The Gaussian function was initially adopted to model the tunneling-induced ground surface settlement, but it could not describe the porosity’s effect on the settlement. To solve the problem, Yu’s fractal theory is implemented to model the porous grand’s geometric property, and the surface settlement profile is modeled by a fractal solitary wave, furthermore, the effects of the maximal surface settlement, the porosity and Yu’s fractal dimension on the settlement’s profile are discussed. The new model offers a new view to predict the morphology of the surface settlement.
{"title":"A fractal model for the tunneling-induced ground surface settlement","authors":"Yuan Mei, Xinyu Tian, Xuejuan Li, Chun-Hui He, Abdulrahman Ali Alsolami","doi":"10.1142/s0218348x23501141","DOIUrl":"https://doi.org/10.1142/s0218348x23501141","url":null,"abstract":"The Gaussian function was initially adopted to model the tunneling-induced ground surface settlement, but it could not describe the porosity’s effect on the settlement. To solve the problem, Yu’s fractal theory is implemented to model the porous grand’s geometric property, and the surface settlement profile is modeled by a fractal solitary wave, furthermore, the effects of the maximal surface settlement, the porosity and Yu’s fractal dimension on the settlement’s profile are discussed. The new model offers a new view to predict the morphology of the surface settlement.","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135463673","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-21DOI: 10.1142/s0218348x23501232
QINGYONG ZHU, JINQUAN HUANG, XIAO XIAO
The C/SiC composite is a promising material for ablation-resistant thermal protection in near-space hypersonic environments. The formation of an SiO 2 oxide layer through passive oxidation on the surface of the composite is a significant factor influencing its performance. It is essential to accurately predict the thickness of the SiO 2 oxide layer and the recession and mass loss of the C/SiC composite during passive oxidation. The SiO 2 oxide layer is a typical porous media exhibiting self-similarity and thus fractal theory can be applied to establish the relation between the oxygen flow rate and microstructural parameters of the oxide layer. The Weierstrass–Mandelbrot (WM) function is employed to simulate the rough interfaces between the SiO 2 oxide layer and the C/SiC composite to evaluate the influence of the fractal dimensions of the oxide layer on the performance of thermal protection of the C/SiC composite. The results show that the C/SiC composite exhibits improved thermal protection performance when accompanied by a lower tortuosity fractal dimension and a higher pore area fractal dimension of the oxide layer. Conversely, the composite demonstrates enhanced ablation resistance with a higher tortuosity fractal dimension and a lower pore area fractal dimension of the oxide layer. The predictions of the calculation model show good agreement with the experimental data and demonstrate the critical influence of microstructural parameters of the oxide layer on passive oxidation of the composite, providing practical implications for designing materials with desired thermal protection or ablation resistance properties.
{"title":"THE INFLUENCE OF FRACTAL DIMENSION OF OXIDE LAYER ON PASSIVE OXIDATION OF THE C/SiC COMPOSITE","authors":"QINGYONG ZHU, JINQUAN HUANG, XIAO XIAO","doi":"10.1142/s0218348x23501232","DOIUrl":"https://doi.org/10.1142/s0218348x23501232","url":null,"abstract":"The C/SiC composite is a promising material for ablation-resistant thermal protection in near-space hypersonic environments. The formation of an SiO 2 oxide layer through passive oxidation on the surface of the composite is a significant factor influencing its performance. It is essential to accurately predict the thickness of the SiO 2 oxide layer and the recession and mass loss of the C/SiC composite during passive oxidation. The SiO 2 oxide layer is a typical porous media exhibiting self-similarity and thus fractal theory can be applied to establish the relation between the oxygen flow rate and microstructural parameters of the oxide layer. The Weierstrass–Mandelbrot (WM) function is employed to simulate the rough interfaces between the SiO 2 oxide layer and the C/SiC composite to evaluate the influence of the fractal dimensions of the oxide layer on the performance of thermal protection of the C/SiC composite. The results show that the C/SiC composite exhibits improved thermal protection performance when accompanied by a lower tortuosity fractal dimension and a higher pore area fractal dimension of the oxide layer. Conversely, the composite demonstrates enhanced ablation resistance with a higher tortuosity fractal dimension and a lower pore area fractal dimension of the oxide layer. The predictions of the calculation model show good agreement with the experimental data and demonstrate the critical influence of microstructural parameters of the oxide layer on passive oxidation of the composite, providing practical implications for designing materials with desired thermal protection or ablation resistance properties.","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135513252","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-20DOI: 10.1142/s0218348x23401941
