On fractional differential equations: the generalised Cattaneo equations

Abstract

The aim of this dissertation is to determine numerical solutions to fractional di usion and fractional Cattaneo equations using nite di erence formula and other de ned schemes. The spatial derivatives and time derivatives of integer order are approximated by a nite di erence approximation. Spatial derivatives of fractional order are approximated using the Gr unwald formula. Fractional time derivatives are approximated using the Gr unwald-Letnikov de nition of the Riemann-Liouville fractional derivative. The resulting di erence schemes are evaluated using Mathematica. The results obtained show that the fractional Cattaneo equaions have propagation and di usive properties. When the fractional exponent is 0:1 with the di usivity coe cient being greater than 0:1 one obtains numerical results that are unstable and display oscillatory behaviour. For other combinations of values, numerical results are stable and consistent with di usive behaviour.