Vo Anh

New Solution and Analytical Techniques of the Implicit Numerical Method for the Anomalous Subdiffusion Equation

A physical-mathematical approach to anomalous diffusion is based on a generalized diffusion equation containing derivatives of fractional order. In this paper, an anomalous subdiffusion equation (ASub-DE) is considered. A new implicit numerical method (INM) and two solution techniques for improving the order of convergence of the INM for solving the ASub-DE are proposed. The stability and convergence of the INM are investigated by the energy method. Some numerical examples are given. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and supporting theoretical results can also be applied to other fractional integro-differential equations and higher-dimensional problems.

A Fourier method for the fractional diffusion equation describing sub-diffusion

In this paper, a fractional partial differential equation (FPDE) describing sub-diffusion 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.

Numerical Methods for the Variable-Order Fractional Advection-Diffusion Equation with a Nonlinear Source Term

In this paper, we consider a variable-order fractional advection-diffusion equation with a nonlinear source term on a finite domain. Explicit and implicit Euler approximations for the equation are proposed. Stability and convergence of the methods are discussed. Moveover, we also present a fractional method of lines, a matrix transfer technique, and an extrapolation method for the equation. Some numerical examples are given, and the results demonstrate the effectiveness of theoretical analysis.

Spectral Analysis of Fractional Kinetic Equations with Random Data

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.

Time fractional advection-dispersion equation

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<α<-1). Using variable transformation, Mellin and Laplace transforms, and properties of H-functions, we derive the complete solution of this time fractional advection-dispersion equation.

Numerical Schemes with High Spatial Accuracy for a Variable-Order Anomalous Subdiffusion Equation

In this paper, we consider a variable-order anomalous subdiffusion equation. A numerical scheme with first order temporal accuracy and fourth order spatial accuracy for the equation is proposed. The convergence, stability, and solvability of the numerical scheme are discussed via the technique of Fourier analysis. Another improved numerical scheme with second order temporal accuracy and fourth order spatial accuracy is also proposed. Some numerical examples are given, and the results demonstrate the effectiveness of theoretical analysis.

Possible long-range dependence in fractional random fields ☆

Abstract Many existing stochastic models have been developed for description and analysis of Markov diffusion. This paper outlines the new concept of α -duality which lays the foundation for an extension of Markov diffusion to fractional diffusion. The theory of Riesz and Bessel potentials and the corresponding potential spaces play a key role in this new approach. We establish the existence of an important subclass of fractional random fields, namely that of fractional Riesz–Bessel motions, which extends the class of fractional Brownian motions. As a result, the scope of Markov diffusion is widened to cover random fields with long-range dependence.

Approximation of the Lévy-Feller advection-dispersion process by random walk and finite difference method

In this paper we present a random walk model for approximating a Levy-Feller advection-dispersion process, governed by the Levy-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 Grunwald-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.

A CRANK-NICOLSON ADI SPECTRAL METHOD FOR A TWO-DIMENSIONAL RIESZ SPACE FRACTIONAL NONLINEAR REACTION-DIFFUSION EQUATION ∗

In this paper, a new alternating direction implicit Galerkin--Legendre spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation is developed. The temporal component is discretized by the Crank--Nicolson method. The detailed implementation of the method is presented. The stability and convergence analysis is strictly proven, which shows that the derived method is stable and convergent of order 2 in time. An optimal error estimate in space is also obtained by introducing a new orthogonal projector. The present method is extended to solve the fractional FitzHugh--Nagumo model. Numerical results are provided to verify the theoretical analysis.

Prediction of protein structural classes by recurrence quantification analysis based on chaos game representation

In this paper, we intend to predict protein structural classes (alpha, beta, alpha+beta, or alpha/beta) for low-homology data sets. Two data sets were used widely, 1189 (containing 1092 proteins) and 25PDB (containing 1673 proteins) with sequence homology being 40% and 25%, respectively. We propose to decompose the chaos game representation of proteins into two kinds of time series. Then, a novel and powerful nonlinear analysis technique, recurrence quantification analysis (RQA), is applied to analyze these time series. For a given protein sequence, a total of 16 characteristic parameters can be calculated with RQA, which are treated as feature representation of protein sequences. Based on such feature representation, the structural class for each protein is predicted with Fishers linear discriminant algorithm. The jackknife test is used to test and compare our method with other existing methods. The overall accuracies with step-by-step procedure are 65.8% and 64.2% for 1189 and 25PDB data sets, respectively. With one-against-others procedure used widely, we compare our method with five other existing methods. Especially, the overall accuracies of our method are 6.3% and 4.1% higher for the two data sets, respectively. Furthermore, only 16 parameters are used in our method, which is less than that used by other methods. This suggests that the current method may play a complementary role to the existing methods and is promising to perform the prediction of protein structural classes.