Analytical solution for the dynamics and optimization of fractional Klein-Gordon equation: an application to quantum particle.

Klein-Gordon equation characterizes spin-particles through neutral charge field within quantum particle. In this context, fractionalized Klein-Gordon equation is investigated for the comparative analysis of the newly presented fractional differential techniques with non-singularity among kernels. The non-singular and non-local kernels of fractional differentiations have been employed on Klein-Gordon equation for the development of governing equation. The analytical solutions of Klein-Gordon equation have been traced out by fractional techniques by means of Laplace transforms and expressed in terms of series form and gamma function. The data analysis of fractionalized Klein-Gordon equation is observed for Pearson's correlation coefficient, probable error and regression analysis. For the sake of comparative analysis of fractional techniques, 2D sketch, 3D pie chart, contour surface with projection and 3D bar sketch have been depicted on the basis of embedded parameters. Our results suggest that varying frequency has reversal trends for quantum wave and de Broglie wave.

Role of copper and alumina for heat transfer in hybrid nanofluid by using Fourier sine transform.

The convection, thermal conductivity, and heat transfer of hybrid nanofluid through nanoparticles has become integral part of several natural and industrial processes. In this manuscript, a new fractionalized model based on hybrid nanofluid is proposed and investigated by employing singular verses and non-singular kernels. The mathematical modeling of hybrid nanofluid is handled via modern fractional definitions of differentiations. The combined Laplace and Fourier Sine transforms have been configurated on the governing equations of hybrid nanofluid. The analytical expression of the governing temperature and velocity equations of hybrid nanofluid have been solved via special functions. For the sake of thermal performance, dimensional analysis of governing equations and suitable boundary conditions based on Mittage-Leffler function have been invoked for the first time in literature. The comparative analysis of heat transfer from hybrid nanofluid has been observed through Caputo-Fabrizio and Atangana-Baleanu differential operators. Finally, our results suggest that volume fraction has the decelerated and accelerated trends of temperature distribution and inclined and declined profile of heat transfer is observed copper and alumina nanoparticles.