The Experts below are selected from a list of 5184 Experts worldwide ranked by ideXlab platform

Anil D Gangal - One of the best experts on this subject based on the ideXlab platform.

  • Calculus on fractal subsets of real line ii conjugacy with Ordinary Calculus
    Fractals, 2011
    Co-Authors: Abhay Parvate, Anil D Gangal
    Abstract:

    Calculus on fractals, or Fα-Calculus, developed in a previous paper, is a Calculus based fractals F ⊂ R, and involves Fα-integral and Fα-derivative of orders α, 0 < α ≤ 1, where α is the dimension of F. The Fα-integral is suitable for integrating functions with fractal support of dimension α, while the Fα-derivative enables us to differentiate functions like the Cantor staircase. Several results in Fα-Calculus are analogous to corresponding results in Ordinary Calculus, such as the Leibniz rule, fundamental theorems, etc. The functions like the Cantor staircase function occur naturally as solutions of Fα-differential equations. Hence the latter can be used to model processes involving fractal space or time, which in particular include a class of dynamical systems exhibiting sublinear behaviour. In this paper we show that, as operators, the Fα-integral and Fα-derivative are conjugate to the Riemann integral and Ordinary derivative respectively. This is accomplished by constructing a map ψ which takes Fα-integrable functions to Riemann integrable functions, such that the corresponding integrals on appropriate intervals have equal values. Under suitable conditions, a restriction of ψ also takes Fα-differentiable functions to ordinarily differentiable functions such that their values at appropriate points are equal. Further, this conjugacy is generalized to one between Sobolev spaces in Ordinary Calculus and Fα-Calculus. This conjugacy is useful, among other things, to find solutions to Fα-differential equations: they can be mapped to Ordinary differential equations, and the solutions of the latter can be transformed back to get those of the former. This is illustrated with a few examples.

  • Calculus ON FRACTAL SUBSETS OF REAL LINE — II: CONJUGACY WITH Ordinary Calculus
    Fractals, 2011
    Co-Authors: Abhay Parvate, Anil D Gangal
    Abstract:

    Calculus on fractals, or Fα-Calculus, developed in a previous paper, is a Calculus based fractals F ⊂ R, and involves Fα-integral and Fα-derivative of orders α, 0 < α ≤ 1, where α is the dimension of F. The Fα-integral is suitable for integrating functions with fractal support of dimension α, while the Fα-derivative enables us to differentiate functions like the Cantor staircase. Several results in Fα-Calculus are analogous to corresponding results in Ordinary Calculus, such as the Leibniz rule, fundamental theorems, etc. The functions like the Cantor staircase function occur naturally as solutions of Fα-differential equations. Hence the latter can be used to model processes involving fractal space or time, which in particular include a class of dynamical systems exhibiting sublinear behaviour. In this paper we show that, as operators, the Fα-integral and Fα-derivative are conjugate to the Riemann integral and Ordinary derivative respectively. This is accomplished by constructing a map ψ which takes Fα-integrable functions to Riemann integrable functions, such that the corresponding integrals on appropriate intervals have equal values. Under suitable conditions, a restriction of ψ also takes Fα-differentiable functions to ordinarily differentiable functions such that their values at appropriate points are equal. Further, this conjugacy is generalized to one between Sobolev spaces in Ordinary Calculus and Fα-Calculus. This conjugacy is useful, among other things, to find solutions to Fα-differential equations: they can be mapped to Ordinary differential equations, and the solutions of the latter can be transformed back to get those of the former. This is illustrated with a few examples.

  • Calculus ON FRACTAL CURVES IN Rn
    Fractals, 2011
    Co-Authors: Abhay Parvate, Seema Satin, Anil D Gangal
    Abstract:

    A new Calculus on fractal curves, such as the von Koch curve, is formulated. We define a Riemann-like integral along a fractal curve F, called Fα-integral, where α is the dimension of F. A derivative along the fractal curve called Fα-derivative, is also defined. The mass function, a measure-like algorithmic quantity on the curves, plays a central role in the formulation. An appropriate algorithm to calculate the mass function is presented to emphasize its algorithmic aspect. Several aspects of this Calculus retain much of the simplicity of Ordinary Calculus. We establish a conjugacy between this Calculus and Ordinary Calculus on the real line. The Fα-integral and Fα-derivative are shown to be conjugate to the Riemann integral and Ordinary derivative respectively. In fact, they can thus be evalutated using the corresponding operators in Ordinary Calculus and conjugacy. Sobolev Spaces are constructed on F, and Fα-differentiability is generalized. Finally we touch upon an example of absorption along fractal paths, to illustrate the utility of the framework in model making.