Rabha W. Ibrahim, Suzan J. Obaiys, Yeliz Karaca, Aydin Secer
Non-local operators of differentiation are bestowed with capabilities of encompassing complex natural into mathematical equations. Symmetry as invariance under a specified group of transformations can allow for the concept to be applied extensively not only to spatial figures but also to abstract objects like mathematical expressions which can be said to be expressions of physical relevance, in particular dynamical equations. Derived from this point of view, it can be noted that the more complex physical problems are, the more complex mathematical operators of differentiation are required. Accordingly, the fractal–fractional operators (FFOs) are expanded into the complex plane in our research which revolves around a unique class of normalized analytic functions in the open unit disk. To bring FFOs (differential and integral) into the normalized class, the study aims to expand and modify them along with the investigation of the FFOs geometrically. The qualities of convexity and starlikeness are implicated in this study where the differential subordination technique serves as the foundation for the inquiry under consideration. Furthermore, a collection of differential FFO inequalities is taken into account, demonstrating that the normalized Fox–Wright function can contain all FFOs. Besides these steps, the concept of Grunsky factors is applied to investigate symmetry, while boundary value issues involving FFOs are probed. Consequently, the related properties and applications can be further developed, which requires the devotion to differential fractional problems and diverse complex problems in relation to viable applications, pointing out the room to modify and upgrade the existing methods for more optimal outcomes in challenging real-world problems.
{"title":"Complex Mathematical Modeling for Advanced Fractal-Fractional Differential Operators within Symmetry","authors":"Rabha W. Ibrahim, Suzan J. Obaiys, Yeliz Karaca, Aydin Secer","doi":"10.1142/s0218348x23401941","DOIUrl":"https://doi.org/10.1142/s0218348x23401941","url":null,"abstract":"Non-local operators of differentiation are bestowed with capabilities of encompassing complex natural into mathematical equations. Symmetry as invariance under a specified group of transformations can allow for the concept to be applied extensively not only to spatial figures but also to abstract objects like mathematical expressions which can be said to be expressions of physical relevance, in particular dynamical equations. Derived from this point of view, it can be noted that the more complex physical problems are, the more complex mathematical operators of differentiation are required. Accordingly, the fractal–fractional operators (FFOs) are expanded into the complex plane in our research which revolves around a unique class of normalized analytic functions in the open unit disk. To bring FFOs (differential and integral) into the normalized class, the study aims to expand and modify them along with the investigation of the FFOs geometrically. The qualities of convexity and starlikeness are implicated in this study where the differential subordination technique serves as the foundation for the inquiry under consideration. Furthermore, a collection of differential FFO inequalities is taken into account, demonstrating that the normalized Fox–Wright function can contain all FFOs. Besides these steps, the concept of Grunsky factors is applied to investigate symmetry, while boundary value issues involving FFOs are probed. Consequently, the related properties and applications can be further developed, which requires the devotion to differential fractional problems and diverse complex problems in relation to viable applications, pointing out the room to modify and upgrade the existing methods for more optimal outcomes in challenging real-world problems.","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135513517","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-20DOI: 10.1142/s0218348x2340203x
M. M. Abelazeem, Raghda A. M. Attia
This study focuses on employing recent and accurate computational techniques, specifically the Sardar-sub equation [Formula: see text] method, to explore novel solitary wave solutions of the Gilson–Pickering [Formula: see text] equation. The GP equation is a mathematical model with implications in fluid dynamics and wave phenomena. It describes the behavior of solitary waves, which are localized disturbances propagating through a medium without changing shape. The physical significance of the [Formula: see text] equation lies in its ability to capture the dynamics of solitary waves in various systems, including water waves, optical fibers, and nonlinear acoustic waves. The study’s findings contribute to the advancement of mathematical modeling approaches and offer valuable insights into solitary wave phenomena. The stability of the constructed solutions is investigated using the properties of the Hamiltonian system. The accuracy of the computational solutions is demonstrated by comparing them with approximate solutions obtained through He’s variational iteration [Formula: see text] method. Furthermore, the effectiveness of the employed computational techniques is validated through comparisons with other existing methods.
