The Experts below are selected from a list of 66 Experts worldwide ranked by ideXlab platform
Mark M Meerschaert - One of the best experts on this subject based on the ideXlab platform.
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a novel numerical method for the time variable fractional order mobile immobile advection dispersion model
Computers & Mathematics With Applications, 2013Co-Authors: Hongmei Zhang, Mantha S Phanikumar, Mark M MeerschaertAbstract:Evolution equations containing fractional Derivatives can provide suitable mathematical models for describing anomalous diffusion and transport dynamics in complex systems that cannot be modeled accurately by normal integer order equations. Recently, researchers have found that many physical processes exhibit fractional order behavior that varies with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. In this paper, we consider the mobile-immobile advection-dispersion model with the Coimbra variable time fractional Derivative which is preferable for modeling dynamical systems and is more efficient from the numerical standpoint. A novel implicit numerical method for the equation is proposed and the stability of the approximation is investigated. As for the convergence of the numerical method, we only consider a special case, i.e., the time fractional Derivative is independent of the time variable t. The case where the time fractional Derivative Depends on both the time variable t and the space variable x will be considered in a future work. Finally, numerical examples are provided to show that the implicit difference approximation is computationally efficient.
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a novel numerical method for the time variable fractional order mobile immobile advection dispersion model
Computers & Mathematics With Applications, 2013Co-Authors: Hongmei Zhang, Mantha S Phanikumar, Fawang Liu, Mark M MeerschaertAbstract:Evolution equations containing fractional Derivatives can provide suitable mathematical models for describing anomalous diffusion and transport dynamics in complex systems that cannot be modeled accurately by normal integer order equations. Recently, researchers have found that many physical processes exhibit fractional order behavior that varies with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. In this paper, we consider the mobile-immobile advection-dispersion model with the Coimbra variable time fractional Derivative which is preferable for modeling dynamical systems and is more efficient from the numerical standpoint. A novel implicit numerical method for the equation is proposed and the stability of the approximation is investigated. As for the convergence of the numerical method, we only consider a special case, i.e., the time fractional Derivative is independent of the time variable t. The case where the time fractional Derivative Depends on both the time variable t and the space variable x will be considered in a future work. Finally, numerical examples are provided to show that the implicit difference approximation is computationally efficient.
Friedrich Sauvigny - One of the best experts on this subject based on the ideXlab platform.
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Sobolev Spaces W 1 , p ( ℝ n , γ ) $W^{1,p}(\mathbb {R}^{n},\gamma )$ Weighted by the Gaussian Normal Distribution γ ( x ) : = 1 π n exp ( − | x | 2 ) $\gamma (x):=\frac {1}{\sqrt {\pi }^{n}}\exp (-|x|^{2})$ and the Spectral Theory
Vietnam Journal of Mathematics, 2020Co-Authors: Friedrich SauvignyAbstract:In the spectral theory it does make a difference, whether we consider differential operators on bounded or unbounded domains. In order to treat eigenvalue problems on the whole Euclidean space, we construct Sobolev spaces over ℝ n $\mathbb {R}^{n}$ , which are weighted by the Gaussian normal distribution . By the methods presented in Chapters 2, 8, and 10 of the treatise F. Sauvigny: Partial Differential Equations 1 and 2, Springer Universitext (2012) , we can prove an analogue of the Sobolev embedding theorem and a Rellich selection theorem for the Sobolev spaces W 0 1 , p ( ℝ n , γ ) $W_{0}^{1,p}(\mathbb {R}^{n},\gamma )$ weighted by γ - with vanishing values towards infinity. We achieve these specific results for our entire Sobolev spaces W 1 , p ( ℝ n , γ ) $W^{1,p}(\mathbb {R}^{n},\gamma )$ , since we concentrate on the Gaussian normal distribution γ as our weight function. Even our notion of the weighted partial Derivative Depends on this weight function. Within the so-called Gauß–Rellich space W 0 1 , 2 ( ℝ n , γ ) $W_{0}^{1,2}(\mathbb {R}^{n},\gamma )$ we shall investigate the discrete spectrum of weighted elliptic operators over ℝ n $\mathbb {R}^{n}$ by spectral methods. There we rely on the treatise F. Sauvigny: Spektraltheorie selbstadjungierter Operatoren im Hilbertraum und elliptischer Differentialoperatoren, Springer Spektrum (2019) . By reflection methods, we solve eigenvalue problems for elliptic differential operators on the sectorial domain ℝ + n $\mathbb {R}_{+}^{n}$ under vanishing and mixed boundary conditions.
