Search

A novel nonlinear fractional order damped Pantographs equation involving input delay is formulated. We explored the controllability of the underlying nonlinear damped dynamical Pantograph equation of fractional order with input delay. The solution of the problem under consideration has initially expressed in the form of the Mittag–Leffler function in two different domains. The controllability Grammians matrices for both cases have been defined. Controllability in the linear case has been shown and applied Schaefer’s fixed point theorem as well as the Arzela-Ascoli theorem to derive necessary conditions for the controllability of the nonlinear case.

In this paper, we investigate gross domestic product model with four different fractional derivatives. We find the solutions of fractional models by utilizing Sumudu transform. We prove the efficiency of the Sumudu transform by some theoretic outcomes and applications. We demonstrate the simulations by some figures. We show the agreement of different fractional derivatives on the model.

Here, we use the modified Kudryashov’s algorithm and addendum to Kudryashov’s technique to obtain new optical solitons of the Ginsburg–Landau fractional complex of non-linearity (CGLE) laws in Kerr, which demonstrate various phenomena in physics such as non-linear waves, the transformation of the secondary phase, superconductivity, surfaces, liquid crystals, and field-theory strings. Dark, bright, singular solitons and combo bright-singular solitons are retrieved. The modified Kudryashov’s algorithm provides dark, singular solitons, and combo bright-singular solitons, although the addendum to Kudryashov’s technique ensures bright and singular solitons. These solutions are practiced for three optional definitions of derivative viz. conformable derivative, beta derivative, and M-truncated. Also shown are graphic representations of the solutions obtained.

The nanofluid technology has added significant contributions towards heat exchange increase and decrease in energy loss. This article focuses, to analyze the heat and mass transfer for two distinct hybrid nanofluids, namely (Ag-CuO/NE), (Ag-CuO/SE), under the impact of magnetic field and Rosseland radiation over a rotating cone embedded in the porous medium. The set of non-linear partial differential equations is derived with help of boundary layer approximation. A suitable and appropriate similarity set is used to obtain a dimensionless set of equations which, are then converted into ordinary differential equations. The problem is tackled numerically with the BVP-4c technique, keeping tolerance at 10−06. The influence of flow study parameters is discussed with help of graphical and tabulated data. The study shows that (Ag-CuO/NE) hybrid nanofluid has produced better heat and mass transfer rates as compared (Ag-CuO/SE) hybrid nanofluid. Therefore, the study suggests more frequent use of (Ag-CuO/NE).

Up to day, the role of ultraviolet rays in producing cancers like melanoma has been considered extensively. However, quantum effects on the evolutions of melanoma has been discussed minorly. In this research, we propose a quantum model for the replication of DNAs within skin cells. We show that ultraviolet rays prevent of imaging information within genes of parent cells on genes of child cells and make some genetic errors in produced DNAs during replications. We compare the model with experimental data and obtain the growth rate of tumour on the skin.