The Experts below are selected from a list of 8397 Experts worldwide ranked by ideXlab platform
Vasily E Tarasov - One of the best experts on this subject based on the ideXlab platform.
-
anisotropic fractal media by Vector Calculus in non integer dimensional space
arXiv: Mathematical Physics, 2015Co-Authors: Vasily E TarasovAbstract:A review of different approaches to describe anisotropic fractal media is proposed. In this paper differentiation and integration non-integer dimensional and multi-fractional spaces are considered as tools to describe anisotropic fractal materials and media. We suggest a generalization of Vector Calculus for non-integer dimensional space by using a product measure method. The product of fractional and non-integer dimensional spaces allows us to take into account the anisotropy of the fractal media in the framework of continuum models. The integration over non-integer-dimensional spaces is considered. In this paper differential operators of first and second orders for fractional space and non-integer dimensional space are suggested. The differential operators are defined as inverse operations to integration in spaces with non-integer dimensions. Non-integer dimensional space that is product of spaces with different dimensions allows us to give continuum models for anisotropic type of the media. The Poisson's equation for fractal medium, the Euler-Bernoulli fractal beam, and the Timoshenko beam equations for fractal material are considered as examples of application of suggested generalization of Vector Calculus for anisotropic fractal materials and media.
-
Vector Calculus in non integer dimensional space and its applications to fractal media
Communications in Nonlinear Science and Numerical Simulation, 2015Co-Authors: Vasily E TarasovAbstract:We suggest a generalization of Vector Calculus for the case of non-integer dimensional space. The first and second orders operations such as gradient, divergence, the scalar and Vector Laplace operators for non-integer dimensional space are defined. For simplification we consider scalar and Vector fields that are independent of angles. We formulate a generalization of Vector Calculus for rotationally covariant scalar and Vector functions. This generalization allows us to describe fractal media and materials in the framework of continuum models with non-integer dimensional space. As examples of application of the suggested Calculus, we consider elasticity of fractal materials (fractal hollow ball and fractal cylindrical pipe with pressure inside and outside), steady distribution of heat in fractal media, electric field of fractal charged cylinder. We solve the correspondent equations for non-integer dimensional space models.
-
toward lattice fractional Vector Calculus
Journal of Physics A, 2014Co-Authors: Vasily E TarasovAbstract:An analog of fractional Vector Calculus for physical lattice models is suggested. We use an approach based on the models of three-dimensional lattices with long-range inter-particle interactions. The lattice analogs of fractional partial derivatives are represented by kernels of lattice long-range interactions, where the Fourier series transformations of these kernels have a power-law form with respect to wave Vector components. In the continuum limit, these lattice partial derivatives give derivatives of non-integer order with respect to coordinates. In the three-dimensional description of the non-local continuum, the fractional differential operators have the form of fractional partial derivatives of the Riesz type. As examples of the applications of the suggested lattice fractional Vector Calculus, we give lattice models with long-range interactions for the fractional Maxwell equations of non-local continuous media and for the fractional generalization of the Mindlin and Aifantis continuum models of gradient elasticity.
