Journal of Applied Mathematics and Computing, 2003
A time fractional advection-dispersion equation is obtained from the standard advection-dispersio... more A time fractional advection-dispersion equation is obtained from the standard advection-dispersion equation by replacing the firstorder derivative in time by a fractional derivative in time of order α(0
In this paper, we consider a space-time fractional advection dispersion equation (STFADE) on a fi... more In this paper, we consider a space-time fractional advection dispersion equation (STFADE) on a finite domain. The STFADE is obtained from the standard advection dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order a 2 (0, 1], and the first-order and second-order space derivatives by the Riemman-Liouville fractional derivatives of order b 2 (0, 1] and of order c 2 (1, 2], respectively. For the space fractional derivatives D b
... Let m I 1 be chosen arbitrary. Then j= (ovg(t)dB(t))ej 2z E ,=E, 2 nI = EZ gi(Bh(ti+&#x27... more ... Let m I 1 be chosen arbitrary. Then j= (ovg(t)dB(t))ej 2z E ,=E, 2 nI = EZ gi(Bh(ti+' ) Bh(ti)) i=0 m n1 nI K2Z ZZ g(()(ti+l)Bh(ti))(B)(tl+l) B)('I))) j=li=0 =lm nI )2 =K2ZE(i=o(Bh(ti+l)B(ti)) j=lmj=lo = K2 T2h #jj=l for all m. Consequently, r dBh(t) ez for (t)ej2z Ef O(t) = E 9(t)dB K2T2h . ...
This paper introduces a fractional heat equation, where the diffusion operator is the composition... more This paper introduces a fractional heat equation, where the diffusion operator is the composition of the Bessel and Riesz potentials. Sharp bounds are obtained for the variance of the spatial and temporal increments of the solution. These bounds establish the degree of singularity of the sample paths of the solution. In the case of unbounded spatial domain, a solution is
We present a spectral representation of the mean-square solution of the fractional kinetic equati... more We present a spectral representation of the mean-square solution of the fractional kinetic equation (also known as fractional diffusion equation) with random initial condition. Gaussian and non-Gaussian limiting distributions of the renormalized solution of the fractional-in-time and in-space kinetic equation are described in terms of multiple stochastic integral representations.
A class of continuous-time models is developed for modelling data with heavy tails and long-range... more A class of continuous-time models is developed for modelling data with heavy tails and long-range dependence. These models are based on the Green function solutions of fractional differential equations driven by Lévy noise. Some exact results on the second- and higher-order characteristics of the equations are obtained. Applications to stochastic volatility of asset prices and macroeconomics are provided.
In this paper we present a random walk model for approximating a Lévy-Feller advection-dispersion... more In this paper we present a random walk model for approximating a Lévy-Feller advection-dispersion process, governed by the Lévy-Feller advection-dispersion differential equation (LFADE). We show that the random walk model converges to LFADE by use of a properly scaled transition to vanishing space and time steps. We propose an explicit finite difference approximation (EFDA) for LFADE, resulting from the Grü nwald-Letnikov discretization of fractional derivatives. As a result of the interpretation of the random walk model, the stability and convergence of EFDA for LFADE in a bounded domain are discussed. Finally, some numerical examples are presented to show the application of the present technique.
Gaussian and non-Gaussian limiting distributions of the rescaled solutions of the fractional (in ... more Gaussian and non-Gaussian limiting distributions of the rescaled solutions of the fractional (in time) di usion-wave equation for Gaussian and non-Gaussian initial data with long-range dependence are described in terms of multiple Wiener-Itô integrals.
The time fractional diffusion equation (tfde) is obtained from the standard diffusion equation by... more The time fractional diffusion equation (tfde) is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order in (0,1). In this work, an explicit finite-difference scheme for tfde is presented. Discrete models of a non-...
Journal of Computational and Applied Mathematics, 2004
The traditional second-order Fokker-Planck equation may not adequately describe the movement of s... more The traditional second-order Fokker-Planck equation may not adequately describe the movement of solute in an aquifer because of large deviation from the dynamics of Brownian motion. Densities of α-stable type have been used to describe the probability distribution of these motions. The resulting governing equation of these motions is similar to the traditional Fokker-Planck equation except that the order α of the highest derivative is fractional.
In this paper, a fractional partial differential equation (FPDE) describing subdiffusion is consi... more In this paper, a fractional partial differential equation (FPDE) describing subdiffusion is considered. An implicit difference approximation scheme (IDAS) for solving a FPDE is presented. We propose a Fourier method for analyzing the stability and convergence of the IDAS, derive the global accuracy of the IDAS, and discuss the solvability. Finally, numerical examples are given to compare with the exact solution for the order of convergence, and simulate the fractional dynamical systems.