{"title":"Investigation of New Solitary Wave Solutions of the Gilson-Pickering Equation Using Advanced Computational Techniques","authors":"M. M. Abelazeem, Raghda A. M. Attia","doi":"10.1142/s0218348x2340203x","DOIUrl":"https://doi.org/10.1142/s0218348x2340203x","url":null,"abstract":"This study focuses on employing recent and accurate computational techniques, specifically the Sardar-sub equation [Formula: see text] method, to explore novel solitary wave solutions of the Gilson–Pickering [Formula: see text] equation. The GP equation is a mathematical model with implications in fluid dynamics and wave phenomena. It describes the behavior of solitary waves, which are localized disturbances propagating through a medium without changing shape. The physical significance of the [Formula: see text] equation lies in its ability to capture the dynamics of solitary waves in various systems, including water waves, optical fibers, and nonlinear acoustic waves. The study’s findings contribute to the advancement of mathematical modeling approaches and offer valuable insights into solitary wave phenomena. The stability of the constructed solutions is investigated using the properties of the Hamiltonian system. The accuracy of the computational solutions is demonstrated by comparing them with approximate solutions obtained through He’s variational iteration [Formula: see text] method. Furthermore, the effectiveness of the employed computational techniques is validated through comparisons with other existing methods.","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135513515","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-20DOI: 10.1142/s0218348x23501098
Wenbing Sun, Haiyang Wan
In this paper, we use the properties of Atangana–Baleanu (AB) fractional calculus and Prabhakar fractional calculus to construct some novel Hermite–Hadamard-type fractional integral inequalities for harmonically convex functions. And these inequalities are represented by the Mittag-Leffler functions. Finally, several special inequalities are established to illustrate the applications of our conclusions in special means.
{"title":"Hermite-Hadamard type inequalities involving several kinds of fractional calculus for harmonically convex functions","authors":"Wenbing Sun, Haiyang Wan","doi":"10.1142/s0218348x23501098","DOIUrl":"https://doi.org/10.1142/s0218348x23501098","url":null,"abstract":"In this paper, we use the properties of Atangana–Baleanu (AB) fractional calculus and Prabhakar fractional calculus to construct some novel Hermite–Hadamard-type fractional integral inequalities for harmonically convex functions. And these inequalities are represented by the Mittag-Leffler functions. Finally, several special inequalities are established to illustrate the applications of our conclusions in special means.","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135513645","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-20DOI: 10.1142/s0218348x23401916
Ahmed Alsaedi, Hana Al-Hutami, Bashir Ahmad
In this paper, we introduce and investigate a new class of nonlinear multi-term impulsive anti-periodic boundary value problems involving Caputo type fractional [Formula: see text]-derivative operators of different orders and the Riemann–Liouville fractional [Formula: see text]-integral operator. The uniqueness of solutions to the given problem is proved with the aid of Banach’s fixed point theorem. Applying a Shaefer-like fixed point theorem, we also obtain an existence result for the problem at hand. Examples are constructed for illustrating the obtained results. The paper concludes with certain interesting observations concerning the reduction of the results proven in the paper to some new results under an appropriate choice of the parameters involved in the governing equation.
