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The aesthetic patterns play a key role in the field of fractals. Due to self similarities in the nature of fractals, researchers used the fractals in many fields of sciences (i.e. in Mathematics, Computer Science, Physics, Image Encryption, Biology and Chemistry). The most studied fractals types are the Mandelbrot sets (MSs) and Julia sets (JSs). To generate fractals, escape criteria is required. In this work, a general escape criteria is proved via extended Jungck-Noor iteration with s -convexity. These results are used in algorithms to present the generation of fractals in extended Jungck-Noor orbit for general complex polynomial f (x) = ∑ i = 0 p a i x i with p ≥ 2 , where a i ∈ ℂ for i = 0 , 1 , 2 , ... , p. The graphics of MSs and JSs are demonstrated in the examples. The variations in MSs and JSs for different values of involved parameters are also shown. [ABSTRACT FROM AUTHOR]

In this exploration, transportation of heat along with homogeneous–heterogeneous reactions for the peristaltic transport phenomenon by considering Ellis fluid model is reported in a symmetric channel comprising compliant walls. Using low Reynolds number and long wavelength assumptions, the governing system of equations of fluid transport problem is abridged and solved using the perturbation technique. The skin friction coefficient and Nusselt number are also incorporated in this contemplation. Particular attention is given to elastic parameters and Brinkman number and plotted their graphs for temperature, concentration, and velocity distribution. Results are deliberated in detail for the various non-Newtonian and Newtonian models. It is initiated that velocity distribution is enhanced for higher values of elastic parameters, which is because less resistance occurs at the channel walls. Relatively reverse comportment is portrayed in the case of concentration profile. The impact of heterogeneous and homogeneous reactions shows reverse bearing on the concentration field. It is comprehended that Brinkman number is an increasing function of viscous dissipation effects, which contribute to enhancing the temperature field for Newtonian and non-Newtonian cases. Moreover, augmentation in skin friction is noticed by escalating the parametric values of damping and rigidity. It is worth mentioning that, fluid velocity grows fastly for the Newtonian model case as compared with the non-Newtonian case by mounting the values of the elastic parameter. Moreover, the temperature field upsurges fastly in a non-Newtonian case as compared with the Newtonian case. The current investigation has an extensive range of use in biomedical science i.e. electromagnetic peristaltic micropumps.

The emergence of Atangana–Baleanu fractional derivative in solving various physical phenomenon is extraordinarily beneficial and provide results that are more realistic. Therefore, this article focused on the application of this fractional operator to magnetohydrodynamic (MHD) movement of second-grade fluid in an inclined heated plate with an inclined magnetic field. The problem is constructed in terms of fractional PDE's subject to physical initial and boundary conditions. The solutions are obtained by the joint application of Laplace and Zakian's numerical algorithm. To explore the effect of various flow parameters, the solutions are plotted in graphs and discussed physically. It is found that the fluid velocity decreases with an increasing fractional parameter α. For α = 1 , the fluid velocity is minimum. Furthermore, it is explored that the strength of the magnetic field is strongest for the inclination angle β = π 2. [ABSTRACT FROM AUTHOR]

The role of iterative algorithms is vital in exploring the diverse domains of science and has proven to be a powerful tool for solving complex computational problems in the most trending branches of computer science. Taking motivation from this fact, we develop and apply a modified four-step iterative algorithm to solve the fixed point problem in the Hadamard spaces using a total asymptotic nonexpansive mapping. MATLAB R2018b is used for numerical experiments to ensure a better convergence rate of the proposed iterative algorithm with existing results.

In this letter, energy transfer of Jeffery–Hamel nanofluid flow in non-parallel walls is investigated analytically using Galerkin method. The effective thermal conductivity and viscosity of nanofluid are calculated by the Maxwell–Garnetts (MG) and Brinkman models, respectively. The influence of the nanofluid volume friction, Reynolds number and angle of the channel on velocity and temperature profiles are investigated. Results show that Nusselt number increases with increase of Reynolds number and nanoparticle volume friction. Also it can be found that skin friction coefficient is an increasing function of Reynolds number, opening angle and nanoparticle volume friction.

