On a Differential Equation with Left and Right Fractional Derivatives

Mathematics Subject Classification: 26A33; 70H03, 70H25, 70S05; 49S05

On a Stochastic Partial Differential Equation with a Noisy Term

2000 Mathematics Subject Classification: 60H15, 60H40

Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term

In this paper, we consider a modified anomalous subdiffusion equation with a nonlinear source term for describing processes that become less anomalous as time progresses by the inclusion of a second fractional time derivative acting on the diffusion term. A new implicit difference method is constructed. The stability and convergence are discussed using a new energy method. Finally, some numerical examples are given. The numerical results demonstrate the effectiveness of theoretical analysis

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. Moreover, 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.

Stability and convergence of an effective numerical method for the time-space fractional Fokker-Planck equation with a nonlinear source term

Fractional Fokker-Planck equations (FFPEs) have gained much interest recently for describing transport dynamics in complex systems that are governed by anomalous diffusion and nonexponential relaxation patterns. However, effective numerical methods and analytic techniques for the FFPE are still in their embryonic state. In this paper, we consider a class of time-space fractional Fokker-Planck equations with a nonlinear source term (TSFFPE-NST), which involve the Caputo time fractional derivative (CTFD) of order α ∈ (0, 1) and the symmetric Riesz space fractional derivative (RSFD) of order μ ∈ (1, 2). Approximating the CTFD and RSFD using the L1-algorithm and shifted Grunwald method, respectively, a computationally effective numerical method is presented to solve the TSFFPE-NST. The stability and convergence of the proposed numerical method are investigated. Finally, numerical experiments are carried out to support the theoretical claims.

Drug diffusion from polymeric delivery devices : a problem with two moving boundaries

An existing model for solvent penetration and drug release from a spherically-shaped polymeric drug delivery device is revisited. The model has two moving boundaries, one that describes the interface between the glassy and rubbery states of polymer, and another that defines the interface between the polymer ball and the pool of solvent. The model is extended so that the nonlinear diffusion coefficient of drug explicitly depends on the concentration of solvent, and the resulting equations are solved numerically using a front-fixing transformation together with a finite difference spatial discretisation and the method of lines. We present evidence that our scheme is much more accurate than a previous scheme. Asymptotic results in the small-time limit are presented, which show how the use of a kinetic law as a boundary condition on the innermost moving boundary dictates qualitative behaviour, the scalings being very different to the similar moving boundary problem that arises from modelling the melting of an ice ball. The implication is that the model considered here exhibits what is referred to as ``non-Fickian'' or Case II diffusion which, together with the initially constant rate of drug release, has certain appeal from a pharmaceutical perspective.

Numerical simulation for the variable-order Galilei invariant advection diffusion equation with a nonlinear source term

In this paper, we consider the variable-order Galilei advection diffusion equation with a nonlinear source term. A numerical scheme with first order temporal accuracy and second order spatial accuracy is developed to simulate the equation. The stability and convergence of the numerical scheme are analyzed. Besides, another numerical scheme for improving temporal accuracy is also developed. Finally, some numerical examples are given and the results demonstrate the effectiveness of theoretical analysis. Keywords: The variable-order Galilei invariant advection diffusion equation with a nonlinear source term; The variable-order Riemann–Liouville fractional partial derivative; Stability; Convergence; Numerical scheme improving temporal accuracy

Industrialized housing in China : a coin with two sides

China today is experiencing a time when housing is needed more than ever and one approach satisfying this need is by industrialization - a streamlined process aimed at generating profits and promoting energy efficiency in the housing sectors. Although large housing programs have been completed in China, few housing projects have been built in an industrialized manner. One contributing factor is that industrialization is not omnipotent and, just as a coin has two sides, not all the outcomes of industrialization are beneficial. In this paper, a preliminary assessment is made of these two sides - the benefits and hindrances of industrialized housing in China - by literature review and survey. Case studies are used to verify the questionnaire survey results and from which the advantages and disadvantages involved are compared. The findings indicate the need for formulating policies to encourage industrialized housing in China and for well-planned R&D themes to be implemented simultaneously with industry practices in the near future.

A revision of Calyptochloa C.E.Hubb. (Poaceae), with two new species and a new subspecies

Thompson, E.J. & Simon, B.K. (2012). A revision of Calyptochloa C.E.Hubb. (Poaceae), with two new species and a new subspecies. Austrobaileya 8(4): 634–652. Two new species of Calyptochloa C.E.Hubb. (Calyptochloa cylindrosperma E.J.Thomps. & B.K.Simon and C. johnsoniana E.J.Thomps. & B.K.Simon) endemic to central Queensland, and a new subspecies of Calyptochloa gracillima C.E.Hubb. (C. gracillima subsp. ipsviciensis E.J.Thomps. & B.K.Simon) endemic to southeast Queensland are described and illustrated.