{"title":"Investigation of a nonlinear multi-term impulsive anti-periodic boundary value problem of fractional <i>q</i>-integro-difference equations","authors":"Ahmed Alsaedi, Hana Al-Hutami, Bashir Ahmad","doi":"10.1142/s0218348x23401916","DOIUrl":"https://doi.org/10.1142/s0218348x23401916","url":null,"abstract":"In this paper, we introduce and investigate a new class of nonlinear multi-term impulsive anti-periodic boundary value problems involving Caputo type fractional [Formula: see text]-derivative operators of different orders and the Riemann–Liouville fractional [Formula: see text]-integral operator. The uniqueness of solutions to the given problem is proved with the aid of Banach’s fixed point theorem. Applying a Shaefer-like fixed point theorem, we also obtain an existence result for the problem at hand. Examples are constructed for illustrating the obtained results. The paper concludes with certain interesting observations concerning the reduction of the results proven in the paper to some new results under an appropriate choice of the parameters involved in the governing equation.","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135513518","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-20DOI: 10.1142/s0218348x23401898
Mohammed Al-Smadi, Shrideh Al-Omari, Sharifah Alhazmi, Yeliz Karaca, Shaher Momani
This paper investigates the dynamics of exact traveling-wave solutions for nonlinear spatial and temporal fractional partial differential equations with conformable order derivatives arising in nonlinear propagation waves of small amplitude including nonlinear fractional modified Benjamin–Bona–Mahony equation, fractional Zakharov–Kuznetsov–Benjamin–Bona–Mahony equation and fractional (2+1)-dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation as well. By utilizing the Sine-Gordon expansion method (SGEM), new real- and complex-valued exact traveling-wave solutions are reported by preferring suitable values of physical free parameters. The nonlinear governing equations are reduced into auxiliary nonlinear ordinary differential equations with aid of fractional traveling-wave transformation, in which the fractional derivative is evaluated in a conformable sense. The productivity process of the proposed method for predicting the desirable solutions is also provided. Some of the obtained solutions are simulated graphically in 3D and contour plots. Meanwhile, the effects of the fractional parameter [Formula: see text] in the space and the time direction are illustrated in 2D plots to ensure the novelty, applicability and credibility of the SGEM. These results reveal that the suggested method is general and adequate for dealing with nonlinear models featuring fractional derivatives and can be employed to analyze wide classes of complex phenomena of partial differential equations occurring in engineering and nonlinear dynamics.
{"title":"Novel Travelling-Wave Solutions of Spatial-Temporal Fractional Model of Dynamical Benjamin-Bona-Mahony System","authors":"Mohammed Al-Smadi, Shrideh Al-Omari, Sharifah Alhazmi, Yeliz Karaca, Shaher Momani","doi":"10.1142/s0218348x23401898","DOIUrl":"https://doi.org/10.1142/s0218348x23401898","url":null,"abstract":"This paper investigates the dynamics of exact traveling-wave solutions for nonlinear spatial and temporal fractional partial differential equations with conformable order derivatives arising in nonlinear propagation waves of small amplitude including nonlinear fractional modified Benjamin–Bona–Mahony equation, fractional Zakharov–Kuznetsov–Benjamin–Bona–Mahony equation and fractional (2+1)-dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation as well. By utilizing the Sine-Gordon expansion method (SGEM), new real- and complex-valued exact traveling-wave solutions are reported by preferring suitable values of physical free parameters. The nonlinear governing equations are reduced into auxiliary nonlinear ordinary differential equations with aid of fractional traveling-wave transformation, in which the fractional derivative is evaluated in a conformable sense. The productivity process of the proposed method for predicting the desirable solutions is also provided. Some of the obtained solutions are simulated graphically in 3D and contour plots. Meanwhile, the effects of the fractional parameter [Formula: see text] in the space and the time direction are illustrated in 2D plots to ensure the novelty, applicability and credibility of the SGEM. These results reveal that the suggested method is general and adequate for dealing with nonlinear models featuring fractional derivatives and can be employed to analyze wide classes of complex phenomena of partial differential equations occurring in engineering and nonlinear dynamics.","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135513513","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Nanofluids are used to achieve maximum thermal performance with the smallest concentration of nanoparticles and stable suspension in conventional fluids. The effectiveness of nanofluids in convection processes is significantly influenced by their increased thermophysical characteristics. Based on the characteristics of nanofluids, this study examines generalized Brinkman-type nanofluid flow in a vertical channel. Three different types of ultrafine solid nanoparticles such as GO, [Formula: see text], and [Formula: see text] are dispersed uniformly in regular water to form nanofluid. Partial differential equations (PDEs) are used to model the phenomena. Fick’s and Fourier’s laws of fractional order were then used to formulate the generalized mathematical model. The exact solution of the generalized mathematical model has been obtained by the joint use of Fourier sine and the Laplace transform (LT) techniques. The obtained solution is represented in Mittag-Leffler function. To analyze the behavior of fluid flow, heat and mass distribution in fluid, the obtained solution was computed numerically and then plotted in response to different physical parameters. It is worth noting from the analysis that the heat transfer efficiency of regular water has been improved by 25% by using GO nanoparticles, 23.98% by using [Formula: see text], and 20.88% by using [Formula: see text].