  • Calculus on Fractal Curves in R^n
    arXiv: Mathematical Physics, 2009
    Co-Authors: Abhay Parvate, Seema Satin, Anil D Gangal
    Abstract:

    A new Calculus on fractal curves, such as the von Koch curve, is formulated. We define a Riemann-like integral along a fractal curve F, called F-alpha-integral, where alpha is the dimension of F. A derivative along the fractal curve called F-alpha-derivative, is also defined. The mass function, a measure-like algorithmic quantity on the curves, plays a central role in the formulation. An appropriate algorithm to calculate the mass function is presented to emphasize algorithmic aspect. Several aspects of this Calculus retain much of the simplicity of Ordinary Calculus. We establish a conjugacy between this Calculus and Ordinary Calculus on the real line. The F-alpha-integral and F-alpha-derivative are shown to be conjugate to the Riemann integral and Ordinary derivative respectively. In fact they can thus be evaluated using the corresponding operators in Ordinary Calculus and conjugacy. Sobolev Spaces are constructed on F, and F-alpha- differentiability is generalized. Finally we touch upon an example of absorption along fractal path to illustrate the utility of the framework in model making.

  • Calculus on fractal subsets of real line - I: formulation
    Fractals, 2009
    Co-Authors: Abhay Parvate, Anil D Gangal
    Abstract:

    A new Calculus based on fractal subsets of the real line is formulated. In this Calculus, an integral of order α, 0 < α ≤ 1, called Fα-integral, is defined, which is suitable to integrate functions with fractal support F of dimension α. Further, a derivative of order α, 0 < α ≤ 1, called Fα-derivative, is defined, which enables us to differentiate functions, like the Cantor staircase, "changing" only on a fractal set. The Fα-derivative is local unlike the classical fractional derivative. The Fα-Calculus retains much of the simplicity of Ordinary Calculus. Several results including analogues of fundamental theorems of Calculus are proved. The integral staircase function, which is a generalization of the functions like the Cantor staircase function, plays a key role in this formulation. Further, it gives rise to a new definition of dimension, the γ-dimension. Spaces of Fα-differentiable and Fα-integrable functions are analyzed. Analogues of Sobolev Spaces are constructed on F and Fα-differentiability is generalized using Sobolev-like construction. Fα-differential equations are equations involving Fα-derivatives. They can be used to model sublinear dynamical systems and fractal time processes, since sublinear behaviors are associated with staircase-like functions which occur naturally as their solutions. As examples, we discuss a fractal-time diffusion equation, and one-dimensional motion of a particle undergoing friction in a fractal medium.

Abhay Parvate - One of the best experts on this subject based on the ideXlab platform.

  • Calculus on fractal subsets of real line ii conjugacy with Ordinary Calculus
    Fractals, 2011
    Co-Authors: Abhay Parvate, Anil D Gangal
    Abstract:

    Calculus on fractals, or Fα-Calculus, developed in a previous paper, is a Calculus based fractals F ⊂ R, and involves Fα-integral and Fα-derivative of orders α, 0 < α ≤ 1, where α is the dimension of F. The Fα-integral is suitable for integrating functions with fractal support of dimension α, while the Fα-derivative enables us to differentiate functions like the Cantor staircase. Several results in Fα-Calculus are analogous to corresponding results in Ordinary Calculus, such as the Leibniz rule, fundamental theorems, etc. The functions like the Cantor staircase function occur naturally as solutions of Fα-differential equations. Hence the latter can be used to model processes involving fractal space or time, which in particular include a class of dynamical systems exhibiting sublinear behaviour. In this paper we show that, as operators, the Fα-integral and Fα-derivative are conjugate to the Riemann integral and Ordinary derivative respectively. This is accomplished by constructing a map ψ which takes Fα-integrable functions to Riemann integrable functions, such that the corresponding integrals on appropriate intervals have equal values. Under suitable conditions, a restriction of ψ also takes Fα-differentiable functions to ordinarily differentiable functions such that their values at appropriate points are equal. Further, this conjugacy is generalized to one between Sobolev spaces in Ordinary Calculus and Fα-Calculus. This conjugacy is useful, among other things, to find solutions to Fα-differential equations: they can be mapped to Ordinary differential equations, and the solutions of the latter can be transformed back to get those of the former. This is illustrated with a few examples.

