Fractal and Fractional

13 pages, 4520 KiB

Open AccessArticle

Chaotic Characteristic Analysis of Dynamic Gravity Model with Fractal Structures via an Improved Conical Volume-Delay Function

by Liumeng Yang, Ruichun He, Jie Wang, Wei Zhou, Hongxing Zhao and Huo Chai

Fractal Fract. 2023, 7(3), 278; https://doi.org/10.3390/fractalfract7030278 - 22 Mar 2023

Viewed by 950

Abstract

Road traffic networks are chaotic and highly complex systems. In this paper, we introduce a dynamic gravity model that characterizes the behaviors of the O-D (origin-destination) traffic, such as equilibrium, period-doubling, chaos, and fractal in discrete time. In cases where the original cost [...] Read more.

Road traffic networks are chaotic and highly complex systems. In this paper, we introduce a dynamic gravity model that characterizes the behaviors of the O-D (origin-destination) traffic, such as equilibrium, period-doubling, chaos, and fractal in discrete time. In cases where the original cost function is used, the trip distribution model might degenerate into an all-or-nothing problem without the capacity constraints. To address this shortcoming, we propose substituting the original cost function with an improved conical volume-delay function. This new function retains some of the properties of the original cost function, and its parameters have the same meaning as those in the original function. Our analysis confirms that the double-constrained dynamic gravity model successfully characterizes complex traffic behavior because of the improved conical volume-delay function. Our analysis further shows that the three-parameter bifurcation diagram based on the period characteristics provides deep insight into the actual state of the road traffic networks. Investigating the properties of the model solutions, we further show that the new model is more effective in addressing the all-or-nothing problem. Full article

(This article belongs to the Special Issue State-of-the-Art in Fractional-Order Neural Networks: Theory, Design and Applications)

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26 pages, 977 KiB

Open AccessArticle

Efficient Spectral Collocation Method for Tempered Fractional Differential Equations

by Tinggang Zhao

Fractal Fract. 2023, 7(3), 277; https://doi.org/10.3390/fractalfract7030277 - 22 Mar 2023

Viewed by 2010

Abstract

Transient anomalous diffusion may be modeled by a tempered fractional diffusion equation. In this paper, we present a spectral collocation method with tempered fractional Jacobi functions (TFJFs) as basis functions and obtain an efficient algorithm to solve tempered-type fractional differential equations. We set [...] Read more.

Transient anomalous diffusion may be modeled by a tempered fractional diffusion equation. In this paper, we present a spectral collocation method with tempered fractional Jacobi functions (TFJFs) as basis functions and obtain an efficient algorithm to solve tempered-type fractional differential equations. We set up the approximation error as

O (N^{μ - ν})

for projection and interpolation by the TFJFs, which shows “spectral accuracy” for a certain class of functions. We derive a recurrence relation to evaluate the collocation differentiation matrix for implementing the spectral collocation algorithm. We demonstrate the effectiveness of the new method for the nonlinear initial and boundary problems, i.e., the fractional Helmholtz equation, and the fractional Burgers equation. Full article

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

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10 pages, 507 KiB

Open AccessArticle

Photothermal Response for the Thermoelastic Bending Effect Considering Dissipating Effects by Means of Fractional Dual-Phase-Lag Theory

by Aloisi Somer, Andressa Novatski, Marcelo Kaminski Lenzi, Luciano Rodrigues da Silva and Ervin Kaminski Lenzi

Fractal Fract. 2023, 7(3), 276; https://doi.org/10.3390/fractalfract7030276 - 22 Mar 2023

Cited by 3 | Viewed by 877

Abstract

We analyze an extension of the dual-phase lag model of thermal diffusion theory to accurately predict the contribution of thermoelastic bending (TE) to the Photoacoustic (PA) signal in a transmission configuration. To achieve this, we adopt the particular case of Jeffrey’s equation, an [...] Read more.

We analyze an extension of the dual-phase lag model of thermal diffusion theory to accurately predict the contribution of thermoelastic bending (TE) to the Photoacoustic (PA) signal in a transmission configuration. To achieve this, we adopt the particular case of Jeffrey’s equation, an extension of the Generalized Cattaneo Equations (GCEs). Obtaining the temperature distribution by incorporating the effects of fractional differential operators enables us to determine the TE effects in solid samples accurately. This study contributes to understanding the mechanisms that contribute to the PA signal and highlights the importance of considering fractional differential operators in the analysis of thermoelastic bending. As a result, we can determine the PA signal’s TE component. Our findings demonstrate that the fractional differential operators lead to a wide range of behaviors, including dissipative effects related to anomalous diffusion. Full article

(This article belongs to the Special Issue Fractional Evolutionary Equations and Modeling of Dissipative Processes)

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30 pages, 1899 KiB

Open AccessArticle

Fractional Gradient Methods via ψ-Hilfer Derivative

by Nelson Vieira, M. Manuela Rodrigues and Milton Ferreira

Fractal Fract. 2023, 7(3), 275; https://doi.org/10.3390/fractalfract7030275 - 21 Mar 2023

Cited by 1 | Viewed by 1092

Abstract

Motivated by the increase in practical applications of fractional calculus, we study the classical gradient method under the perspective of the

ψ

-Hilfer derivative. This allows us to cover several definitions of fractional derivatives that are found in the literature in our study. [...] Read more.

Motivated by the increase in practical applications of fractional calculus, we study the classical gradient method under the perspective of the

ψ

-Hilfer derivative. This allows us to cover several definitions of fractional derivatives that are found in the literature in our study. The convergence of the

ψ

-Hilfer continuous fractional gradient method was studied both for strongly and non-strongly convex cases. Using a series representation of the target function, we developed an algorithm for the

ψ

-Hilfer fractional order gradient method. The numerical method obtained by truncating higher-order terms was tested and analyzed using benchmark functions. Considering variable order differentiation and step size optimization, the

ψ

-Hilfer fractional gradient method showed better results in terms of speed and accuracy. Our results generalize previous works in the literature. Full article

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

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29 pages, 21253 KiB

Open AccessArticle

Optimal FOPI Error Voltage Control Dead-Time Compensation for PMSM Servo System

by Fumin Li, Ying Luo, Xin Luo, Pengchong Chen and Yangquan Chen

Fractal Fract. 2023, 7(3), 274; https://doi.org/10.3390/fractalfract7030274 - 21 Mar 2023

Cited by 3 | Viewed by 1391

Abstract

This paper proposed a dead-time compensation method with fractional-order proportional integral (FOPI) error voltage control. The disturbance voltages caused by the power devices’ dead time and non-ideal switching characteristics are compensated for with the FOPI controller and fed to the reference voltage. In [...] Read more.

