Abstract
We depart from the well-known one-dimensional Fisher’s equation from population dynamics and consider an extension of this model using Riesz fractional derivatives in space. Positive and bounded initial-boundary data are imposed on a closed and bounded domain, and a fully discrete form of this fractional initial-boundary-value problem is provided next using fractional centered differences. The fully discrete population model is implicit and linear, so a convenient vector representation is readily derived. Under suitable conditions, the matrix representing the implicit problem is an inverse-positive matrix. Using this fact, we establish that the discrete population model is capable of preserving the positivity and the boundedness of the discrete initial-boundary conditions. Moreover, the computational solubility of the discrete model is tackled in the closing remarks.
Highlights
The development in recent decades of fractional calculus has led to important discoveries in many scientific areas [1, 2]
Research in the physical sciences has developed toward the construction of a physically meaningful calculus of variations for fractional systems [3, 4]
The model is an extension of Fisher’s equation from population dynamics in which the diffusion term is expressed as a Riesz fractional derivative
Summary
The development in recent decades of fractional calculus has led to important discoveries in many scientific areas [1, 2]. In the field of differential/difference equations, the determination of theorems on the existence and the uniqueness of solutions of fractional systems is nowadays an area of analytic importance [6] In this context, the determination of the properties of the relevant solutions of fractional systems is a transited avenue of research, with the conditions of positivity and boundedness being of particular interest [7]. The model is an extension of Fisher’s equation from population dynamics in which the diffusion term is expressed as a Riesz fractional derivative Both the continuous and the discrete Riesz fractional operators are presented in that section, together with some useful lemmas quoted from the standard literature. This note closes with a section of concluding remarks and directions of future research
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