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The current research studies a fractal Duffing oscillator in the presence of periodic force. To find an analytic solution for this oscillator, the aspects explained in the following are considered. First, we obtain an alternative unforced fractal fourthorder equation and then convert it into a continuous space. Therefore, the nonperturbative (NP) approach is used to calculate the analytic solution for the alternate equation in the second-order form after reducing its rank. It is seen that the analytical and numerical solutions agree very well. The computations reveal that for every value of the fraction parameter, the approximation and numerical solutions are identical. The present study gives reliability in the technique of reducing the order of differential equations. Furthermore, the required periodic solution is also obtained by Galerkin’s technique. In contrast to the traditional technique, which works to transform the variable and is valid only in the absence of external forces, if there is an external force, it leads to significant mathematical difficulties. The current technique works on the operator, which is simple and effective when investigating fractal oscillators with external forces, easy to obtain analytic solutions, and doesn't lead to any mathematical difficulties. [ABSTRACT FROM AUTHOR]

The present work studies the bio-convection movement of an incompressible non-Newtonian Sutterby liquid on a stretchable surface in the existence of both nanomaterials and microbes. The liquid flows throughout a leaky region and is affected by a homogeneous vertical magnetic field. The energy spread is established by Ohmic and non-Newtonian dissipations in addition to an exponentially-space heat source, while the nanomaterials transmission is reachable with chemical reaction. The physical configuration is covered by force, temperature, nano-volume fractions, and microbes’ formulae together with the appropriate border criteria. The novel aspect of this work is motivated due to its consideration of the exponential distribution of viscosity with temperature, concentration of microorganisms, and nanoparticles. Furthermore, the involvement of microbes in the flow across a stretched surface added another innovative feature, in light of its wide application range. The foremost format of nonlinear partial differential formulae is transformed into ordinary ones providing suitable match converters. These formulae are scrutinized by the fourth-order Runge-Kutta numerical technique with the support of shooting criteria. Consequently, arithmetical and graphical foundations of the objective distributions are achieved. The conclusions are examined, and significant results are summarized. Several important physiognomies are completed from the outcomes. The heat profile improves the efficient factors, which is an excellent rule that may be employed in various implications. Microbes’ accumulation is found to increase with the increase of viscosity variation, whereas it decreases with the growth of Peclet, Lewis numbers, and the bio convection constants. Such findings may be useful in expecting the behavior of these microscopic organisms through similar flows. [ABSTRACT FROM AUTHOR]

The purpose of this study is to look at how gold nanoparticles affect the circulation near wavy biological cell walls. Nonlinear thermal radiation was found to enhance the heat transfer rates of nanofluid flow by numerical calculations. The mathematical model was a temporally magnetized non-Newtonian Casson micropolar nanofluid flow through a heated vertical wavy surface. The importance of predicting heat and mass transfer for irregular surfaces cannot be overstated, as irregular surfaces are common in many applications, including refrigerator condensers and flat-plate solar collectors. For this reason, it is imperative to study heat and mass transfer in complex geometries. Furthermore, the fluid temperature factors like nanofluid viscosity and microrotation viscosity were taken into account. A graph comparing the published data and the present numerical computation revealed an exact match. A physical interpretation of images was provided to describe the phenomenon of blood flow by heat transfer according to various circumstances. In medical treatment, especially cancer therapy, these results are crucial. Gold nanoparticles are among the best particles because they are stable metallic nanoparticles with excellent catalytic, magnetic, and optical properties. The investigation's findings showed that as time-steps grow, each profile's effectiveness tends to decrease, moving the unstable condition closer to the steady state. Whereas, the sphere-shaped nanoparticles have a significant effect on temperature profile change, column-shaped nanoparticles have less effect. Local skin friction rises and the local Nusselt number falls when the values of the two surface amplitude parameters rise. [ABSTRACT FROM AUTHOR]

The primary goal of the present study is to investigate how to obtain a periodic solution for a fractal Mathieu–Duffing oscillator. To achieve this, the fractal oscillator in the fractal space has been transformed into a damping Mathieu–Duffing equation in the continuous space by employing a new modification of He's definition of the fractal derivative. The required analytical periodic solution has been based on the rank upgrade technique (RUT) presented. The RUT successfully generates a periodic solution without sacrificing the damping coefficient by creating an alternate equation, aside from any challenges in managing the impact of the linear damping component. The homotopy perturbation method (HPM) has been used to find the required periodic solution for the alternate equation. A comparison of the numerical solutions of the original equation and the alternative equation showed good agreement. The stability behavior in the non-resonance case as well as in the sub-harmonic resonance case has also been discussed. Further, another method, "the non-perturbative approach", that deals with the obtained equation has been introduced. [ABSTRACT FROM AUTHOR]

The purpose of the present communication is to investigate the flow of a radiative electromagnetic-Casson nanofluid past a stretching sheet under the impacts of a chemical reaction and nonlinear thermal radiation. To enrich the blood flow, a modulated viscosity/thermal conductivity dependent temperature/nanoparticles concentration parameter is included in the governing equations. The system of PDEs is transformed to ODEs by invoking similarity transformations and then solved numerically by the well-known fourth-order Runge-Kutta integration scheme based on shooting approach. The main factors affecting the Casson fluid's temperature profiles are revealed. [ABSTRACT FROM AUTHOR]

In this article, the main objective is to employ the homotopy perturbation method (HPM) as an alternative to classical perturbation methods for solving nonlinear equations having periodic coefficients. As a simple example, the nonlinear damping Mathieu equation has been investigated. In this investigation, two nonlinear solvability conditions are imposed. One of them was imposed in the first-order homotopy perturbation and used to study the stability behavior at resonance and non-resonance cases. The next level of the perturbation approaches another solvability condition and is applied to obtain the unknowns become clear in the solution for the firstorder solvability condition. The approach assumed here is so significant for solving many parametric nonlinear equations that arise within the engineering and nonlinear science. [ABSTRACT FROM AUTHOR]

In this paper, a numerical analysis to study thermal radiation effect on magnetohydrodynamic (MHD) unsteady mixed-convection flow over a moving vertical cylinder with constant heat flux through a porous medium in the presence of transversal uniform magnetic field has been performed. The boundary layer equations are transformed into a linear algebraic system by a Crank–Nicolson type of implicit finite-difference method. The stability condition of the scheme was studied. Numerical results are presented with various sets of parameters for both air and water. The results show many interesting effects on velocity, temperature and concentration profiles on local as well as average shear stress, rate of heat and mass transfer. Also, the velocity and temperature profiles are compared with the available results in the literature and it is found to be in good agreement. Copyright © 2007 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]