Pub Date : 2021-01-12DOI: 10.5772/INTECHOPEN.95573
N. Gupta, N. Kanth
Many physical and engineering problems can be modeled using partial differential equations such as heat transfer through conduction process in steady and unsteady state. Perturbation methods are analytical approximation method to understand physical phenomena which depends on perturbation quantity. Homotopy perturbation method (HPM) was proposed by Ji Huan He. HPM is considered as effective method in solving partial differential equations. The solution obtained by HPM converges to exact solution, which are in the form of an infinite function series. Biazar and Eslami proposed new homotopy perturbation method (NHPM) in which construction of an appropriate homotopy equation and selection of appropriate initial approximation guess are two important steps. In present work, heat flow analysis has been done on a rod of length L and diffusivity α using HPM and NHPM. The solution obtained using different perturbation methods are compared with the solution obtained from most common analytical method separation of variables.
{"title":"Application of Perturbation Theory in Heat Flow Analysis","authors":"N. Gupta, N. Kanth","doi":"10.5772/INTECHOPEN.95573","DOIUrl":"https://doi.org/10.5772/INTECHOPEN.95573","url":null,"abstract":"Many physical and engineering problems can be modeled using partial differential equations such as heat transfer through conduction process in steady and unsteady state. Perturbation methods are analytical approximation method to understand physical phenomena which depends on perturbation quantity. Homotopy perturbation method (HPM) was proposed by Ji Huan He. HPM is considered as effective method in solving partial differential equations. The solution obtained by HPM converges to exact solution, which are in the form of an infinite function series. Biazar and Eslami proposed new homotopy perturbation method (NHPM) in which construction of an appropriate homotopy equation and selection of appropriate initial approximation guess are two important steps. In present work, heat flow analysis has been done on a rod of length L and diffusivity α using HPM and NHPM. The solution obtained using different perturbation methods are compared with the solution obtained from most common analytical method separation of variables.","PeriodicalId":381248,"journal":{"name":"A Collection of Papers on Chaos Theory and Its Applications","volume":"58 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123931075","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-04DOI: 10.5772/intechopen.94852
V. Dzyuba, R. Romashko
An analytical method based on the Green’s function for describing the electromagnetic field, scalar-vector and phase characteristics of the acoustic field in a stationary isotropic and arbitrarily inhomogeneous medium is proposed. The method uses, in the case of an electromagnetic field, the wave equation proposed by the author for the electric vector of the electromagnetic field, which is valid for dielectric and magnetic inhomogeneous media with conductivity. In the case of an acoustic field, the author uses the wave equation proposed by the author for the particle velocity vector and the well-known equation for acoustic pressure in an inhomogeneous stationary medium. The approach used allows one to reduce the problem of solving differential wave equations in an arbitrarily inhomogeneous medium to the problem of taking an integral.
{"title":"Green’s Function Method for Electromagnetic and Acoustic Fields in Arbitrarily Inhomogeneous Media","authors":"V. Dzyuba, R. Romashko","doi":"10.5772/intechopen.94852","DOIUrl":"https://doi.org/10.5772/intechopen.94852","url":null,"abstract":"An analytical method based on the Green’s function for describing the electromagnetic field, scalar-vector and phase characteristics of the acoustic field in a stationary isotropic and arbitrarily inhomogeneous medium is proposed. The method uses, in the case of an electromagnetic field, the wave equation proposed by the author for the electric vector of the electromagnetic field, which is valid for dielectric and magnetic inhomogeneous media with conductivity. In the case of an acoustic field, the author uses the wave equation proposed by the author for the particle velocity vector and the well-known equation for acoustic pressure in an inhomogeneous stationary medium. The approach used allows one to reduce the problem of solving differential wave equations in an arbitrarily inhomogeneous medium to the problem of taking an integral.","PeriodicalId":381248,"journal":{"name":"A Collection of Papers on Chaos Theory and Its Applications","volume":"275 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122750651","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-12-02DOI: 10.5772/intechopen.94295
L. M. Saha
Chaotic phenomena and presence of complexity in various nonlinear dynamical systems extensively discussed in the context of recent researches. Discrete as well as continuous dynamical systems both considered here. Visualization of regularity and chaotic motion presented through bifurcation diagrams by varying a parameter of the system while keeping other parameters constant. In the processes, some perfect indicator of regularity and chaos discussed with appropriate examples. Measure of chaos in terms of Lyapunov exponents and that of complexity as increase in topological entropies discussed. The methodology to calculate these explained in details with exciting examples. Regular and chaotic attractors emerging during the study are drawn and analyzed. Correlation dimension, which provides the dimensionality of a chaotic attractor discussed in detail and calculated for different systems. Results obtained presented through graphics and in tabular form. Two techniques of chaos control, pulsive feedback control and asymptotic stability analysis, discussed and applied to control chaotic motion for certain cases. Finally, a brief discussion held for the concluded investigation.