  • Calculus ON FRACTAL SUBSETS OF REAL LINE — II: CONJUGACY WITH Ordinary Calculus
    Fractals, 2011
    Co-Authors: Abhay Parvate, Anil D Gangal
    Abstract:

    Calculus on fractals, or Fα-Calculus, developed in a previous paper, is a Calculus based fractals F ⊂ R, and involves Fα-integral and Fα-derivative of orders α, 0 < α ≤ 1, where α is the dimension of F. The Fα-integral is suitable for integrating functions with fractal support of dimension α, while the Fα-derivative enables us to differentiate functions like the Cantor staircase. Several results in Fα-Calculus are analogous to corresponding results in Ordinary Calculus, such as the Leibniz rule, fundamental theorems, etc. The functions like the Cantor staircase function occur naturally as solutions of Fα-differential equations. Hence the latter can be used to model processes involving fractal space or time, which in particular include a class of dynamical systems exhibiting sublinear behaviour. In this paper we show that, as operators, the Fα-integral and Fα-derivative are conjugate to the Riemann integral and Ordinary derivative respectively. This is accomplished by constructing a map ψ which takes Fα-integrable functions to Riemann integrable functions, such that the corresponding integrals on appropriate intervals have equal values. Under suitable conditions, a restriction of ψ also takes Fα-differentiable functions to ordinarily differentiable functions such that their values at appropriate points are equal. Further, this conjugacy is generalized to one between Sobolev spaces in Ordinary Calculus and Fα-Calculus. This conjugacy is useful, among other things, to find solutions to Fα-differential equations: they can be mapped to Ordinary differential equations, and the solutions of the latter can be transformed back to get those of the former. This is illustrated with a few examples.

  • Calculus ON FRACTAL CURVES IN Rn
    Fractals, 2011
    Co-Authors: Abhay Parvate, Seema Satin, Anil D Gangal
    Abstract:

    A new Calculus on fractal curves, such as the von Koch curve, is formulated. We define a Riemann-like integral along a fractal curve F, called Fα-integral, where α is the dimension of F. A derivative along the fractal curve called Fα-derivative, is also defined. The mass function, a measure-like algorithmic quantity on the curves, plays a central role in the formulation. An appropriate algorithm to calculate the mass function is presented to emphasize its algorithmic aspect. Several aspects of this Calculus retain much of the simplicity of Ordinary Calculus. We establish a conjugacy between this Calculus and Ordinary Calculus on the real line. The Fα-integral and Fα-derivative are shown to be conjugate to the Riemann integral and Ordinary derivative respectively. In fact, they can thus be evalutated using the corresponding operators in Ordinary Calculus and conjugacy. Sobolev Spaces are constructed on F, and Fα-differentiability is generalized. Finally we touch upon an example of absorption along fractal paths, to illustrate the utility of the framework in model making.

  • Calculus on Fractal Curves in R^n
    arXiv: Mathematical Physics, 2009
    Co-Authors: Abhay Parvate, Seema Satin, Anil D Gangal
    Abstract:

    A new Calculus on fractal curves, such as the von Koch curve, is formulated. We define a Riemann-like integral along a fractal curve F, called F-alpha-integral, where alpha is the dimension of F. A derivative along the fractal curve called F-alpha-derivative, is also defined. The mass function, a measure-like algorithmic quantity on the curves, plays a central role in the formulation. An appropriate algorithm to calculate the mass function is presented to emphasize algorithmic aspect. Several aspects of this Calculus retain much of the simplicity of Ordinary Calculus. We establish a conjugacy between this Calculus and Ordinary Calculus on the real line. The F-alpha-integral and F-alpha-derivative are shown to be conjugate to the Riemann integral and Ordinary derivative respectively. In fact they can thus be evaluated using the corresponding operators in Ordinary Calculus and conjugacy. Sobolev Spaces are constructed on F, and F-alpha- differentiability is generalized. Finally we touch upon an example of absorption along fractal path to illustrate the utility of the framework in model making.

  • Calculus on fractal subsets of real line - I: formulation
    Fractals, 2009
    Co-Authors: Abhay Parvate, Anil D Gangal
    Abstract:

    A new Calculus based on fractal subsets of the real line is formulated. In this Calculus, an integral of order α, 0 < α ≤ 1, called Fα-integral, is defined, which is suitable to integrate functions with fractal support F of dimension α. Further, a derivative of order α, 0 < α ≤ 1, called Fα-derivative, is defined, which enables us to differentiate functions, like the Cantor staircase, "changing" only on a fractal set. The Fα-derivative is local unlike the classical fractional derivative. The Fα-Calculus retains much of the simplicity of Ordinary Calculus. Several results including analogues of fundamental theorems of Calculus are proved. The integral staircase function, which is a generalization of the functions like the Cantor staircase function, plays a key role in this formulation. Further, it gives rise to a new definition of dimension, the γ-dimension. Spaces of Fα-differentiable and Fα-integrable functions are analyzed. Analogues of Sobolev Spaces are constructed on F and Fα-differentiability is generalized using Sobolev-like construction. Fα-differential equations are equations involving Fα-derivatives. They can be used to model sublinear dynamical systems and fractal time processes, since sublinear behaviors are associated with staircase-like functions which occur naturally as their solutions. As examples, we discuss a fractal-time diffusion equation, and one-dimensional motion of a particle undergoing friction in a fractal medium.