{"title":"Fractional Model of Brinkman-Type Nanofluid Flow with Fractional Order Fourier's and Fick's Laws","authors":"Saqib Murtaza, Poom Kumam, Zubair Ahmad, Kanokwan Sitthithakerngkiet, Thana Sutthibutpong","doi":"10.1142/s0218348x23401990","DOIUrl":"https://doi.org/10.1142/s0218348x23401990","url":null,"abstract":"Nanofluids are used to achieve maximum thermal performance with the smallest concentration of nanoparticles and stable suspension in conventional fluids. The effectiveness of nanofluids in convection processes is significantly influenced by their increased thermophysical characteristics. Based on the characteristics of nanofluids, this study examines generalized Brinkman-type nanofluid flow in a vertical channel. Three different types of ultrafine solid nanoparticles such as GO, [Formula: see text], and [Formula: see text] are dispersed uniformly in regular water to form nanofluid. Partial differential equations (PDEs) are used to model the phenomena. Fick’s and Fourier’s laws of fractional order were then used to formulate the generalized mathematical model. The exact solution of the generalized mathematical model has been obtained by the joint use of Fourier sine and the Laplace transform (LT) techniques. The obtained solution is represented in Mittag-Leffler function. To analyze the behavior of fluid flow, heat and mass distribution in fluid, the obtained solution was computed numerically and then plotted in response to different physical parameters. It is worth noting from the analysis that the heat transfer efficiency of regular water has been improved by 25% by using GO nanoparticles, 23.98% by using [Formula: see text], and 20.88% by using [Formula: see text].","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135513514","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-20DOI: 10.1142/s0218348x23402041
Yeliz Karaca
Fractional calculus and fractional-order calculus are arranged in lineage as regards the mathematical models with complexity-theoretical tenets capable of capturing the subtle molecular dynamics by the integration of power-law convolution kernels into time- and space-related derivatives emerging in equations concerning the Magnetic Resonance Imaging (MRI) phenomena to which the fractional models of diffusion and relaxation are applied. Endowed with an intricate level of complexity and a unique physical and structural scaffolding at molecular and cellular levels with numerous synapses forming elaborate neural networks which entail in-depth probing and computing of patterns and signatures in individual cells and neurons, human brain as a heterogeneous medium is constituted of tissues with cells of different sizes and shapes, distributed across an extra-cellular space. Characterization of the unique brain cells is sought after to unravel the connections between different cells and tissues for accurate, reliable, robust and optimal models and computing. Accordingly, Diffusion Magnetic Resonance Imaging (DMRI), as a noninvasive and experimental imaging technique with clinical and research applications, provides a measure related to the diffusion characteristics of water in biological tissues, particularly in the brain tissues. Compatible with these aspects and beyond the diffusion coefficients’ measurement, DMRI technique aims to exceed the spatial resolution of the MRI images and draw inferences from the microstructural properties of the related medium. Thus, novel tools become essential for the description of the biological (organelles, membranes, macromolecules and so on) and neurological (axons, dendrites, neurons and so forth) tissues’ complexity. Mathematical model-based computational analyses with multifaceted methods to extract information from the DMRI with SpinDoctor into neuronal dynamics can provide quantitative parametric instruments in order to reflect the tissue properties focusing on the precise link between the tissue microstructure and signals acquired by employing advanced medical imaging technologies. Coalesced with accurate neuron geometry models as well as numerical DMRI simulations, a novel extended and multifaceted predictive mathematical model based on SpinDoctor and Bloch–Torrey partial differential equation (BTPDE) with the Caputo fractional-order derivative (FOD) with three-parameter [Formula: see text] Mittag-Leffler function (MLF) has been proposed and developed in our study by extending for the application on Brain Neuron Spin Unit dataset with the relevant multi-stage application-related steps. The feedforward neural networks (FFNNs) with BFGS Quasi-Newton equation, as one of the artificial neural network (ANN) algorithms, are applied on BTPDE with Caputo fractional-order derivative for the neurons and their algorithmic complexity is computed by building a BTPDE with Caputo FOD Neuron model based on different fractional