Journal of Applied Mathematics and Computing, 2003
A time fractional advection-dispersion equation is obtained from the standard advection-dispersio... more A time fractional advection-dispersion equation is obtained from the standard advection-dispersion equation by replacing the firstorder derivative in time by a fractional derivative in time of order α(0
In this paper, we consider a space-time fractional advection dispersion equation (STFADE) on a fi... more In this paper, we consider a space-time fractional advection dispersion equation (STFADE) on a finite domain. The STFADE is obtained from the standard advection dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order a 2 (0, 1], and the first-order and second-order space derivatives by the Riemman-Liouville fractional derivatives of order b 2 (0, 1] and of order c 2 (1, 2], respectively. For the space fractional derivatives D b
... Let m I 1 be chosen arbitrary. Then j= (ovg(t)dB(t))ej 2z E ,=E, 2 nI = EZ gi(Bh(ti+&#x27... more ... Let m I 1 be chosen arbitrary. Then j= (ovg(t)dB(t))ej 2z E ,=E, 2 nI = EZ gi(Bh(ti+' ) Bh(ti)) i=0 m n1 nI K2Z ZZ g(()(ti+l)Bh(ti))(B)(tl+l) B)('I))) j=li=0 =lm nI )2 =K2ZE(i=o(Bh(ti+l)B(ti)) j=lmj=lo = K2 T2h #jj=l for all m. Consequently, r dBh(t) ez for (t)ej2z Ef O(t) = E 9(t)dB K2T2h . ...
This paper introduces a fractional heat equation, where the diffusion operator is the composition... more This paper introduces a fractional heat equation, where the diffusion operator is the composition of the Bessel and Riesz potentials. Sharp bounds are obtained for the variance of the spatial and temporal increments of the solution. These bounds establish the degree of singularity of the sample paths of the solution. In the case of unbounded spatial domain, a solution is
We present a spectral representation of the mean-square solution of the fractional kinetic equati... more We present a spectral representation of the mean-square solution of the fractional kinetic equation (also known as fractional diffusion equation) with random initial condition. Gaussian and non-Gaussian limiting distributions of the renormalized solution of the fractional-in-time and in-space kinetic equation are described in terms of multiple stochastic integral representations.
A class of continuous-time models is developed for modelling data with heavy tails and long-range... more A class of continuous-time models is developed for modelling data with heavy tails and long-range dependence. These models are based on the Green function solutions of fractional differential equations driven by Lévy noise. Some exact results on the second- and higher-order characteristics of the equations are obtained. Applications to stochastic volatility of asset prices and macroeconomics are provided.
In this paper we present a random walk model for approximating a Lévy-Feller advection-dispersion... more In this paper we present a random walk model for approximating a Lévy-Feller advection-dispersion process, governed by the Lévy-Feller advection-dispersion differential equation (LFADE). We show that the random walk model converges to LFADE by use of a properly scaled transition to vanishing space and time steps. We propose an explicit finite difference approximation (EFDA) for LFADE, resulting from the Grü nwald-Letnikov discretization of fractional derivatives. As a result of the interpretation of the random walk model, the stability and convergence of EFDA for LFADE in a bounded domain are discussed. Finally, some numerical examples are presented to show the application of the present technique.
Gaussian and non-Gaussian limiting distributions of the rescaled solutions of the fractional (in ... more Gaussian and non-Gaussian limiting distributions of the rescaled solutions of the fractional (in time) di usion-wave equation for Gaussian and non-Gaussian initial data with long-range dependence are described in terms of multiple Wiener-Itô integrals.
The time fractional diffusion equation (tfde) is obtained from the standard diffusion equation by... more The time fractional diffusion equation (tfde) is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order in (0,1). In this work, an explicit finite-difference scheme for tfde is presented. Discrete models of a non-...
Journal of Computational and Applied Mathematics, 2004
The traditional second-order Fokker-Planck equation may not adequately describe the movement of s... more The traditional second-order Fokker-Planck equation may not adequately describe the movement of solute in an aquifer because of large deviation from the dynamics of Brownian motion. Densities of α-stable type have been used to describe the probability distribution of these motions. The resulting governing equation of these motions is similar to the traditional Fokker-Planck equation except that the order α of the highest derivative is fractional.
In this paper, a fractional partial differential equation (FPDE) describing subdiffusion is consi... more In this paper, a fractional partial differential equation (FPDE) describing subdiffusion is considered. An implicit difference approximation scheme (IDAS) for solving a FPDE is presented. We propose a Fourier method for analyzing the stability and convergence of the IDAS, derive the global accuracy of the IDAS, and discuss the solvability. Finally, numerical examples are given to compare with the exact solution for the order of convergence, and simulate the fractional dynamical systems.
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