Bruce J. West - One of the best experts on this subject based on the ideXlab platform.

  • Colloquium: Fractional Calculus view of complexity: A tutorial
    Reviews of Modern Physics, 2014
    Co-Authors: Bruce J. West
    Abstract:

    The fractional Calculus has been part of the mathematics and science literature for 310 years. However, it is only in the past decade or so that it has drawn the attention of mainstream science as a way to describe the dynamics of complex phenomena with long-term memory, spatial heterogeneity, along with nonstationary and nonergodic statistics. The most recent application encompasses complex networks, which require new ways of thinking about the world. Part of the new cognition is provided by the fractional Calculus description of temporal and topological complexity. Consequently, this Colloquium is not so much a tutorial on the mathematics of the fractional Calculus as it is an exploration of how complex phenomena in the physical, social, and life sciences that have eluded traditional mathematical modeling become less mysterious when certain historical assumptions such as differentiability are discarded and the Ordinary Calculus is replaced with the fractional Calculus. Exemplars considered include the fractional differential equations describing the dynamics of viscoelastic materials, turbulence, foraging, and phase transitions in complex social networks.

Kiran M. Kolwankar - One of the best experts on this subject based on the ideXlab platform.

  • Local Fractional Calculus: a Calculus for Fractal Space-Time
    Fractals, 1999
    Co-Authors: Kiran M. Kolwankar, Anil D Gangal
    Abstract:

    Recently, new notions such as local fractional derivatives and local fractional differential equations were introduced. Here we argue that these developments provide a possible Calculus to deal with phenomena in fractal space-time. We show how the usual Calculus is generalized to deal with non Lipschitz functions. We also indicate how a definition of a fractal measure arises from these developments much the same way as the Lebesgue measure from Ordinary Calculus.

  • Definition of fractal measures arising from fractional Calculus
    arXiv: Chaotic Dynamics, 1998
    Co-Authors: Kiran M. Kolwankar, Anil D Gangal
    Abstract:

    It is wellknown that the Ordinary Calculus is inadequate to handle fractal structures and processes and another suitable Calculus needs to be developed for this purpose. Recently it was realized that fractional Calculus with suitable constructions does offer such a possibility. This makes it necessary to have a definition of fractal measures based on the fractional Calculus so that the fractals can be naturally incorporated in the Calculus. With this motivation a definition of fractal measure has been systematically developed using the concepts of fractional Calculus. It has been demonstrated that such a definition naturally arises in the solution of an equation describing diffusion in fractal time.

Pranab Ghosh - One of the best experts on this subject based on the ideXlab platform.

  • Evolution of Compact Binary Populations in Globular Clusters: A Boltzmann Study. II. Introducing Stochasticity
    The Astrophysical Journal, 2008
    Co-Authors: Sambaran Banerjee, Pranab Ghosh
    Abstract:

    We continue the exploration that we began in Paper I of using the Boltzmann scheme to study the evolution of compact binary populations of globular clusters, introducing in this paper our method of handling the stochasticity inherent in the dynamical processes of binary formation, destruction, and hardening in globular clusters. We describe these stochastic processes as "Wiener processes," whereupon the Boltzmann equation becomes a stochastic partial differential equation, the solution of which involves the use of "Itō Calculus" (this use being the first, to our knowledge, in this subject), in addition to Ordinary Calculus. As in Paper I, we focus on the evolution of (1) the number of X-ray binaries NXB in globular clusters and (2) the orbital period distribution of these binaries. We show that, although the details of the fluctuations in the above quantities differ from one "realization" to another of the stochastic processes, the general trends follow those found in the continuous-limit study of Paper I, and the average result over many such realizations is very close to the continuous-limit result. We investigate the dependence of NXB found by these calculations on two essential globular cluster properties, namely, the star-star and star-binary encounter rate parameters Γ and γ, for which we coined the name "Verbunt parameters" in Paper I. We compare our computed results with those from Chandra observations of Galactic globular clusters, showing that the expected scalings of NXB with the Verbunt parameters are in good agreement with those observed. We indicate additional features that can be incorporated into the scheme in the future, as well as how more elaborate problems can be tackled.