This paper proposed a dead-time compensation method with fractional-order proportional integral (FOPI) error voltage control. The disturbance voltages caused by the power devices’ dead time and non-ideal switching characteristics are compensated for with the FOPI controller and fed to the reference voltage. In this paper, the actual error voltage is calculated based on the model and actual voltage of the permanent magnet synchronous motor. Considering the parameter error of the permanent magnet synchronous motor and the voltage error caused by the dead-time effect, a FOPI controller is used to calculate the compensation voltage. An improved particle swarm optimization (PSO) algorithm is utilized to design the parameters of the FOPI controller in order to eliminate the dead-time effect, and the optimal fitness function is designed. Compared with other optimization algorithms, the improved PSO algorithm can achieve faster convergence speed in the error voltage controller parameter design. The proposed dead-time compensation method can improve the performance of the current response and eliminate the dead-time effect. This method also eliminates all harmonic disturbances and has a good suppression effect on high-frequency harmonics. The simulation and experimental results show that the dead-time compensation method using optimal FOPI error voltage control makes the current ripple smaller and the response speed faster than that of the traditional optimal integer-order PI control, thus demonstrating the effectiveness and advantages of the proposed method. Full article

(This article belongs to the Special Issue Fractional Order Systems with Application to Electrical Power Engineering)

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18 pages, 5466 KiB

Open AccessArticle

Fractal-Based Approaches to Pore Structure Investigation and Water Saturation Prediction from NMR Measurements: A Case Study of the Gas-Bearing Tight Sandstone Reservoir in Nanpu Sag

by Weibiao Xie, Qiuli Yin, Jingbo Zeng, Guiwen Wang, Cheng Feng and Pan Zhang

Fractal Fract. 2023, 7(3), 273; https://doi.org/10.3390/fractalfract7030273 - 21 Mar 2023

Cited by 5 | Viewed by 1307

Abstract

Pore space of tight sandstone samples exhibits fractal characteristics. Nuclear magnetic resonance is an effective method for pore size characterization. This paper focuses on fractal characteristics of pore size from nuclear magnetic resonance (NMR) of tight sandstone samples. The relationship between the fractal [...] Read more.

Pore space of tight sandstone samples exhibits fractal characteristics. Nuclear magnetic resonance is an effective method for pore size characterization. This paper focuses on fractal characteristics of pore size from nuclear magnetic resonance (NMR) of tight sandstone samples. The relationship between the fractal dimension from NMR with pore structure and water saturation is parameterized by analyzing experimental data. Based on it, a pore structure characterization and classification method for water-saturated tight sandstone and a water saturation prediction method in a gas-bearing sandstone reservoir have been proposed. To verify the models, the fractal dimension from NMR of 19 tight sandstone samples selected from the gas-bearing tight sandstone reservoir of Shahejie Formation in Nanpu Sag and that of 16 of them under different water saturation states are analyzed. The application result of new methods in the gas-bearing tight sandstone reservoir of Shahejie Formation in Nanpu Sag shows consistency with experimental data. This paper has facilitated the development of the NMR application by providing a non-electrical logging idea in reservoir quality evaluation and water saturation prediction. It provides a valuable scientific resource for reservoir engineering and petrophysics of unconventional reservoir types, such as tight sandstone, low porosity, and low permeability sandstone, shale, and carbonate rock reservoirs. Full article

(This article belongs to the Topic Petroleum and Gas Engineering)

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14 pages, 2569 KiB

Open AccessArticle

Gray Level Co-Occurrence Matrix, Fractal and Wavelet Analyses of Discrete Changes in Cell Nuclear Structure following Osmotic Stress: Focus on Machine Learning Methods

by Igor Pantic, Svetlana Valjarevic, Jelena Cumic, Ivana Paunkovic, Tatjana Terzic and Peter R. Corridon

Fractal Fract. 2023, 7(3), 272; https://doi.org/10.3390/fractalfract7030272 - 20 Mar 2023

Cited by 11 | Viewed by 1464

Abstract

In this work, we demonstrate that it is possible to create supervised machine-learning models using a support vector machine and random forest algorithms to separate yeast cells exposed to hyperosmotic stress from intact cells. We performed fractal, gray level co-occurrence matrix (GLCM), and [...] Read more.

In this work, we demonstrate that it is possible to create supervised machine-learning models using a support vector machine and random forest algorithms to separate yeast cells exposed to hyperosmotic stress from intact cells. We performed fractal, gray level co-occurrence matrix (GLCM), and discrete wavelet transform analyses on digital micrographs of nuclear regions of interest of a total of 2000 Saccharomyces cerevisiae cells: 1000 exposed to hyperosmotic environments and 1000 control cells. For each nucleus, we calculated values for fractal dimension, angular second moment, inverse difference moment, textural contrast, correlation feature, textural variance, and discrete wavelet coefficient energy. The support vector machine achieved an acceptable classification accuracy of 71.7% in predicting whether the cell belonged to the experimental or control group. The random forest model performed better than the support vector machine, with a classification accuracy of 79.8%. These findings can serve as a starting point for developing AI-based methods that use GLCM, fractal, and wavelet data to classify damaged and healthy cells and make predictions about various physiological and pathological phenomena associated with osmotic stress. Full article

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10 pages, 340 KiB

Open AccessArticle

High-Order Nonlinear Functional Differential Equations: New Monotonic Properties and Their Applications

by Hail S. Alrashdi, Osama Moaaz, Ghada AlNemer and Elmetwally M. Elabbasy

Fractal Fract. 2023, 7(3), 271; https://doi.org/10.3390/fractalfract7030271 - 20 Mar 2023

Viewed by 803

Abstract

We provide streamlined criteria for evaluating the oscillatory behavior of solutions to a class of higher-order functional differential equations in the non-canonical case. We use a comparison approach with first-order equations that have standard oscillation criteria. Normally, in the non-canonical situation, the oscillation [...] Read more.