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Sobolev Spaces W 1 , p ( ℝ n , γ ) $W^{1,p}(\mathbb {R}^{n},\gamma )$ Weighted by the Gaussian Normal Distribution γ ( x ) : = 1 π n exp ( − | x | 2 ) $\gamma (x):=\frac {1}{\sqrt {\pi }^{n}}\exp (-|x|^{2})$ and the Spectral Theory
Vietnam Journal of Mathematics, 2020Co-Authors: Friedrich SauvignyAbstract:In the spectral theory it does make a difference, whether we consider differential operators on bounded or unbounded domains. In order to treat eigenvalue problems on the whole Euclidean space, we construct Sobolev spaces over ℝ n $\mathbb {R}^{n}$ , which are weighted by the Gaussian normal distribution . By the methods presented in Chapters 2, 8, and 10 of the treatise F. Sauvigny: Partial Differential Equations 1 and 2, Springer Universitext (2012) , we can prove an analogue of the Sobolev embedding theorem and a Rellich selection theorem for the Sobolev spaces W 0 1 , p ( ℝ n , γ ) $W_{0}^{1,p}(\mathbb {R}^{n},\gamma )$ weighted by γ - with vanishing values towards infinity. We achieve these specific results for our entire Sobolev spaces W 1 , p ( ℝ n , γ ) $W^{1,p}(\mathbb {R}^{n},\gamma )$ , since we concentrate on the Gaussian normal distribution γ as our weight function. Even our notion of the weighted partial Derivative Depends on this weight function. Within the so-called Gauß–Rellich space W 0 1 , 2 ( ℝ n , γ ) $W_{0}^{1,2}(\mathbb {R}^{n},\gamma )$ we shall investigate the discrete spectrum of weighted elliptic operators over ℝ n $\mathbb {R}^{n}$ by spectral methods. There we rely on the treatise F. Sauvigny: Spektraltheorie selbstadjungierter Operatoren im Hilbertraum und elliptischer Differentialoperatoren, Springer Spektrum (2019) . By reflection methods, we solve eigenvalue problems for elliptic differential operators on the sectorial domain ℝ + n $\mathbb {R}_{+}^{n}$ under vanishing and mixed boundary conditions.
Hongmei Zhang - One of the best experts on this subject based on the ideXlab platform.
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a novel numerical method for the time variable fractional order mobile immobile advection dispersion model
Computers & Mathematics With Applications, 2013Co-Authors: Hongmei Zhang, Mantha S Phanikumar, Mark M MeerschaertAbstract:Evolution equations containing fractional Derivatives can provide suitable mathematical models for describing anomalous diffusion and transport dynamics in complex systems that cannot be modeled accurately by normal integer order equations. Recently, researchers have found that many physical processes exhibit fractional order behavior that varies with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. In this paper, we consider the mobile-immobile advection-dispersion model with the Coimbra variable time fractional Derivative which is preferable for modeling dynamical systems and is more efficient from the numerical standpoint. A novel implicit numerical method for the equation is proposed and the stability of the approximation is investigated. As for the convergence of the numerical method, we only consider a special case, i.e., the time fractional Derivative is independent of the time variable t. The case where the time fractional Derivative Depends on both the time variable t and the space variable x will be considered in a future work. Finally, numerical examples are provided to show that the implicit difference approximation is computationally efficient.