-
fractional Vector Calculus
2010Co-Authors: Vasily E TarasovAbstract:The Calculus of derivatives and integrals of non-integer order go back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. The fractional Calculus has a long history from 1695, when the derivative of order α = 0.5 was described by Leibniz (Oldham and Spanier, 1974; Samko et al., 1993; Ross, 1975). The history of fractional Vector Calculus (FVC) is not so long. It has only 10 years and can be reduced to the papers (Ben Adda, 1997, 1998a, b, 2001; Engheta, 1998; Veliev and Engheta, 2004; Ivakhnychenko and Veliev, 2004; Naqvi and Abbas, 2004; Naqvi et al., 2006; Hussain and Naqvi, 2006; Hussain et al., 2006; Meerschaert et al., 2006; Yong et al., 2003; Kazbekov, 2005) and (Tarasov, 2005a, b, d, e, 2006a, b, c, 2007, 2008). There are some fundamental problems of consistent formulations of FVC that can be solved by using a fractional generalization of the fundamental theorem of Calculus (Tarasov, 2008). We define the fractional differential and integral Vector operations. The fractional Green’s, Stokes’ and Gauss’ theorems are formulated. The proofs of these theorems are realized for simplest regions. A fractional generalization of exterior differential Calculus of differential forms is discussed. A consistent FVC can be used in fractional statistical mechanics (Tarasov, 2006c, 2007), fractional electrodynamics (Engheta, 1998; Veliev and Engheta, 2004; Ivakhnychenko and Veliev, 2004; Naqvi and Abbas, 2004; Naqvi et al., 2006; Hussain and Naqvi, 2006; Hussain et al., 2006; Tarasov, 2005d, 2006a, b, 2005e) and fractional hydrodynamics (Meerschaert et al., 2006; Tarasov, 2005c). Fractional Vector Calculus is very important to describe processes in complex media (Carpinteri and Mainardi, 1997).
-
fractional Vector Calculus and fractional maxwell s equations
arXiv: Mathematical Physics, 2009Co-Authors: Vasily E TarasovAbstract:The theory of derivatives and integrals of non-integer order goes back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. The history of fractional Vector Calculus (FVC) has only 10 years. The main approaches to formulate a FVC, which are used in the physics during the past few years, will be briefly described in this paper. We solve some problems of consistent formulations of FVC by using a fractional generalization of the Fundamental Theorem of Calculus. We define the differential and integral Vector operations. The fractional Green's, Stokes' and Gauss's theorems are formulated. The proofs of these theorems are realized for simplest regions. A fractional generalization of exterior differential Calculus of differential forms is discussed. Fractional nonlocal Maxwell's equations and the corresponding fractional wave equations are considered.
Kun Zhou - One of the best experts on this subject based on the ideXlab platform.
-
a nonlocal Vector Calculus nonlocal volume constrained problems and nonlocal balance laws
Mathematical Models and Methods in Applied Sciences, 2013Co-Authors: Max D Gunzburger, Richard B Lehoucq, Kun ZhouAbstract:A Vector Calculus for nonlocal operators is developed, including the definition of nonlocal divergence, gradient, and curl operators and the derivation of the corresponding adjoint operators. Nonlo...
-
analysis and approximation of nonlocal diffusion problems with volume constraints
Siam Review, 2012Co-Authors: Qiang Du, Max D Gunzburger, Richard B Lehoucq, Kun ZhouAbstract:A recently developed nonlocal Vector Calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a linear integral equation on bounded domains in $\mbRn$. The nonlocal Vector Calculus also enables striking analogies to be drawn between the nonlocal model and classical models for diffusion, including a notion of nonlocal flux. The ubiquity of the nonlocal operator in applications is illustrated by a number of examples ranging from continuum mechanics to graph theory. In particular, it is shown that fractional Laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion that we consider. The numerous applications elucidate different interpretations of the operator and the associated governing equations. For example, a probabilistic perspective explains that the nonlocal spatial operator appearing in our model corresponds to the infinitesimal generator for a symmetric jump process. Sufficie...
Hong Qin - One of the best experts on this subject based on the ideXlab platform.
