil, chemical and petrochemical industries is one of the most important challenges. Performance of capacitance based two phase flow meters highly depends on the fluid properties. Fluctuation of the liquid phase properties such as density, due to temperature and pressure changes, would cause massive errors in determination of the void fraction. A common approach to fix this problem is periodic recalibration of the system, which is a tedious task. The aim of this study is proposing a method based on artificial intelligence (AI), which offers the advantage of intelligent measuring of the void fraction regardless of the liquid phase changes without the need for recalibration. To train AI, a data set for different liquid phases is required. Although it is possible to obtain the required data from experiments, it is time-consuming and also incorporates its own specific safety laboratory consideration, particularly working with flammable liquids such as gasoline, oil and gasoil. So, COMSOL Multiphysics software was used to model a homogenous regime of two-phase flow with five different liquid phases and void fractions. To validate the simulation geometry, initially an experimental setup including a concave sensor to measure the capacitance by LCR meter for the case that water used as the liquid phase, was established. After validation of the simulated geometry for concave sensor, a ring sensor was also simulated to investigate the best sensor type. It was found that the concave type has a better sensitivity. Therefore, the concave type was used to measure the capacitance for different liquid phases and void fractions inside the pipe. Finally, simulated data were used to train a Multi-Layer Perceptron (MLP) neural network model in MATLAB software. The trained MLP model was able to predict the void fraction independent of the liquid phase density changes with a Mean Absolute Error (MAE) of 1.74. Full article

(This article belongs to the Special Issue Fractal and Fractional Analysis and Non-conventional Methods for Solid and Fluid Mechanics)

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14 pages, 331 KiB

Open AccessArticle

Investigation of a Coupled System of Hilfer–Hadamard Fractional Differential Equations with Nonlocal Coupled Hadamard Fractional Integral Boundary Conditions

by Bashir Ahmad and Shorog Aljoudi

Fractal Fract. 2023, 7(2), 178; https://doi.org/10.3390/fractalfract7020178 - 10 Feb 2023

Cited by 8 | Viewed by 1497

Abstract

We investigate the existence criteria for solutions of a nonlinear coupled system of Hilfer–Hadamard fractional differential equations of different orders complemented with nonlocal coupled Hadamard fractional integral boundary conditions. The desired results are accomplished with the aid of standard fixed-point theorems. We emphasize [...] Read more.

We investigate the existence criteria for solutions of a nonlinear coupled system of Hilfer–Hadamard fractional differential equations of different orders complemented with nonlocal coupled Hadamard fractional integral boundary conditions. The desired results are accomplished with the aid of standard fixed-point theorems. We emphasize that the fixed point approach is one of the effective methods to establish the existence results for boundary value problems. Examples illustrating the obtained results are constructed. Full article

(This article belongs to the Special Issue New Insights into Nonlinear Coupled Differential Equations with Its Applications)

15 pages, 763 KiB

Open AccessArticle

Noise Spectral of GML Noise and GSR Behaviors for FGLE with Random Mass and Random Frequency

by Lini Qiu, Guitian He, Yun Peng, Hui Cheng and Yujie Tang

Fractal Fract. 2023, 7(2), 177; https://doi.org/10.3390/fractalfract7020177 - 10 Feb 2023

Cited by 3 | Viewed by 1231

Abstract

Due to the interest of anomalous diffusion phenomena and their application, our work has widely studied a fractional-order generalized Langevin Equation (FGLE) with a generalized Mittag–Leffler (GML) noise. Significantly, the spectral of GML noise involving three parameters is well addressed. Furthermore, the spectral [...] Read more.

Due to the interest of anomalous diffusion phenomena and their application, our work has widely studied a fractional-order generalized Langevin Equation (FGLE) with a generalized Mittag–Leffler (GML) noise. Significantly, the spectral of GML noise involving three parameters is well addressed. Furthermore, the spectral amplification (SPA) of an FGLE has also been investigated. The generalized stochastic resonance (GSR) phenomenon for FGLE only influenced by GML noise has been found. Furthermore, material GSR for FGLE influenced by two types of noise has been studied. Moreover, it is found that the GSR behaviors of the FGLE could also be induced by the fractional orders of the FGLE. Full article

(This article belongs to the Special Issue Application of Fractional-Calculus in Physical Systems)

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17 pages, 1775 KiB

Open AccessArticle

Analytical Computational Scheme for Multivariate Nonlinear Time-Fractional Generalized Biological Population Model

by Mohammad Alaroud, Abedel-Karrem Alomari, Nedal Tahat and Anuar Ishak

Fractal Fract. 2023, 7(2), 176; https://doi.org/10.3390/fractalfract7020176 - 10 Feb 2023

Cited by 6 | Viewed by 1509

Abstract

This work provides exact and analytical approximate solutions for a non-linear time-fractional generalized biology population model (FGBPM) with suitable initial data under the time-Caputo fractional derivative, in view of a novel effective and applicable scheme, based upon elegant amalgamation between the Laplace transform [...] Read more.

This work provides exact and analytical approximate solutions for a non-linear time-fractional generalized biology population model (FGBPM) with suitable initial data under the time-Caputo fractional derivative, in view of a novel effective and applicable scheme, based upon elegant amalgamation between the Laplace transform operator and the generalized power series method. The solution form obtained by the proposed algorithm of considered FGBPM is an infinite multivariable convergent series toward the exact solutions for the integer fractional order. Some applications of the posed model are tested to confirm the theoretical aspects and highlight the superiority of the proposed scheme in predicting the analytical approximate solutions in closed forms compared to other existing analytical methods. Associated figure representations and the results are displayed in different dimensional graphs. Numerical analyses are performed, and discussions regarding the errors and the convergence of the scheme are presented. The simulations and results report that the proposed modern scheme is, indeed, direct, applicable, and effective to deal with a wide range of non-linear time multivariable fractional models. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Their Applications)

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Fractal Fract., Volume 7, Issue 2 (February 2023) – 107 articles

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