Fractional Calculus in Population Dynamics

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Ashlin Powell Harris
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Middle Tennessee State University
Research in recent decades has incorporated fractional derivatives into partial differential equation models of natural phenomena. This generalisation to a non-integer order provides a way to describe anomalous diffusion within fractal spaces. However, most numerical methods developed for the integer order are not suited for efficient computation of these systems. In this work, we develop a method to numerically solve a multi-component and multidimensional space-fractional system. For space discretization, we apply a Fourier spectral method that is suited for multidimensional systems. Efficient approximation of time-stepping is accomplished with an exponential time differencing approach. We consider the convergence and stability of the methods and observe the effect of different fractional parameters. While the scope of this research is limited to the dynamics of biological systems, these same techniques may be applied to other disciplines.
A Dissertation Submitted in Partial Fullment of the Requirements for the Degree of Doctor of Computational Science