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Within the context of fractional calculus, we investigate novel mathematical possibilities. In this context, using the fractional dispersion relations for the fractional wave equation, we explore a class of the generalized fractional wave equation numerically. Some important classes of differential equations in the theory of wave studies are Drinfeld–Sokolov–Wilson and Shallow Water equations. In this effort, the natural transform decomposition technique has been implemented to investigate the explicit result of fractional-order coupled schemes of Drinfeld–Sokolov–Wilson and Shallow Water coupled systems. The proposed method is obtained by coupling the Natural transform with the Adomian decomposition process. The current technique significantly works to find the approximate solution without any discretization or constraining parameter assumptions. The obtained numerical and graphical outcomes by the devised technique are compared with the available exact result to verify the convergence of the method. For mathematical calculations, the Mathematica software package is used. [ABSTRACT FROM AUTHOR]

The study of nonlinear phenomena associated with physical phenomena is a hot topic in the present era. The fundamental aim of this paper is to find the iterative solution for generalized quintic complex Ginzburg–Landau (GCGL) equation using fractional natural decomposition method (FNDM) within the frame of fractional calculus. We consider the projected equations by incorporating the Caputo fractional operator and investigate two examples for different initial values to present the efficiency and applicability of the FNDM. We presented the nature of the obtained results defined in three distinct cases and illustrated with the help of surfaces and contour plots for the particular value with respect to fractional order. Moreover, to present the accuracy and capture the nature of the obtained results, we present plots with different fractional order, and these plots show the essence of incorporating the fractional concept into the system exemplifying nonlinear complex phenomena. The present investigation confirms the efficiency and applicability of the considered method and fractional operators while analyzing phenomena in science and technology. [ABSTRACT FROM AUTHOR]

We start by presenting a brief summary of fractional quantum mechanics, as means to convey a motivation towards fractional quantum cosmology. Subsequently, such application is made concrete with the assistance of a case study. Specifically, we investigate and then discuss a model of stiff matter in a spatially flat homogeneous and isotropic universe. A new quantum cosmological solution, where fractional calculus implications are explicit, is presented and then contrasted with the corresponding standard quantum cosmology setting. [ABSTRACT FROM AUTHOR]

Bees are the main contributors of pollination and it is predicted that disruption of pollination causes some serious problems in economics, agriculture and ecology. It has been reported that honeybee colony collapse has increased dramatically for some time in different parts of the world. In order to investigate the causes of colony collapse, we establish a fractional multi-order honeybee colony population model. Two different models with Caputo and Caputo–Fabrizio differentiation operators have been compared. We justify the existence and uniqueness of the solution. Stability analysis has been presented. We find the numerical values of the fractional order ( p , q ) so that the numerical results are close to the experimental findings more than the integer-order case. [ABSTRACT FROM AUTHOR]

This paper deals with the study of heat transfer and thermal damage in triple layer skin tissue using fractional bioheat model. Here, we consider three types of heating viz. sinusoidal heat flux, constant temperature and constant heat flux heating on skin surface. An implicit finite difference scheme is obtained by approximating fractional time derivative by quadrature formula and space derivative by central difference formula. The temperature profiles and thermal damage in the skin tissue are obtained to study the effect of fractional parameter on diffusion process for constant temperature and heat flux boundary heating on skin surface. A parametric study for sinusoidal heat flux at skin surface has also been made. [ABSTRACT FROM AUTHOR]

In this paper, we have developed a fractional hyperbolic bioheat transfer (FHBHT) model by applying fractional Taylor series formula to the single-phase-lag constitutive relation. A new hybrid numerical scheme that combines the multi-resolution and multi-scale computational property of Legendre wavelets based on fractional operational matrix has been used to find the numerical solution of the present problem. This study demonstrates that FHBHT model can provide a unified approach for analyzing heat transfer within living biological tissues, as standard hyperbolic bioheat transfer (SHBHT) and Pennes models are particular cases of FHBHT model. The effect of phase lag time and order of fractional derivative on temperature distribution within living biological tissues for both SHBHT and FHBHT models have been studied and shown graphically. It has been observed that thermal signal propagates more easily with larger values of order of fractional derivative within living biological tissues. The time interval for achieving temperature range of thermal treatment for different models have been studied and compared. It is least for Pennes model, highest for FHBHT model and in between them for SHBHT model. The whole analysis is presented in dimensionless form. [ABSTRACT FROM AUTHOR]

In this paper, we present an analysis and modeling of the electrical diffusion equation using the fractional calculus approach. This alternative representation for the current density is expressed in terms of the Caputo derivatives, the order for the space domain is and for the time domain is . We present solutions for the full fractional equation involving space and time fractional derivatives using numerical methods based on Fourier variable separation. The case with spatial fractional derivatives leads to Levy flight type phenomena, while the time fractional equation is related to sub- or super diffusion. We show that the mathematical concept of fractional derivatives can be useful to understand the behavior of semiconductors, the design of solar panels, electrochemical phenomena and the description of anomalous complex processes. [ABSTRACT FROM AUTHOR]