Caputo Fractional Time Research Articles

Fractional fluctuation effects on the light scattered by a viscoelastic suspension

We generalize fluctuating hydrodynamics to study the effect of fractional time derivatives on the light-scattering spectrum of a suspension in a viscoelastic solvent under an external density gradient. Viscoelasticity introduces additional memory effects into the fluctuating hydrodynamic equations, causing the time scales associated with the mesoscopic variables and those of the microscopic events to be no longer well separated. This situation is taken into account by introducing Caputo's fractional time derivative into the description. The structure factor of the suspension is calculated, and we find that its nonequilibrium correction is an odd function of the frequency. It exhibits a shift towards negative frequencies proportional to the magnitude of the imposed gradient. We consider solvents that are described by Maxwell's or power-law rheological equations of state. The fractional structure factor is compared with the nonfractional one, and it is found that the ratio of the former to the latter may be positive and up to two orders of magnitude for both types of viscoelasticity. This prediction of our model calculation suggests that this relative change might be measurable.

The Homotopy Analysis Method for Fractional Cauchy Reaction-Diffusion Problems

In this article, homotopy analysis method is successfully applied to obtain the approximate analytical solutions of the characteristic Cauchy reaction-diffusion equation with fractional time derivative. The beauty of the article is the wonderful application of Caputo fractional order time derivative. The linear interactions of the merging populations are examined using perturbation theory and the method of matched asymptotic expansions. The solutions of the problem for different particular cases are presented graphically.