We provide streamlined criteria for evaluating the oscillatory behavior of solutions to a class of higher-order functional differential equations in the non-canonical case. We use a comparison approach with first-order equations that have standard oscillation criteria. Normally, in the non-canonical situation, the oscillation test requires three independent conditions, but we provide criteria with two-conditions without checking the additional conditions. Lastly, we give examples to highlight the significance of the findings. Full article

(This article belongs to the Special Issue Operators of Fractional Calculus and Their Multidisciplinary Applications, 2nd Edition)

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18 pages, 348 KiB

Open AccessArticle

Some New Applications of the Faber Polynomial Expansion Method for Generalized Bi-Subordinate Functions of Complex Order γ Defined by q-Calculus

by Mohammad Faisal Khan and Mohammed AbaOud

Fractal Fract. 2023, 7(3), 270; https://doi.org/10.3390/fractalfract7030270 - 18 Mar 2023

Viewed by 769

Abstract

This work examines a new subclass of generalized bi-subordinate functions of complex order

γ

connected to the q-difference operator. We obtain the upper bounds

ρ_{m}

for generalized bi-subordinate functions of complex order

γ

using the Faber polynomial expansion technique. Additionally, we [...] Read more.

This work examines a new subclass of generalized bi-subordinate functions of complex order

γ

connected to the q-difference operator. We obtain the upper bounds

ρ_{m}

for generalized bi-subordinate functions of complex order

γ

using the Faber polynomial expansion technique. Additionally, we find coefficient bounds

|ρ_{2}|

and Feke–Sezgo problems

|ρ_{3} - ρ_{2}^{2}|

for the functions in the newly defined class, subject to gap series conditions. Using the Faber polynomial expansion method, we show some results that illustrate diverse uses of the Ruschewey q differential operator. The findings in this paper generalize those from previous efforts by a number of prior researchers. Full article

(This article belongs to the Special Issue Fractional Calculus and Hypergeometric Functions in Complex Analysis)

17 pages, 1421 KiB

Open AccessArticle

Quantum Weighted Fractional-Order Transform

by Tieyu Zhao and Yingying Chi

Fractal Fract. 2023, 7(3), 269; https://doi.org/10.3390/fractalfract7030269 - 18 Mar 2023

Cited by 1 | Viewed by 896

Abstract

Quantum Fourier transform (QFT) transformation plays a very important role in the design of many quantum algorithms. Fractional Fourier transform (FRFT), as an extension of the Fourier transform, is particularly important due to the design of its quantum algorithm. [...] Read more.

Quantum Fourier transform (QFT) transformation plays a very important role in the design of many quantum algorithms. Fractional Fourier transform (FRFT), as an extension of the Fourier transform, is particularly important due to the design of its quantum algorithm. In this paper, a new reformulation of the weighted fractional Fourier transform (WFRFT) is proposed in order to realize quantum FRFT; however, we found that this reformulation can be applied to other transformations, and therefore, this paper presents the weighted fractional Hartley transform (WFRHT). For the universality of application, we further propose a general weighted fractional-order transform (WFRT). When designing the quantum circuits, we realized the quantum WFRFT via QFT and quantum phase estimation (QPE). Moreover, after extending our design to the WFRHT, we were able to formulate the quantum WFRHT. Finally, in accordance with the research results, we designed the quantum circuit of the general WFRT, and subsequently proposed the quantum WFRT. The research in this paper has great value as a reference for the design and application of quantum algorithms. Full article

(This article belongs to the Special Issue Recent Developments in Fractional Quantum Mechanics)

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18 pages, 1117 KiB

Open AccessArticle

Dynamic Event-Triggered Consensus for Fractional-Order Multi-Agent Systems without Intergroup Balance Condition

by Bingrui Xu and Bing Li

Fractal Fract. 2023, 7(3), 268; https://doi.org/10.3390/fractalfract7030268 - 17 Mar 2023

Cited by 1 | Viewed by 931

Abstract

This paper deals with the problem of group consensus for a fractional-order multi-agent system (FOMAS) without considering the intergroup balance condition. By adopting a dynamic event-triggered mechanism, the updating frequency of control input is significantly reduced while the consensus performance is maintained. By [...] Read more.

This paper deals with the problem of group consensus for a fractional-order multi-agent system (FOMAS) without considering the intergroup balance condition. By adopting a dynamic event-triggered mechanism, the updating frequency of control input is significantly reduced while the consensus performance is maintained. By utilizing the Lyapunov direct method and the properties of a fractional-order derivative, several novel criteria are presented for analyzing the Mittag–Leffler stability of error systems and excluding the Zeno behavior in the triggering mechanism. An example and its simulations are demonstrated to prove the validity of the theoretical results. Full article

(This article belongs to the Special Issue Advances in Fractional-Order Neural Networks, Volume II)

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19 pages, 2451 KiB

Open AccessArticle

Cross-Correlation Multifractal Analysis of Technological Innovation, Financial Market and Real Economy Indices

by Jinchuan Ke, Yu Duan, Chao Xu and Yue Zhang

Fractal Fract. 2023, 7(3), 267; https://doi.org/10.3390/fractalfract7030267 - 17 Mar 2023

Cited by 2 | Viewed by 1324

Abstract

Technological innovation, the financial market, and the real economy are mutually promoting and restricting. Considering the interference of market-noise information, this paper applies the wavelet-denoising method of the soft- and hard-threshold compromise functions to process the original information so as to eliminate the [...] Read more.

