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Transient Space-fractional Diffusion with a Power-law Superdiffusivity : Approximate Integral-balance Approach

Hristov J.

Fundamenta Informaticae

2017

Vol. 151, nr 1/4

371--388

This paper focuses on an approximate analytical solution of an initial-boundary value problem of spatial-fractional partial differential diffusion equation with RiemannLiouville fractional derivative in space. The spatial correlation of the superdiffusion coefficient as a power-law has been discussed in cases of fast and slow spatial superdiffusion. Approximate closed form solutions in terms of non-linear similarity variable are based on the integral-balance method and series expansion of the assumed parabolic profile with undefined exponent. The law of the spatial and temporal propagation of the solution was the primary issue and discussed in two cases: fast and slow superdiffussion.

Lifetime of a soluble solid particle in a stagnant medium: approximate analytical modelling involving fractional (half-time) derivatives

Hristov J.

Polish Journal of Chemical Technology

2013

Vol. 15, nr 3

74-77

Approximate analytical solutions concerning lifetime of soluble solid particles in an unbounded stagnant medium have been developed by simple application of fractional half-time derivative in the Riemann-Liouville sense to express the relationship between the net surface mass flux and the concentration at the interface. The solutions start with the initial formulation of Rice and Do on the time-depletion of the radius of a spherical particle expressed through terms including the solubility parameter as the only key parameter controlling the process of dissolution. The two approximate developed solutions use different scaling and dimensionless variables: The 1st solution is developed by an introduction of a similarity variable [xxx] while the 2nd solution applies the classical scaling using the initial sphere radius as a length scale that leads to dimensionless radius r = R/R0 and time τ = Dt/R02. Both solutions provide approximate relationships close to that of Rice and Do.