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a novel numerical method for the time variable fractional order mobile immobile advection dispersion model
Computers & Mathematics With Applications, 2013Co-Authors: Hongmei Zhang, Mantha S Phanikumar, Fawang Liu, Mark M MeerschaertAbstract:Evolution equations containing fractional Derivatives can provide suitable mathematical models for describing anomalous diffusion and transport dynamics in complex systems that cannot be modeled accurately by normal integer order equations. Recently, researchers have found that many physical processes exhibit fractional order behavior that varies with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. In this paper, we consider the mobile-immobile advection-dispersion model with the Coimbra variable time fractional Derivative which is preferable for modeling dynamical systems and is more efficient from the numerical standpoint. A novel implicit numerical method for the equation is proposed and the stability of the approximation is investigated. As for the convergence of the numerical method, we only consider a special case, i.e., the time fractional Derivative is independent of the time variable t. The case where the time fractional Derivative Depends on both the time variable t and the space variable x will be considered in a future work. Finally, numerical examples are provided to show that the implicit difference approximation is computationally efficient.
Bruce J. West - One of the best experts on this subject based on the ideXlab platform.
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Aging and rejuvenation with fractional Derivatives
Physical Review E, 2004Co-Authors: Gerardo Aquino, Mauro Bologna, Paolo Grigolini, Bruce J. WestAbstract:We discuss a dynamic procedure that makes fractional Derivatives emerge in the time asymptotic limit of non-Poisson processes. We find that two-state fluctuations, with an inverse power-law distribution of waiting times, finite first moment, and divergent second moment, namely, with the power indexm in the interval 2 , m , 3, yield a generalized master equation equivalent to the sum of an ordinary Markov contribution and a fractional Derivative term. We show that the order of the fractional Derivative Depends on the age of the process under study. If the system is infinitely old, the order of the fractional Derivative, o, is given by o =3 ˛ m. A brand new system is characterized by the degree o =m ˛ 2. If the system is prepared at time ˛ ta, 0 and the observation begins at time t = 0, we derive the following scenario. For times 0 , t ! ta the system is satisfactorily described by the fractional Derivative with o =3˛ m. Upon time increase the system undergoes a rejuvenation process that in the time limit t @ ta yields o =m ˛ 2. The intermediate time regime is probably incompatible with a picture based on fractional Derivatives, or, at least, with a mono-order fractional Derivative.
Mantha S Phanikumar - One of the best experts on this subject based on the ideXlab platform.
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a novel numerical method for the time variable fractional order mobile immobile advection dispersion model
Computers & Mathematics With Applications, 2013Co-Authors: Hongmei Zhang, Mantha S Phanikumar, Mark M MeerschaertAbstract:Evolution equations containing fractional Derivatives can provide suitable mathematical models for describing anomalous diffusion and transport dynamics in complex systems that cannot be modeled accurately by normal integer order equations. Recently, researchers have found that many physical processes exhibit fractional order behavior that varies with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. In this paper, we consider the mobile-immobile advection-dispersion model with the Coimbra variable time fractional Derivative which is preferable for modeling dynamical systems and is more efficient from the numerical standpoint. A novel implicit numerical method for the equation is proposed and the stability of the approximation is investigated. As for the convergence of the numerical method, we only consider a special case, i.e., the time fractional Derivative is independent of the time variable t. The case where the time fractional Derivative Depends on both the time variable t and the space variable x will be considered in a future work. Finally, numerical examples are provided to show that the implicit difference approximation is computationally efficient.
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a novel numerical method for the time variable fractional order mobile immobile advection dispersion model
Computers & Mathematics With Applications, 2013Co-Authors: Hongmei Zhang, Mantha S Phanikumar, Fawang Liu, Mark M MeerschaertAbstract:Evolution equations containing fractional Derivatives can provide suitable mathematical models for describing anomalous diffusion and transport dynamics in complex systems that cannot be modeled accurately by normal integer order equations. Recently, researchers have found that many physical processes exhibit fractional order behavior that varies with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. In this paper, we consider the mobile-immobile advection-dispersion model with the Coimbra variable time fractional Derivative which is preferable for modeling dynamical systems and is more efficient from the numerical standpoint. A novel implicit numerical method for the equation is proposed and the stability of the approximation is investigated. As for the convergence of the numerical method, we only consider a special case, i.e., the time fractional Derivative is independent of the time variable t. The case where the time fractional Derivative Depends on both the time variable t and the space variable x will be considered in a future work. Finally, numerical examples are provided to show that the implicit difference approximation is computationally efficient.