Technological innovation, the financial market, and the real economy are mutually promoting and restricting. Considering the interference of market-noise information, this paper applies the wavelet-denoising method of the soft- and hard-threshold compromise functions to process the original information so as to eliminate the noise information, and combines multifractal detrended cross-correlation analysis with the sliding-window approach, focusing on the change in the Hurst index and the parameter change in the multifractal spectrum to explore the interaction in between. The research results show that there is a certain cross-correlation among technological-innovation, financial-market, and real-economy indices. Firstly, the cross-correlation among them has significant multifractal characteristics rather than single-fractal characteristics. Secondly, the fractal characteristics reveal the long memory of the interaction among the three indices. Thirdly, there are also obvious differences in the degree of local chaos and volatility of the interaction. Fourthly, the cross-correlation among technological-innovation, financial-market, and real-economy indices has significant multifractal characteristics rather than single-fractal characteristics. In comparison, the cross-correlation multifractal characteristics among technological innovation, the financial market, and the real economy are time-varying, and the cross-correlation multifractal characteristics between the technological-innovation index and the real-economy index are the most obvious. Full article

(This article belongs to the Special Issue Application of Fractal Processes and Fractional Derivatives in Finance)

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10 pages, 299 KiB

Open AccessBrief Report

An Application of the Prabhakar Fractional Operator to a Subclass of Analytic Univalent Function

by M. Indushree and Madhu Venkataraman

Fractal Fract. 2023, 7(3), 266; https://doi.org/10.3390/fractalfract7030266 - 17 Mar 2023

Cited by 1 | Viewed by 936

Abstract

Fractional differential operators have recently been linked with numerous other areas of science, technology and engineering studies. For a real variable, the class of fractional differential and integral operators is evaluated. In this study, we look into the Prabhakar fractional differential operator, which [...] Read more.

Fractional differential operators have recently been linked with numerous other areas of science, technology and engineering studies. For a real variable, the class of fractional differential and integral operators is evaluated. In this study, we look into the Prabhakar fractional differential operator, which is the most applicable fractional differential operator in a complex domain. In terms of observing a group of normalized analytical functions, we express the operator. In the open unit disc, we deal with its geometric performance. Applying the Prabhakar fractional differential operator

_{d}^{c} θ_{α, β}^{γ, ω}

to a subclass of analytic univalent function results in the creation of a new subclass of mathematical functions:

W

(

γ, ω, α, β, θ

,m,c,z,p,q). We obtain the characteristic, neighborhood and convolution properties for this class. Some of these properties are extensions of defined results. Full article

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

11 pages, 686 KiB

Open AccessArticle

Mathematical Model of Heat Conduction for a Semi-Infinite Body, Taking into Account Memory Effects and Spatial Correlations

by Vetlugin D. Beybalaev, Abutrab A. Aliverdiev, Amuchi Z. Yakubov, Said A. Ninalalov and Anise A. Amirova

Fractal Fract. 2023, 7(3), 265; https://doi.org/10.3390/fractalfract7030265 - 16 Mar 2023

Cited by 1 | Viewed by 953

Abstract

One of the promising approaches to the description of many physical processes is the use of the fractional derivative mathematical apparatus. Fractional dimensions very often arise when modeling various processes in fractal (multi-scale and self-similar) environments. In a fractal medium, in contrast to [...] Read more.

One of the promising approaches to the description of many physical processes is the use of the fractional derivative mathematical apparatus. Fractional dimensions very often arise when modeling various processes in fractal (multi-scale and self-similar) environments. In a fractal medium, in contrast to an ordinary continuous medium, a randomly wandering particle moves away from the reference point more slowly since not all directions of motion become available to it. The slowdown of the diffusion process in fractal media is so significant that physical quantities begin to change more slowly than in ordinary media.This effect can only be taken into account with the help of integral and differential equations containing a fractional derivative with respect to time. Here, the problem of heat and mass transfer in media with a fractal structure was posed and analytically solved when a heat flux was specified on one of the boundaries. The second initial boundary value problem for the heat equation with a fractional Caputo derivative with respect to time and the Riesz derivative with respect to the spatial variable was studied. A theorem on the semigroup property of the fractional Riesz derivative was proved. To find a solution, the problem was reduced to a boundary value problem with boundary conditions of the first kind. The solution to the problem was found by applying the Fourier transform in the spatial variable and the Laplace transform in time. A computational experiment was carried out to analyze the obtained solutions. Graphs of the temperature distribution dependent on the coordinate and time were constructed. Full article

(This article belongs to the Special Issue Recent Advances in Fractal and Fractional Calculus Theory and Its Applications)

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17 pages, 1049 KiB

Open AccessArticle

Improved Particle Swarm Optimization Fractional-System Identification Algorithm for Electro-Optical Tracking System

by Tong Guo, Jiuqiang Deng, Yao Mao and Xi Zhou

Fractal Fract. 2023, 7(3), 264; https://doi.org/10.3390/fractalfract7030264 - 16 Mar 2023

Viewed by 1054

Abstract

When an electro-optical tracking system (ETS) needs higher control precision, system identification can be considered to improve the accuracy of the system, so as to improve its control effect. The fractional system model of ETS can describe the characteristics of the system better [...] Read more.

When an electro-optical tracking system (ETS) needs higher control precision, system identification can be considered to improve the accuracy of the system, so as to improve its control effect. The fractional system model of ETS can describe the characteristics of the system better and improve the accuracy of the system model. Therefore, this paper presents a fractional system identification algorithm for ETS that is based on an improved particle swarm optimization algorithm. The existence of the fractional order system of ETS was verified by identification experiments, and the fractional order system model was obtained. Under the same conditions, PI controllers were designed based on a fractional order system and an integer order system, respectively. The results verify the superiority of fractional order system in ETS. Full article

(This article belongs to the Special Issue Fractional-Order Control: Design, Stability Analysis, and Implementation)

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17 pages, 480 KiB

Open AccessArticle

Refinable Trapezoidal Method on Riemann–Stieltjes Integral and Caputo Fractional Derivatives for Non-Smooth Functions

by Gopalakrishnan Karnan and Chien-Chang Yen

Fractal Fract. 2023, 7(3), 263; https://doi.org/10.3390/fractalfract7030263 - 15 Mar 2023

Viewed by 1351

Abstract

The Caputo fractional

α

-derivative,

0 < α < 1

, for non-smooth functions with

1 + α

regularity is calculated by numerical computation. Let I be an interval and

D_{α} (I)

be the set of all functions [...] Read more.