{"title":"Chaos and Complexity Dynamics of Evolutionary Systems","authors":"L. M. Saha","doi":"10.5772/intechopen.94295","DOIUrl":"https://doi.org/10.5772/intechopen.94295","url":null,"abstract":"Chaotic phenomena and presence of complexity in various nonlinear dynamical systems extensively discussed in the context of recent researches. Discrete as well as continuous dynamical systems both considered here. Visualization of regularity and chaotic motion presented through bifurcation diagrams by varying a parameter of the system while keeping other parameters constant. In the processes, some perfect indicator of regularity and chaos discussed with appropriate examples. Measure of chaos in terms of Lyapunov exponents and that of complexity as increase in topological entropies discussed. The methodology to calculate these explained in details with exciting examples. Regular and chaotic attractors emerging during the study are drawn and analyzed. Correlation dimension, which provides the dimensionality of a chaotic attractor discussed in detail and calculated for different systems. Results obtained presented through graphics and in tabular form. Two techniques of chaos control, pulsive feedback control and asymptotic stability analysis, discussed and applied to control chaotic motion for certain cases. Finally, a brief discussion held for the concluded investigation.","PeriodicalId":381248,"journal":{"name":"A Collection of Papers on Chaos Theory and Its Applications","volume":"381 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124751972","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-11-17DOI: 10.5772/intechopen.94491
P. Bracken
The concept of integrability of a quantum system is developed and studied. By formulating the concepts of quantum degree of freedom and quantum phase space, a realization of the dynamics is achieved. For a quantum system with a dynamical group G in one of its unitary irreducible representative carrier spaces, the quantum phase space is a finite topological space. It is isomorphic to a coset space G/R by means of the unitary exponential mapping, where R is the maximal stability subgroup of a fixed state in the carrier space. This approach has the distinct advantage of exhibiting consistency between classical and quantum integrability. The formalism will be illustrated by studying several quantum systems in detail after this development.
{"title":"Classical and Quantum Integrability: A Formulation That Admits Quantum Chaos","authors":"P. Bracken","doi":"10.5772/intechopen.94491","DOIUrl":"https://doi.org/10.5772/intechopen.94491","url":null,"abstract":"The concept of integrability of a quantum system is developed and studied. By formulating the concepts of quantum degree of freedom and quantum phase space, a realization of the dynamics is achieved. For a quantum system with a dynamical group G in one of its unitary irreducible representative carrier spaces, the quantum phase space is a finite topological space. It is isomorphic to a coset space G/R by means of the unitary exponential mapping, where R is the maximal stability subgroup of a fixed state in the carrier space. This approach has the distinct advantage of exhibiting consistency between classical and quantum integrability. The formalism will be illustrated by studying several quantum systems in detail after this development.","PeriodicalId":381248,"journal":{"name":"A Collection of Papers on Chaos Theory and Its Applications","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121353550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper describes how chaos theory was implemented to explain a behavioral aspect in an information system. The chaos theory was developed from the physical sciences and has been widely applied to many fields. However, this theory may also be applied to the social sciences. For certain types of human behavior, the chaos theory could comprehensively explain the phenomena of the use of information and communications technology (ICT). It means that this theory could clarify all the different kinds of human interactions with ICT. When the researchers used the chaos theory integratively, they could explain the distressed behavior of ICT users comprehensively. This theory argues that an individual acts randomly, even though the system is deterministic. When individuals use ICT, they could get technostress due to either the information systems or other users. This paper explains that ICT users could use information systems, with their complicated procedures and outputs. They were also probably disturbed by other users. The users, furthermore, experience chaotic pressures through their experiential values. This paper shows that users’ behavior when facing chaotic pressure depends upon their personality dimensions. The authors finally propose a new paradigm that this chaos theory could explain the chaotic actions of ICT users.