In this work, a generalized unsteady magnetohydrodynamic transport of a rate type fluid near an unbounded upright plate is analyzed under Newtonian heating and non-uniform velocity conditions. The vertical plate is suspended in a porous medium and is encountering the radiation effects. Three different fractional models for Maxwell fluid are established by using the modern definitions of Atangana–Baleanu, Caputo–Fabrizio, and Caputo fractional operators. Triple fractional analysis is conducted to reach out to the solutions of consequent flow and energy equations. Laplace transform and Stehfest's numerical algorithm are jointly applied to solve each fractional model. Shear stress and heat transfer rate are measured at the solid–fluid interface in the form of skin friction coefficient and Nusselt number respectively. The physical significance of associated parameters in velocity and energy boundary layers is investigated and graphs are provided to discuss the relevant physical arguments. A tabular analysis is performed to demonstrate the influence of physical parameters on shear stress and heat transfer rate. An empirical comparison between fractional and classical solutions indicates that fractional operators provide a better explanation of the physical features of the model. It is also analyzed that for Newtonian heating and non-uniform velocity conditions, the Atangana–Baleanu fractional operator is the finest fractional model to describe the memory effect of velocity and energy distribution. The velocity of a Maxwell fluid is always higher for constant velocity condition as compared to the ramped velocity condition. Furthermore, the heat transfer rate declines for increasing values of the fractional parameter α and a minimum value is witnessed for the classical model but, it follows an inverse trend when the radiation parameter R d increases. [ABSTRACT FROM AUTHOR]

Mittag-Leffler function is a natural generalization of the exponential function. Recent applications of Mittag-Leffler function have reshaped the scientific literature due to its fractional effects that cannot be obtained by using exponential function. Present motivation is to define a new special function by modification in the original gamma function with Mittag-Leffler function. Properties of this modified function are discussed by investigating a new series representation involving delta function. Hence, the results are also validated with the earlier obtained results for gamma function as special cases. Furthermore, the new function is used to generate a probability density function, and its statistical properties are explored. Similar properties of existing distributions can be deduced.

• Brinkman and Maxwell–Garnetts (MG) representations have been used to observe the strength of two important properties namely; viscosity and thermal conductivity of under consideration fluid. • The movement of fluid has demonstrated in the form of partial differential equations by means of physical circumstances in view of cosine oscillations of the plate. • Expressions for velocity and temperature fields have been obtained explicitly by making use of Laplace transform. Zakian's explicit formula has been exploited to testify the results by inversion of Laplace transform. • All the involved parameters for velocity and temperature profiles have been analyzed in view of their graphical representations. • Results of Nusselt number have been computed in tabular form. The results showed that a rise in nanoparticle volume fraction leads to heat transfer enrichment. In this work, a new idea of Atangana-Baleanu fractional derivative has been applied to study heat transfer due to free convection in non-Newtonian nanofluids over an infinite vertical plate. Sodium alginate (SA-NaAlg) has been taken as non-Newtonian base fluid and a suspension of molybdenum disulphide (MoS2) nanoparticles has been used in it. Brinkman and Maxwell–Garnetts (MG) representations have been applied to observe the strength of two important properties namely; viscosity and thermal conductivity. The problem is modeled in the form of partial differential equations with imposed conditions. The plate transfers a cosine type oscillations to the fluid with a constant temperature. Expressions for velocity and temperature fields have been obtained explicitly by making use of the Laplace transform technique. Zakian's explicit formula has been used to testify the results by inversion of Laplace transform. All the involved parameters for velocity and temperature profiles have been analyzed because of their graphical representations. Results of Nusselt number have been computed in tabular form. The results showed that a rise in nanoparticle volume fraction leads to heat transfer enrichment. [ABSTRACT FROM AUTHOR]