The Caputo fractional

α

-derivative,

0 < α < 1

, for non-smooth functions with

1 + α

regularity is calculated by numerical computation. Let I be an interval and

D_{α} (I)

be the set of all functions

f (x)

which satisfy

f (x) = f (c) + f^{'} (c) (x - a) + g_{c} (x) (x - c) {| (x - c) |}^{α}

, where

x, c \in I

and

g_{c} (x)

is a continuous function for each c. We first extend the trapezoidal method on the set

D_{α} (I)

and rewrite the integrand of the Caputo fractional integral as a product of two differentiable functions. In this approach, the non-smooth function and the singular kernel could have the same impact. The trapezoidal method using the Riemann–Stieltjes integral (TRSI) depends on the regularity of the two functions in the integrand. Numerical simulations demonstrated that the order of accuracy cannot be increased as the number of zones increases using the uniform discretization. However, for a fixed coarsest grid discretization, a refinable mesh approach was employed; the corresponding results show that the order of accuracy is

k α

, where k is a refinable scale. Meanwhile, the application of the product of two differentiable functions can also be applied to some Riemann–Liouville fractional differential equations. Finally, the stable numerical scheme is shown. Full article

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

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12 pages, 2121 KiB

Open AccessArticle

Cerofolini’s Model and the Fractal Adsorption Isotherms

by Gianina Dobrescu, Florica Papa, Daniela Culita, Ioan Balint and Niculae I. Ionescu

Fractal Fract. 2023, 7(3), 262; https://doi.org/10.3390/fractalfract7030262 - 14 Mar 2023

Viewed by 1024

Abstract

The close link between the roughness of a surface and its adsorptive properties in Cerofolini’s model yields, with an adequate choice of adsorption energy, the well-known Dubinin-Radushkevich or Freundlich adsorption isotherms. Assuming fractal behavior concerning both energetic and geometric surface heterogeneities described by [...] Read more.

The close link between the roughness of a surface and its adsorptive properties in Cerofolini’s model yields, with an adequate choice of adsorption energy, the well-known Dubinin-Radushkevich or Freundlich adsorption isotherms. Assuming fractal behavior concerning both energetic and geometric surface heterogeneities described by the power-law expressions and fractal dimensions, the paper will develop some fractal adsorption isotherms. Using our theoretical approach, fractal isotherms will provide insights not only into the fractal behavior of the surface geometry but also into the fractal energetic heterogeneities, implying that a sorbent does not need to be porous to apply a fractal isotherm: adsorption on “flat” surfaces can also be described by fractal isotherms and fractal dimensions related to energetic disorders. For example, the theory will be applied to computing the energetic fractal dimensions of some nanoparticle catalysts, Rh/Al₂O₃, Rh/TiO₂, and Rh/WO₃. Full article

(This article belongs to the Special Issue Fractal Analysis and Fractal Dimension in Materials Chemistry)

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19 pages, 378 KiB

Open AccessArticle

On Discrete Weighted Lorentz Spaces and Equivalent Relations between Discrete ℓ_p-Classes

by Ravi P. Agarwal, Safi S. Rabie and Samir H. Saker

Fractal Fract. 2023, 7(3), 261; https://doi.org/10.3390/fractalfract7030261 - 14 Mar 2023

Viewed by 943

Abstract

In this paper, we study some relations between different weights in the classes

B_{p},

B_{p}^{*},

M_{p}

and

M_{p}^{*}

that characterize the boundedness of the Hardy operator and the adjoint Hardy operator. We also prove that [...] Read more.

In this paper, we study some relations between different weights in the classes

B_{p},

B_{p}^{*},

M_{p}

and

M_{p}^{*}

that characterize the boundedness of the Hardy operator and the adjoint Hardy operator. We also prove that these classes generate the same weighted Lorentz space

Λ_{p}

. These results will be proven by using the properties of classes

B_{p},

B_{p}^{*},

M_{p}

and

M_{p}^{*}

, including the self-improving properties and also the properties of the generalized Hardy operator

H_{p}

, the adjoint operator

S_{q}

and some fundamental relations between them connecting their composition to their sum. Full article

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

18 pages, 2273 KiB

Open AccessArticle

Approximate Solutions for Time-Fractional Fornberg–Whitham Equation with Variable Coefficients

by Fahad Alsidrani, Adem Kılıçman and Norazak Senu

Fractal Fract. 2023, 7(3), 260; https://doi.org/10.3390/fractalfract7030260 - 14 Mar 2023

Cited by 1 | Viewed by 1225

Abstract

In this research, three numerical methods, namely the variational iteration method, the Adomian decomposition method, and the homotopy analysis method are considered to achieve an approximate solution for a third-order time-fractional partial differential Equation (TFPDE). The equation is obtained from the classical (FW) [...] Read more.

In this research, three numerical methods, namely the variational iteration method, the Adomian decomposition method, and the homotopy analysis method are considered to achieve an approximate solution for a third-order time-fractional partial differential Equation (TFPDE). The equation is obtained from the classical (FW) equation by replacing the integer-order time derivative with the Caputo fractional derivative of order

η = (0, 1]

with variable coefficients. We consider homogeneous boundary conditions to find the approximate solutions for the bounded space variable

l < χ < L

and

l, L \in R

. To confirm the effectiveness of the proposed methods of non-integer order

η

, the computation of two test problems was presented. A comparison is made between the obtained results of the (VIM), (ADM), and (HAM) through tables and graphs. The numerical results demonstrate the effectiveness of the three numerical methods. Full article

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

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17 pages, 1004 KiB

Open AccessArticle

The Fractional Analysis of a Nonlinear mKdV Equation with Caputo Operator

by Haifa A. Alyousef, Rasool Shah, Nehad Ali Shah, Jae Dong Chung, Sherif M. E. Ismaeel and Samir A. El-Tantawy

Fractal Fract. 2023, 7(3), 259; https://doi.org/10.3390/fractalfract7030259 - 14 Mar 2023

Cited by 6 | Viewed by 1116

Abstract

In this study, we aim to provide reliable methods for the initial value problem of the fractional modified Korteweg–de Vries (mKdV) equations. Fractional differential equations are essential for more precise simulation of numerous processes. The hybrid Yang transformation decomposition method (YTDM) and Yang [...] Read more.