{"title":"Fractional Calculus Operators - Bloch-Torrey Partial Differential Equation - Artificial Neural Networks-Computational Complexity Modeling of the Micro-Macrostructural Brain Tissues with Diffusion MRI Signal Processing and Neuronal Multicomponents","authors":"Yeliz Karaca","doi":"10.1142/s0218348x23402041","DOIUrl":"https://doi.org/10.1142/s0218348x23402041","url":null,"abstract":"Fractional calculus and fractional-order calculus are arranged in lineage as regards the mathematical models with complexity-theoretical tenets capable of capturing the subtle molecular dynamics by the integration of power-law convolution kernels into time- and space-related derivatives emerging in equations concerning the Magnetic Resonance Imaging (MRI) phenomena to which the fractional models of diffusion and relaxation are applied. Endowed with an intricate level of complexity and a unique physical and structural scaffolding at molecular and cellular levels with numerous synapses forming elaborate neural networks which entail in-depth probing and computing of patterns and signatures in individual cells and neurons, human brain as a heterogeneous medium is constituted of tissues with cells of different sizes and shapes, distributed across an extra-cellular space. Characterization of the unique brain cells is sought after to unravel the connections between different cells and tissues for accurate, reliable, robust and optimal models and computing. Accordingly, Diffusion Magnetic Resonance Imaging (DMRI), as a noninvasive and experimental imaging technique with clinical and research applications, provides a measure related to the diffusion characteristics of water in biological tissues, particularly in the brain tissues. Compatible with these aspects and beyond the diffusion coefficients’ measurement, DMRI technique aims to exceed the spatial resolution of the MRI images and draw inferences from the microstructural properties of the related medium. Thus, novel tools become essential for the description of the biological (organelles, membranes, macromolecules and so on) and neurological (axons, dendrites, neurons and so forth) tissues’ complexity. Mathematical model-based computational analyses with multifaceted methods to extract information from the DMRI with SpinDoctor into neuronal dynamics can provide quantitative parametric instruments in order to reflect the tissue properties focusing on the precise link between the tissue microstructure and signals acquired by employing advanced medical imaging technologies. Coalesced with accurate neuron geometry models as well as numerical DMRI simulations, a novel extended and multifaceted predictive mathematical model based on SpinDoctor and Bloch–Torrey partial differential equation (BTPDE) with the Caputo fractional-order derivative (FOD) with three-parameter [Formula: see text] Mittag-Leffler function (MLF) has been proposed and developed in our study by extending for the application on Brain Neuron Spin Unit dataset with the relevant multi-stage application-related steps. The feedforward neural networks (FFNNs) with BFGS Quasi-Newton equation, as one of the artificial neural network (ANN) algorithms, are applied on BTPDE with Caputo fractional-order derivative for the neurons and their algorithmic complexity is computed by building a BTPDE with Caputo FOD Neuron model based on different fractional","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135513516","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-19DOI: 10.1142/s0218348x23401886
Abdel-Haleem Abdel-Aty
In this paper, we study the interaction of Nitrogen Vacancies in Diamond (NVD) with quantized cavity field. The system is explored analytically and the effect of the system parameters is analyzed. The stability of a quantum system with influencing factors is investigated using the Mandal Parameter. The generated correlation between the NVD and the quantized cavity field is quantified using the negativity. This study also investigates geometric phase and its dependence on the system parameters. The results show that this system holds great potential applications in quantum computation and quantum memory. Additionally, the features of the system can be controlled by the system parameters.
{"title":"Engineering Geometric Phase and Correlation Dynamics of Nitrogen Vacancies in Diamond Interacting with two Nanocavities","authors":"Abdel-Haleem Abdel-Aty","doi":"10.1142/s0218348x23401886","DOIUrl":"https://doi.org/10.1142/s0218348x23401886","url":null,"abstract":"In this paper, we study the interaction of Nitrogen Vacancies in Diamond (NVD) with quantized cavity field. The system is explored analytically and the effect of the system parameters is analyzed. The stability of a quantum system with influencing factors is investigated using the Mandal Parameter. The generated correlation between the NVD and the quantized cavity field is quantified using the negativity. This study also investigates geometric phase and its dependence on the system parameters. The results show that this system holds great potential applications in quantum computation and quantum memory. Additionally, the features of the system can be controlled by the system parameters.","PeriodicalId":55144,"journal":{"name":"Fractals-Complex Geometry Patterns and Scaling in Nature and Society","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135667695","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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