Numerical simulation of the Riesz fractional diffusion equation with a nonlinear source term

In this paper, A Riesz fractional diffusion equation with a nonlinear source term (RFDE-NST) is considered. This equation is commonly used to model the growth and spreading of biological species. According to the equivalent of the Riemann-Liouville(R-L) and Gr¨unwald-Letnikov(GL) fractional derivative definitions, an implicit difference approximation (IFDA) for the RFDE-NST is derived. We prove the IFDA is unconditionally stable and convergent. In order to evaluate the efficiency of the IFDA, a comparison with a fractional method of lines (FMOL) is used. Finally, two numerical examples are presented to show that the numerical results are in good agreement with our theoretical analysis.

Participatory Learning in the Early Years : Research and Pedagogy

The early years are an important period for learning, but the questions surrounding participatory learning amongst toddlers remain under-examined. This book presents the latest theoretical and research perspectives about how ECEC (Early Childhood Education and Care) contexts promote democracy and citizenship through participatory learning approaches. The contributors provide insight into national policies, provisions, and practices and advance our understandings of theory and research on toddlers’ experiences for democratic participation across a number of countries, including the UK, Australia, New Zealand, the United States, Canada, Sweden, and Norway.