In this study, we aim to provide reliable methods for the initial value problem of the fractional modified Korteweg–de Vries (mKdV) equations. Fractional differential equations are essential for more precise simulation of numerous processes. The hybrid Yang transformation decomposition method (YTDM) and Yang homotopy perturbation method (YHPM) are employed in a very simple and straightforward manner to handle the current problems. The derivative of fractional order is displayed in a Caputo form operator. To illustrate the conclusion given from the findings, a few numerical cases are taken into account for their approximate analytical solutions. We looked at two cases and contrasted them with the actual result to validate the methodologies. These techniques create recurrence relations representing the proposed problem’s solution. It is possible to find the series solutions to the given problems, and these solutions have components that converge to precise solutions more quickly. Tables and graphs are used to describe the new results, which demonstrate the present methods’ adequate accuracy. The actual and estimated outcomes are demonstrated in graphs and tables to be quite similar, demonstrating the usefulness of the proposed approaches. The innovation of the current work resides in the application of effective methods that require less calculation and achieve a greater level of accuracy. Additionally, the suggested approaches can be applied in the future to resolve other nonlinear fractional problems, which will be a scientific contribution to the research community. Full article

(This article belongs to the Special Issue Mathematical Modelling of Real Phenomena Based on Fractional Derivatives)

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19 pages, 930 KiB

Open AccessArticle

Dynamical Analysis of Generalized Tumor Model with Caputo Fractional-Order Derivative

by Ausif Padder, Laila Almutairi, Sania Qureshi, Amanullah Soomro, Afroz Afroz, Evren Hincal and Asifa Tassaddiq

Fractal Fract. 2023, 7(3), 258; https://doi.org/10.3390/fractalfract7030258 - 13 Mar 2023

Cited by 21 | Viewed by 1921

Abstract

In this study, we perform a dynamical analysis of a generalized tumor model using the Caputo fractional-order derivative. Tumor growth models are widely used in biomedical research to understand the dynamics of tumor development and to evaluate potential treatments. The Caputo fractional-order derivative [...] Read more.

In this study, we perform a dynamical analysis of a generalized tumor model using the Caputo fractional-order derivative. Tumor growth models are widely used in biomedical research to understand the dynamics of tumor development and to evaluate potential treatments. The Caputo fractional-order derivative is a mathematical tool that is recently being applied to model biological systems, including tumor growth. We present a detailed mathematical analysis of the generalized tumor model with the Caputo fractional-order derivative and examine its dynamical behavior. Our results show that the Caputo fractional-order derivative provides a more accurate description of the tumor growth dynamics compared to classical integer-order derivatives. We also provide a comprehensive stability analysis of the tumor model and show that the fractional-order derivative allows for a more nuanced understanding of the stability of the system. The least-square curve fitting method fits several biological parameters, including the fractional-order parameter

α

. In conclusion, our study provides new insights into the dynamics of tumor growth and highlights the potential of the Caputo fractional-order derivative as a valuable tool in biomedical research. The results of this study shall have significant implications for the development of more effective treatments for tumor growth and the design of more accurate mathematical models of tumor development. Full article

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

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17 pages, 4376 KiB

Open AccessArticle

The Hidden Dimension of Façades: Fractal Analysis Reveals Composition Rules in Classical and Renaissance Architecture

by Vilmos Katona

Fractal Fract. 2023, 7(3), 257; https://doi.org/10.3390/fractalfract7030257 - 11 Mar 2023

Cited by 2 | Viewed by 1834

Abstract

This study uses fractal analysis to measure the detailed intensity of well-known Classical and Renaissance façades. The study develops a method to understand their interrelated design principles more comprehensively. With this evaluation tool, one can observe intrinsic connections that support the historical continuity [...] Read more.

This study uses fractal analysis to measure the detailed intensity of well-known Classical and Renaissance façades. The study develops a method to understand their interrelated design principles more comprehensively. With this evaluation tool, one can observe intrinsic connections that support the historical continuity and point out balancing composition protocols, such as the ‘compensation rule’, that regulated design for centuries. The calculations offer mathematical constants to identify Classical and Renaissance details by plasticity rates. Finally, we base this method on spatial evaluation. Our calculations involve depth, which connects planar front views with the haptic reality of the façades’ tectonic layers. The article also discusses the cultural and urban implications of our results. Full article

(This article belongs to the Section Geometry)

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16 pages, 6383 KiB

Open AccessArticle

Dual-Loop Voltage–Current Control of a Fractional-Order Buck-Boost Converter Using a Fractional-Order PI^λ Controller

by Lingling Xie, Di Wan and Rui Qin

Fractal Fract. 2023, 7(3), 256; https://doi.org/10.3390/fractalfract7030256 - 11 Mar 2023

Cited by 2 | Viewed by 1145

Abstract

Based on the fact that the inductor and capacitor are of a non-integer order by nature, to provide a more accurate theoretical basis for the optimal control of the converter, the fractional-order model of the Buck-Boost converter in the continuous mode of current [...] Read more.

