CONVERGENCE OF THE MATRIX TRANSFORMATION METHOD FOR THE FINITE DIFFERENCE APPROXIMATION OF FRACTIONAL ORDER DIFFUSION PROBLEMS

Numerical solution of fractional order diffusion problems with homogeneous Dirichlet boundary conditions is investigated on a square domain. An appropriate extension is applied to have a well-posed problem on R 2 and the solution on the square is regarded as a localization. For the numerical approximation a finite difference method is applied combined with the matrix transformation method. Here the discrete fractional Laplacian is approximated with a matrix power instead of computing the complicated approxima- tions of fractional order derivatives. The spatial convergence of this method is proved and demonstrated in some numerical experiments

Models of space-fractional diffusion: a critical review

Space-fractional diffusion problems are investigated from the modeling point of view. It is pointed out that the elementwise power of the Laplacian operator in R n is an inadequate model of fractional diffusion. Also, the approach with fractional calculus using zero extension is not a proper model of homogeneous Dirichlet boundary conditions. At the time, the spectral definition of the fractional Dirichlet Laplacian seems to be in many aspects a proper model of fractional diffusion

A finite difference method for fractional diffusion equations with Neumann boundary conditions

A finite difference numerical method is investigated for fractional order diffusion problems in one space dimension. The basis of the mathematical model and the numerical approximation is an appropriate extension of the initial values, which incorporates homogeneous Dirichlet or Neumann type boundary conditions. The well-posedness of the obtained initial value problem is proved and it is pointed out that each extensions is compatible with the original boundary conditions. Accordingly, a finite difference scheme is constructed for the Neumann problem using the shifted Grünwald--Letnikov approximation of the fractional order derivatives, which is based on infinite many basis points. The corresponding matrix is expressed in a closed form and the convergence of an appropriate implicit Euler scheme is proved

Interaction Of As(III) with Thiolate-Containing Molecules

The aqueous solutions of arsenous acid with thiolate containing organic ligands such as the meso and racemic forms of 1,4-dithiol-butane-2,3-diol, (dithioerythritol – dte and dithiothreitol - dtt) as well as 2,3-dimercaptopropanol (called also British anti-Lewisite (BAL) or Dimercaprol) were investigated. pH-mertric titrations were performed in solutions with different molar ratios of As(III) and the ligands. The pKa values for As(OH)3, and the ligands determined from these data were in good agreement with the literature data. In all investigated systems containing both As(OH)3 and one of the ligands, the deprotonation steps appeared at a higher pH in the titration curves, than in those of the individual components. This unusual observation was explained by the condensation reactions between the reagents taking place in the pH < 8 range. In some of these systems the pH-metry was combined with NMR and UV spectroscopic measurements. We observed the complexes with 1:1 As(III):ligand composition as being the major species in aqueous solutions. In the case of As(III)-dte system we could crystallize the complex of 1:1 composition from ethanolic solution