{"title":"The Chaotic Behavior of ICT Users","authors":"Sumiyana Sumiyana, Sriwidharmanely Sriwidharmanely","doi":"10.5772/intechopen.94443","DOIUrl":"https://doi.org/10.5772/intechopen.94443","url":null,"abstract":"This paper describes how chaos theory was implemented to explain a behavioral aspect in an information system. The chaos theory was developed from the physical sciences and has been widely applied to many fields. However, this theory may also be applied to the social sciences. For certain types of human behavior, the chaos theory could comprehensively explain the phenomena of the use of information and communications technology (ICT). It means that this theory could clarify all the different kinds of human interactions with ICT. When the researchers used the chaos theory integratively, they could explain the distressed behavior of ICT users comprehensively. This theory argues that an individual acts randomly, even though the system is deterministic. When individuals use ICT, they could get technostress due to either the information systems or other users. This paper explains that ICT users could use information systems, with their complicated procedures and outputs. They were also probably disturbed by other users. The users, furthermore, experience chaotic pressures through their experiential values. This paper shows that users’ behavior when facing chaotic pressure depends upon their personality dimensions. The authors finally propose a new paradigm that this chaos theory could explain the chaotic actions of ICT users.","PeriodicalId":381248,"journal":{"name":"A Collection of Papers on Chaos Theory and Its Applications","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133981920","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-10-29DOI: 10.5772/intechopen.94173
J. Mohapatra
The main purpose of this chapter is to describe the application of perturbation expansion techniques to the solution of differential equations. Approximate expressions are generated in the form of asymptotic series. These may not and often do not converge but in a truncated form of only two or three terms, provide a useful approximation to the original problem. These analytical techniques provide an alternative to the direct computer solution. Before attempting to solve these problems numerically, one should have an awareness of the perturbation approach.
{"title":"Perturbation Expansion to the Solution of Differential Equations","authors":"J. Mohapatra","doi":"10.5772/intechopen.94173","DOIUrl":"https://doi.org/10.5772/intechopen.94173","url":null,"abstract":"The main purpose of this chapter is to describe the application of perturbation expansion techniques to the solution of differential equations. Approximate expressions are generated in the form of asymptotic series. These may not and often do not converge but in a truncated form of only two or three terms, provide a useful approximation to the original problem. These analytical techniques provide an alternative to the direct computer solution. Before attempting to solve these problems numerically, one should have an awareness of the perturbation approach.","PeriodicalId":381248,"journal":{"name":"A Collection of Papers on Chaos Theory and Its Applications","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122510457","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-10-12DOI: 10.5772/intechopen.93736
V. Gaganis
Equations of State (EoS) live at the heart of all thermodynamic calculations in chemical engineering applications as they allow for the determination of all related fluid properties such as vapor pressure, density, enthalpy, specific heat, and speed of sound, in an accurate and consistent way. Both macroscopic EoS models such as the classic cubic EoS models as well as models based on statistical mechanics and developed by means of perturbation theory are available. Under suitable pressure and temperature conditions, fluids of known composition may split in more than one phases, usually vapor and liquid while solids may also be present, each one exhibiting its own composition. Therefore, computational methods are utilized to calculate the number and the composition of the equilibrium phases at which a feed composition will potentially split so as to estimate their thermodynamic properties by means of the EoS. This chapter focuses on two of the most pronounced EoS models, the cubic ones and those based on statistical mechanics incorporating perturbation analysis. Subsequently, it describes the existing algorithms to solve phase behavior problems that rely on the classic rigorous thermodynamics context as well as modern trends that aim at accelerating computations.