Based on the fact that the inductor and capacitor are of a non-integer order by nature, to provide a more accurate theoretical basis for the optimal control of the converter, the fractional-order model of the Buck-Boost converter in the continuous mode of current is established according to the fractional-order calculus theory. The fractional-order PI^λ control system of the fractional-order Buck-Boost converter is designed to compare the performance of the integer-order PI controller with the fractional-order controller. Secondly, the sparrow search algorithm is applied to the optimal design of the fractional-order PI^λ control system of the fractional-order Buck-Boost converter to improve the system’s phase margin, stability, and robustness. Finally, the simulation is verified on the Matlab/Simulink simulation platform and compared with the integer-order PI controller. Full article

(This article belongs to the Section Engineering)

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14 pages, 468 KiB

Open AccessArticle

On Exact Solutions of Some Space–Time Fractional Differential Equations with M-truncated Derivative

by Ayten Özkan, Erdoĝan Mehmet Özkan and Ozgur Yildirim

Fractal Fract. 2023, 7(3), 255; https://doi.org/10.3390/fractalfract7030255 - 10 Mar 2023

Cited by 9 | Viewed by 1189

Abstract

In this study, the extended

G^{'} / G

method is used to investigate the space–time fractional Burger-like equation and the space–time-coupled Boussinesq equation with M-truncated derivative, which have an important place in fluid dynamics. This method is efficient and produces soliton solutions. [...] Read more.

In this study, the extended

G^{'} / G

method is used to investigate the space–time fractional Burger-like equation and the space–time-coupled Boussinesq equation with M-truncated derivative, which have an important place in fluid dynamics. This method is efficient and produces soliton solutions. A symbolic computation program called Maple was used to implement the method in a dependable and effective way. There are also a few graphs provided for the solutions. Using the suggested method to solve these equations, we have provided many new exact solutions that are distinct from those previously found. By offering insightful explanations of many nonlinear systems, the study’s findings add to the body of literature. The results revealed that the suggested method is a valuable mathematical tool and that using a symbolic computation program makes these tasks simpler, more dependable, and quicker. It is worth noting that it may be used for a wide range of nonlinear evolution problems in mathematical physics. The study’s findings may have an influence on how different physical problems are interpreted. Full article

(This article belongs to the Special Issue Advances in Nonlinear Differential Equations with Applications)

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17 pages, 580 KiB

Open AccessArticle

Mittag–Leffler Functions in Discrete Time

by Ferhan M. Atıcı, Samuel Chang and Jagan Mohan Jonnalagadda

Fractal Fract. 2023, 7(3), 254; https://doi.org/10.3390/fractalfract7030254 - 10 Mar 2023

Viewed by 1015

Abstract

In this paper, we give an efficient way to calculate the values of the Mittag–Leffler (h-ML) function defined in discrete time

h N

, where

h > 0

is a real number. We construct a matrix equation that represents an iteration [...] Read more.

In this paper, we give an efficient way to calculate the values of the Mittag–Leffler (h-ML) function defined in discrete time

h N

, where

h > 0

is a real number. We construct a matrix equation that represents an iteration scheme obtained from a fractional h-difference equation with an initial condition. Fractional h-discrete operators are defined according to the Nabla operator and the Riemann–Liouville definition. Some figures and examples are given to illustrate this new calculation technique for the h-ML function in discrete time. The h-ML function with a square matrix variable in a square matrix form is also given after proving the Putzer algorithm. Full article

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

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16 pages, 1523 KiB

Open AccessArticle

Control and Synchronization of a Novel Realizable Nonlinear Chaotic System

by Mohammed Almuzaini and Abdullah Alzahrani

Fractal Fract. 2023, 7(3), 253; https://doi.org/10.3390/fractalfract7030253 - 10 Mar 2023

Cited by 4 | Viewed by 1100

Abstract

The study proposes a novel chaotic system with a cubic non-linear term. Different system characteristics are investigated including equilibria, stability, invariance, dissipation, Lyapunov dimension, and Lyapunov exponents. Also, the electronic circuit and Signal flow graph of the system are carried out to show [...] Read more.

The study proposes a novel chaotic system with a cubic non-linear term. Different system characteristics are investigated including equilibria, stability, invariance, dissipation, Lyapunov dimension, and Lyapunov exponents. Also, the electronic circuit and Signal flow graph of the system are carried out to show the applicability of the chaotic system. Lyapunov stability theorem converts the system’s chaotic behavior to unstable trivial fixed point. The study also focuses on demonstrating complete synchronization between two similar novel chaotic systems. According to Lyapunov stability theorem, simple application in secure communication was developed by employing the chaos synchronization results. Numerical simulations for the systems are performed for establishing the synchronization strategy effectiveness and proposed control. Full article

(This article belongs to the Special Issue Fractional-Order Chaotic System: Control and Synchronization)

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18 pages, 1454 KiB

Open AccessArticle

Solitary Wave Solutions to a Fractional Model Using the Improved Modified Extended Tanh-Function Method

by Mohammed Bakheet Almatrafi

Fractal Fract. 2023, 7(3), 252; https://doi.org/10.3390/fractalfract7030252 - 10 Mar 2023

Cited by 13 | Viewed by 1145

Abstract

Nonlinear fractional partial differential equations (NLFPDEs) are widely used in simulating a variety of phenomena arisen in several disciplines such as applied mathematics, engineering, physics, and a wide range of other applications. Solitary wave solutions of NLFPDEs have become a significant tool in [...] Read more.

Nonlinear fractional partial differential equations (NLFPDEs) are widely used in simulating a variety of phenomena arisen in several disciplines such as applied mathematics, engineering, physics, and a wide range of other applications. Solitary wave solutions of NLFPDEs have become a significant tool in understanding the long-term dynamics of these events. This article primarily focuses on using the improved modified extended tanh-function algorithm to determine certain traveling wave solutions to the space-time fractional symmetric regularized long wave (SRLW) equation, which is used to discuss space-charge waves, shallow water waves, etc. The Jumarie’s modified Riemann-Liouville derivative is successfully used to deal with the fractional derivatives, which appear in the SRLW problem. We find many traveling wave solutions on the form of trigonometric, hyperbolic, complex, and rational functions. Furthermore, the performance of the employed technique is investigated in comparison to other techniques such as the Oncoming

e x p (- Θ (q))

-expansion method and the extended Jacobi elliptic function expansion strategy. Some obtained results are graphically displayed to show their physical features. The findings of this article demonstrate that the used approach enables us to handle more NLFPDEs that emerge in mathematical physics. Full article

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

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14 pages, 2476 KiB

Open AccessArticle

Wear Model of a Mechanical Seal Based on Piecewise Fractal Theory

by Xingya Ni, Jianjun Sun, Chenbo Ma and Yuyan Zhang

Fractal Fract. 2023, 7(3), 251; https://doi.org/10.3390/fractalfract7030251 - 10 Mar 2023

Cited by 3 | Viewed by 1261

Abstract

In this research, a model is proposed using piecewise fractal theory that considers abrasive and adhesive wear. The model demonstrated improved wear computation accuracy of a contact mechanical seal end face. The validity of the model was established by comparing the simulation results [...] Read more.