-
vest abstract Vector Calculus simplification in mathematica
Computer Physics Communications, 2014Co-Authors: Jonathan Squire, J W Burby, Hong QinAbstract:Abstract We present a new package, VEST (Vector Einstein Summation Tools), that performs abstract Vector Calculus computations in Mathematica. Through the use of index notation, VEST is able to reduce three-dimensional scalar and Vector expressions of a very general type to a well defined standard form. In addition, utilizing properties of the Levi-Civita symbol, the program can derive types of multi-term Vector identities that are not recognized by reduction, subsequently applying these to simplify large expressions. In a companion paper Burby et al. (2013) [12] , we employ VEST in the automation of the calculation of high-order Lagrangians for the single particle guiding center system in plasma physics, a computation which illustrates its ability to handle very large expressions. VEST has been designed to be simple and intuitive to use, both for basic checking of work and more involved computations. Program summary Program title: VEST (Vector Einstein Summation Tools) Catalogue identifier: AEQN_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEQN_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 10469 No. of bytes in distributed program, including test data, etc.: 72539 Distribution format: tar.gz Programming language: Mathematica. Computer: Any computer running Mathematica. Operating system: Linux, Unix, Windows, Mac OS X. RAM: Usually under 10 Mbytes Classification: 5, 12, 19. Nature of problem: Large scale Vector Calculus computations Solution method: Reduce expressions to standard form in index notation, automatic derivation of multi-term Vector identities. Restrictions: Current version cannot derive Vector identities without cross products or curl Additional comments: Intuitive user input and output in a combination of Vector and index notation Running time: Reduction to standard form is usually less than one second. Simplification of very large expressions can take much longer.
Max D Gunzburger - One of the best experts on this subject based on the ideXlab platform.
-
regularity analyses and approximation of nonlocal variational equality and inequality problems
Journal of Mathematical Analysis and Applications, 2019Co-Authors: Olena Burkovska, Max D GunzburgerAbstract:Abstract We consider linear and obstacle problems driven by a nonlocal integral operator, for which nonlocal interactions are restricted to a ball of finite radius. These types of operators are used to model anomalous diffusion and, for a special choice of the integral kernels, reduce to the fractional Laplace operator on a bounded domain. By means of a nonlocal Vector Calculus we recast the problems in a weak form, leading to corresponding nonlocal variational equality and inequality problems. We prove optimal regularity results for both problems, including a higher regularity of the solution and the Lagrange multiplier. Based on the regularity results, we analyze the convergence of finite element approximations for a linear problem and illustrate the theoretical findings by numerical results.
-
a generalized nonlocal Vector Calculus
Zeitschrift für Angewandte Mathematik und Physik, 2015Co-Authors: Bacim Alali, Kuo Liu, Max D GunzburgerAbstract:A nonlocal Vector Calculus was introduced in Du et al. (Math Model Meth Appl Sci 23:493–540, 2013) that has proved useful for the analysis of the peridynamics model of nonlocal mechanics and nonlocal diffusion models. A formulation is developed that provides a more general setting for the nonlocal Vector Calculus that is independent of particular nonlocal models. It is shown that general nonlocal Calculus operators are integral operators with specific integral kernels. General nonlocal Calculus properties are developed, including nonlocal integration by parts formula and Green’s identities. The nonlocal Vector Calculus introduced in Du et al. (Math Model Meth Appl Sci 23:493–540, 2013) is shown to be recoverable from the general formulation as a special example. This special nonlocal Vector Calculus is used to reformulate the peridynamics equation of motion in terms of the nonlocal gradient operator and its adjoint. A new example of nonlocal Vector Calculus operators is introduced, which shows the potential use of the general formulation for general nonlocal models.
-
a nonlocal Vector Calculus nonlocal volume constrained problems and nonlocal balance laws
Mathematical Models and Methods in Applied Sciences, 2013Co-Authors: Max D Gunzburger, Richard B Lehoucq, Kun ZhouAbstract:A Vector Calculus for nonlocal operators is developed, including the definition of nonlocal divergence, gradient, and curl operators and the derivation of the corresponding adjoint operators. Nonlo...
-
analysis and approximation of nonlocal diffusion problems with volume constraints
Siam Review, 2012Co-Authors: Qiang Du, Max D Gunzburger, Richard B Lehoucq, Kun ZhouAbstract:A recently developed nonlocal Vector Calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a linear integral equation on bounded domains in $\mbRn$. The nonlocal Vector Calculus also enables striking analogies to be drawn between the nonlocal model and classical models for diffusion, including a notion of nonlocal flux. The ubiquity of the nonlocal operator in applications is illustrated by a number of examples ranging from continuum mechanics to graph theory. In particular, it is shown that fractional Laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion that we consider. The numerous applications elucidate different interpretations of the operator and the associated governing equations. For example, a probabilistic perspective explains that the nonlocal spatial operator appearing in our model corresponds to the infinitesimal generator for a symmetric jump process. Sufficie...