{"title":"Perturbation Theory and Phase Behavior Calculations Using Equation of State Models","authors":"V. Gaganis","doi":"10.5772/intechopen.93736","DOIUrl":"https://doi.org/10.5772/intechopen.93736","url":null,"abstract":"Equations of State (EoS) live at the heart of all thermodynamic calculations in chemical engineering applications as they allow for the determination of all related fluid properties such as vapor pressure, density, enthalpy, specific heat, and speed of sound, in an accurate and consistent way. Both macroscopic EoS models such as the classic cubic EoS models as well as models based on statistical mechanics and developed by means of perturbation theory are available. Under suitable pressure and temperature conditions, fluids of known composition may split in more than one phases, usually vapor and liquid while solids may also be present, each one exhibiting its own composition. Therefore, computational methods are utilized to calculate the number and the composition of the equilibrium phases at which a feed composition will potentially split so as to estimate their thermodynamic properties by means of the EoS. This chapter focuses on two of the most pronounced EoS models, the cubic ones and those based on statistical mechanics incorporating perturbation analysis. Subsequently, it describes the existing algorithms to solve phase behavior problems that rely on the classic rigorous thermodynamics context as well as modern trends that aim at accelerating computations.","PeriodicalId":381248,"journal":{"name":"A Collection of Papers on Chaos Theory and Its Applications","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130707320","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-12-27DOI: 10.5772/intechopen.93864
A. Gecow
The research concerns the dynamics of complex autonomous Kauffman networks. The article defines and shows using simulation experiments half-chaotic networks, which exhibit features much more similar to typically modeled systems like a living, technological or social than fully random Kauffman networks. This represents a large change in the widely held view taken of the dynamics of complex systems. Current theory predicts that random autonomous systems can be either ordered or chaotic with fast phase transition between them. The theory uses shift of finite, discrete networks to infinite and continuous space. This move loses important features like e.g. attractor length, making description too simplified. Modeled adapted systems are not fully random, they are usually stable, but the estimated parameters are usually “chaotic”, they place the fully random networks in the chaotic regime, far from the narrow phase transition. I show that among the not fully random systems with “chaotic parameters”, a large third state called half-chaos exists. Half-chaotic system simultaneously exhibits small (ordered) and large (chaotic) reactions for small disturbances in similar share. The discovery of half-chaos frees modeling of adapted systems from sharp restrictions; it allows to use “chaotic parameters” and get a nearly stable system more similar to modeled one. It gives a base for identity criterion of an evolving object, simplifies the definition of basic Darwinian mechanism and changes “life on the edge of chaos” to “life evolves in the half-chaos of not fully random systems”.
{"title":"Life Is Not on the Edge of Chaos but in a Half-Chaos of Not Fully Random Systems. Definition and Simulations of the Half-Chaos in Complex Networks","authors":"A. Gecow","doi":"10.5772/intechopen.93864","DOIUrl":"https://doi.org/10.5772/intechopen.93864","url":null,"abstract":"The research concerns the dynamics of complex autonomous Kauffman networks. The article defines and shows using simulation experiments half-chaotic networks, which exhibit features much more similar to typically modeled systems like a living, technological or social than fully random Kauffman networks. This represents a large change in the widely held view taken of the dynamics of complex systems. Current theory predicts that random autonomous systems can be either ordered or chaotic with fast phase transition between them. The theory uses shift of finite, discrete networks to infinite and continuous space. This move loses important features like e.g. attractor length, making description too simplified. Modeled adapted systems are not fully random, they are usually stable, but the estimated parameters are usually “chaotic”, they place the fully random networks in the chaotic regime, far from the narrow phase transition. I show that among the not fully random systems with “chaotic parameters”, a large third state called half-chaos exists. Half-chaotic system simultaneously exhibits small (ordered) and large (chaotic) reactions for small disturbances in similar share. The discovery of half-chaos frees modeling of adapted systems from sharp restrictions; it allows to use “chaotic parameters” and get a nearly stable system more similar to modeled one. It gives a base for identity criterion of an evolving object, simplifies the definition of basic Darwinian mechanism and changes “life on the edge of chaos” to “life evolves in the half-chaos of not fully random systems”.","PeriodicalId":381248,"journal":{"name":"A Collection of Papers on Chaos Theory and Its Applications","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127063809","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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