In this research, a model is proposed using piecewise fractal theory that considers abrasive and adhesive wear. The model demonstrated improved wear computation accuracy of a contact mechanical seal end face. The validity of the model was established by comparing the simulation results with experimental data and the conventional MB and Archard models. The loading effect and surface morphology parameters involved in wear were also investigated. Results show that severe wear occurs with increasing load ratio, large fractal dimensions, and a higher scale coefficient of the surface morphology. The findings offer a novel method for precisely calculating and projecting the amount of wear of a contact mechanical seal and predicting its wear behavior. Full article

(This article belongs to the Section Geometry)

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22 pages, 6940 KiB

Open AccessArticle

Robust Trajectory Tracking Control for Serial Robotic Manipulators Using Fractional Order-Based PTID Controller

by Banu Ataşlar-Ayyıldız

Fractal Fract. 2023, 7(3), 250; https://doi.org/10.3390/fractalfract7030250 - 09 Mar 2023

Cited by 6 | Viewed by 1257

Abstract

The design of advanced robust control is crucial for serial robotic manipulators under various uncertainties and disturbances in case of the forceful performance needs of industrial robotic applications. Therefore, this work has proposed the design and implementation of a fractional order proportional tilt [...] Read more.

The design of advanced robust control is crucial for serial robotic manipulators under various uncertainties and disturbances in case of the forceful performance needs of industrial robotic applications. Therefore, this work has proposed the design and implementation of a fractional order proportional tilt integral derivative (FOPTID) controller in joint space for a 3-DOF serial robotic manipulator. The proposed controller has been designed based on the fractional calculus concept to fulfill trajectory tracking with high accuracy and also reduce effects from disturbances and uncertainties. The parameters of the controller have been optimized with a GWO–PSO algorithm, which is a hybrid tuning method, by considering the time integral performance criterion. The superior and contribution of the GWO–PSO-based FOPTID controller has been demonstrated by comparing the results with those offered by PID, FOPID and PTID control strategies tuned by the GWO–PSO. The examination of the results showed that the proposed controller, which is based on the GWO–PSO algorithm, demonstrates better trajectory tracking performance and increased robustness against various trajectories, external disturbances, and joint frictions as compared to other controllers under the same operating conditions. In terms of the trajectory tracking performance for robustness, the superiority of the proposed controllers tuned by GWO–PSO has been confirmed by 20.2% to 44.5% reductions in the joint tracking errors. Moreover, for assessing the energy consumption of the tuned controllers, the total energy consumption of the proposed controller for all joints has been remarkably reduced by 2.45% as compared to others. Consequently, the results illustrated that the proposed controller is robust and stable and sustains against the continuous disturbance. Full article

(This article belongs to the Special Issue Applications of Fractional-Order Calculus in Robotics)

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29 pages, 472 KiB

Open AccessArticle

Cauchy–Dirichlet Problem to Semilinear Multi-Term Fractional Differential Equations

by Nataliya Vasylyeva

Fractal Fract. 2023, 7(3), 249; https://doi.org/10.3390/fractalfract7030249 - 09 Mar 2023

Cited by 2 | Viewed by 2645

Abstract

In this paper, we analyze the well-posedness of the Cauchy–Dirichlet problem to an integro-differential equation on a multidimensional domain

Ω \subset R^{n}

in the unknown

u = u (x, t)

, [...] Read more.

In this paper, we analyze the well-posedness of the Cauchy–Dirichlet problem to an integro-differential equation on a multidimensional domain

Ω \subset R^{n}

in the unknown

u = u (x, t)

,

D_{t}^{ν_{0}} (ϱ_{0} u) - D_{t}^{ν_{1}} (ϱ_{1} u) - L_{1} u - \int_{0}^{t} K (t - s) L_{2} u (x, s) d s = f (x, t) + g (u), 0 < ν_{1} < ν_{0} < 1,

where

D_{t}^{ν_{i}}

are the Caputo fractional derivatives,

ϱ_{i} = ϱ_{i} (x, t)

with

ϱ_{0} \geq μ_{0} > 0

, and

L_{i}

are uniform elliptic operators with time-dependent smooth coefficients. The principal feature of this equation is related to the integro-differential operator

D_{t}^{ν_{0}} (ϱ_{0} u) - D_{t}^{ν_{1}} (ϱ_{1} u)

, which (under certain assumption on the coefficients) can be rewritten in the form of a generalized fractional derivative with a non-positive kernel. A particular case of this equation describes oxygen delivery through capillaries to tissue. First, under proper requirements on the given data in the linear model and certain relations between

ν_{0}

and

ν_{1}

, we derive a priori estimates of a solution in Sobolev–Slobodeckii spaces that gives rise to providing the Hölder regularity of the solution. Exploiting these estimates and constructing appropriate approximate solutions, we prove the global strong solvability to the corresponding linear initial-boundary value problem. Finally, obtaining a priori estimates in the fractional Hölder classes and assuming additional conditions on the coefficients

ϱ_{0}

and

ϱ_{1}

and the nonlinearity

g (u)

, the global one-valued classical solvability to the nonlinear model is claimed with the continuation argument method. Full article

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

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Fractal Fract., Volume 7, Issue 3 (March 2023) – 73 articles

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