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Discrete fractional calculus is suggested in diffusion modeling in porous media. A variable-order fractional diffusion equation is proposed on discrete time scales. A function of the variable order is constructed by a chaotic map. The model shows some new random behaviors in comparison with other variable-order cases. [ABSTRACT FROM AUTHOR]

The fundamental description of relaxation (T1 and T2) in nuclear magnetic resonance (NMR) is provided by the Bloch equation, an integer-order ordinary differential equation that interrelates precession of magnetization with time- and space-dependent relaxation. In this paper, we propose a fractional order Bloch equation that includes an extended model of time delays. The fractional time derivative embeds in the Bloch equation a fading power law form of system memory while the time delay averages the present value of magnetization with an earlier one. The analysis shows different patterns in the stability behavior for T1 and T2 relaxation. The T1 decay is stable for the range of delays tested (1 μsec to 200 μsec), while the T2 relaxation in this extended model exhibits a critical delay (typically 100 μsec to 200 μsec) above which the system is unstable. Delays arise in NMR in both the system model and in the signal excitation and detection processes. Therefore, by adding extended time delay to the fractional derivative model for the Bloch equation, we believe that we can develop a more appropriate model for NMR resonance and relaxation. [ABSTRACT FROM AUTHOR]

In this paper, we apply the Mickens nonstandard discretization method to solve a class of initial-boundary value fractional partial differential equations with variable coefficients on a finite domain, and thereby increase the accuracy of the solutions. We examine the case when a left-handed and a right-handed fractional spatial derivative may be present in the partial differential equation. Two numerical examples using this method are presented and compared successfully with the exact analytical solutions. [ABSTRACT FROM AUTHOR]