-
A non-local Vector Calculus,non-local volume-constrained problems,and non-local balance laws
2011Co-Authors: Max D Gunzburger, Richard B Lehoucq, K. ZhouAbstract:A Vector Calculus for nonlocal operators is developed, including the definition of nonlocal divergence, gradient, and curl operators and the derivation of the corresponding adjoints operators. Nonlocal analogs of several theorems and identities of the Vector Calculus for differential operators are also presented. Relationships between the nonlocal operators and their differential counterparts are established, first in a distributional sense and then in a weak sense by considering weighted integrals of the nonlocal adjoint operators. The nonlocal Calculus gives rise to volume-constrained problems that are analogous to elliptic boundary-value problems for differential operators; this is demonstrated via some examples. Another application is posing abstract nonlocal balance laws and deriving the corresponding nonlocal field equations.
Richard B Lehoucq - One of the best experts on this subject based on the ideXlab platform.
-
a nonlocal Vector Calculus nonlocal volume constrained problems and nonlocal balance laws
Mathematical Models and Methods in Applied Sciences, 2013Co-Authors: Max D Gunzburger, Richard B Lehoucq, Kun ZhouAbstract:A Vector Calculus for nonlocal operators is developed, including the definition of nonlocal divergence, gradient, and curl operators and the derivation of the corresponding adjoint operators. Nonlo...
-
analysis and approximation of nonlocal diffusion problems with volume constraints
Siam Review, 2012Co-Authors: Qiang Du, Max D Gunzburger, Richard B Lehoucq, Kun ZhouAbstract:A recently developed nonlocal Vector Calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a linear integral equation on bounded domains in $\mbRn$. The nonlocal Vector Calculus also enables striking analogies to be drawn between the nonlocal model and classical models for diffusion, including a notion of nonlocal flux. The ubiquity of the nonlocal operator in applications is illustrated by a number of examples ranging from continuum mechanics to graph theory. In particular, it is shown that fractional Laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion that we consider. The numerous applications elucidate different interpretations of the operator and the associated governing equations. For example, a probabilistic perspective explains that the nonlocal spatial operator appearing in our model corresponds to the infinitesimal generator for a symmetric jump process. Sufficie...
-
A non-local Vector Calculus,non-local volume-constrained problems,and non-local balance laws
2011Co-Authors: Max D Gunzburger, Richard B Lehoucq, K. ZhouAbstract:A Vector Calculus for nonlocal operators is developed, including the definition of nonlocal divergence, gradient, and curl operators and the derivation of the corresponding adjoints operators. Nonlocal analogs of several theorems and identities of the Vector Calculus for differential operators are also presented. Relationships between the nonlocal operators and their differential counterparts are established, first in a distributional sense and then in a weak sense by considering weighted integrals of the nonlocal adjoint operators. The nonlocal Calculus gives rise to volume-constrained problems that are analogous to elliptic boundary-value problems for differential operators; this is demonstrated via some examples. Another application is posing abstract nonlocal balance laws and deriving the corresponding nonlocal field equations.
-
a nonlocal Vector Calculus with application to nonlocal boundary value problems
Multiscale Modeling & Simulation, 2010Co-Authors: Max D Gunzburger, Richard B LehoucqAbstract:We develop a Calculus for nonlocal operators that mimics Gauss's theorem and Green's identities of the classical Vector Calculus. The operators we define do not involve derivatives. We then apply the nonlocal Calculus to define weak formulations of nonlocal “boundary-value” problems that mimic the Dirichlet and Neumann problems for second-order scalar elliptic partial differential equations. For the nonlocal problems, we derive a fundamental solution and Green's functions, demonstrate that weak formulations of the nonlocal “boundary-value” problems are well posed, and show how, under appropriate limits, the nonlocal problems reduce to their local analogues.
