A second-order accurate numerical approximation for the Riesz space fractional advection-dispersion equation

In this paper, we consider a space Riesz fractional advection-dispersion equation. The equation is obtained from the standard advection-diffusion equation by replacing the ¯rst-order and second-order space derivatives by the Riesz fractional derivatives of order β 1 Є (0; 1) and β2 Є(1; 2], respectively. Riesz fractional advection and dispersion terms are approximated by using two fractional centered difference schemes, respectively. A new weighted Riesz fractional ¯nite difference approximation scheme is proposed. When the weighting factor Ѳ = 1/2, a second- order accurate numerical approximation scheme for the Riesz fractional advection-dispersion equation is obtained. Stability, consistency and convergence of the numerical approximation scheme are discussed. A numerical example is given to show that the numerical results are in good agreement with our theoretical analysis.

Numerical analysis of the time variable fractional order mobile-immobile advection-dispersion model

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 time variable fractional order mobile-immobile advection-dispersion model. Numerical methods and analyses of stability and convergence for the fractional partial differential equations are quite limited and difficult to derive. This motivates us to develop efficient numerical methods as well as stability and convergence of the implicit numerical methods for the fractional order mobile immobile advection-dispersion model. In the paper, we use the Coimbra variable time fractional derivative which is more efficient from the numerical standpoint and is preferable for modeling dynamical systems. An implicit Euler approximation for the equation is proposed and then the stability of the approximation are investigated. As for the convergence of the numerical scheme we only consider a special case, i.e. the time fractional derivative is independent of time variable t. The case where the time fractional derivative depends both the time variable t and the space variable x will be considered in the future work. Finally, numerical examples are provided to show that the implicit Euler approximation is computationally efficient.

Numerical techniques for the variable order time fractional diffusion equation

In this paper we consider the variable order time fractional diffusion equation. We adopt the Coimbra variable order (VO) time fractional operator, which defines a consistent method for VO differentiation of physical variables. The Coimbra variable order fractional operator also can be viewed as a Caputo-type definition. Although this definition is the most appropriate definition having fundamental characteristics that are desirable for physical modeling, numerical methods for fractional partial differential equations using this definition have not yet appeared in the literature. Here an approximate scheme is first proposed. The stability, convergence and solvability of this numerical scheme are discussed via the technique of Fourier analysis. Numerical examples are provided to show that the numerical method is computationally efficient. Crown Copyright © 2012 Published by Elsevier Inc. All rights reserved.

Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation

Anomalous subdiffusion equations have in recent years received much attention. In this paper, we consider a two-dimensional variable-order anomalous subdiffusion equation. Two numerical methods (the implicit and explicit methods) are developed to solve the equation. Their stability, convergence and solvability are investigated by the Fourier method. Moreover, the effectiveness of our theoretical analysis is demonstrated by some numerical examples. © 2011 American Mathematical Society.

Association between prostinogen (KLK15) genetic variants and prostate cancer risk and aggressiveness in Australia and a meta-analysis of GWAS data

Background: Kallikrein 15 (KLK15)/Prostinogen is a plausible candidate for prostate cancer susceptibility. Elevated KLK15 expression has been reported in prostate cancer and it has been described as an unfavorable prognostic marker for the disease. Objectives: We performed a comprehensive analysis of association of variants in the KLK15 gene with prostate cancer risk and aggressiveness by genotyping tagSNPs, as well as putative functional SNPs identified by extensive bioinformatics analysis. Methods and Data Sources: Twelve out of 22 SNPs, selected on the basis of linkage disequilibrium pattern, were analyzed in an Australian sample of 1,011 histologically verified prostate cancer cases and 1,405 ethnically matched controls. Replication was sought from two existing genome wide association studies (GWAS): the Cancer Genetic Markers of Susceptibility (CGEMS) project and a UK GWAS study. Results: Two KLK15 SNPs, rs2659053 and rs3745522, showed evidence of association (p, 0.05) but were not present on the GWAS platforms. KLK15 SNP rs2659056 was found to be associated with prostate cancer aggressiveness and showed evidence of association in a replication cohort of 5,051 patients from the UK, Australia, and the CGEMS dataset of US samples. A highly significant association with Gleason score was observed when the data was combined from these three studies with an Odds Ratio (OR) of 0.85 (95% CI = 0.77-0.93; p = 2.7610 24). The rs2659056 SNP is predicted to alter binding of the RORalpha transcription factor, which has a role in the control of cell growth and differentiation and has been suggested to control the metastatic behavior of prostate cancer cells. Conclusions: Our findings suggest a role for KLK15 genetic variation in the etiology of prostate cancer among men of European ancestry, although further studies in very large sample sets are necessary to confirm effect sizes.