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In this paper, we explore how to generate solitary, peakon, periodic, cuspon and kink wave solution of the well-known partial differential equation Korteweg–de Vries (KdV) by using exp-function and modified exp-function methods. The presented methods construct more efficiently almost all types of soliton solution of KdV equation that can be rarely seen in the history. These methods appear to be straightforward and symbolic calculations are used to solve the problem. All resulting answers are verified for accuracy using the symbolic computation program with M a p l e. To show the physical appearance of the model, 3D plots of all the generated solutions are then displayed. The obtained solutions revealed the compatibility of the proposed techniques which provide the general solution with some free parameters. This is the key benefit of these methods over the other methods. [ABSTRACT FROM AUTHOR]

This study investigates analytically the stability analysis of a nonlinear convective boundary layer (BL) flow of the nanofluids GO-EG and GO-W on a shrinking surface in the presence of a magnetic field and viscous dissipation. We provide a second-order nonlinear ordinary differential equation (NODE) for temperature distribution (TD) and a third-order NODE for velocity profile (VP) coupling based on the thermophysical characteristics of the base fluid and nanofluid as well as similarity transformations (STs) in the fundamental governing equations of momentum and energy. The analytical method HAM is used to answer the equations that have been gathered. In many different industries, such as manufacturing, automotive, microelectronics, and defense, cooling of various types of equipment and devices has very important challenges. To fulfill the demands of the engineering and industrial industries, it is hoped that this issue would significantly improve the heat transformation ratio. [ABSTRACT FROM AUTHOR]

In this study, the Falkner–Skan stream (FSS) of ZnO-EG across a moving wedge is investigated using artificial neural networks backpropagated using a Bayesian regularization strategy (ANN-BRS). The PDEs of the Falkner–Skan are converted into a set of ordinary differential equations (ODEs). The reference dataset is created using the mathematical solver in Mathematica and includes moving wedge boundaries, radiation boundaries, nanoparticle volume division boundaries and Falkner–Skan power-regulation boundaries for all proposed scenarios (ANN-BRS). Examined is the effect of practical cut off points on the stream field and temperature profiles, including the radiation limit, moving wedge limit, nanoparticle volume division and Falkner–Skan power. When nanoparticles are present, viscosity and heat conductivity rise, which can be physically explained. Increases in Falkner–Skan power have both positive and negative effects. When the settings are adjusted to their highest values, the maximum rate of heat transmission is realized. The effects of radiation boundary and so on are identical due to the linear augment. As a result, as the parameters are increased, the rate of heat transmission increases. Based on the intended data points that were obtained, the estimated answer is calculated for every scenario utilizing the testing, training and validation procedures. The mean square error data (MSE), error histogram and regression analysis are used to validate the performance of (ANN-BRS). [ABSTRACT FROM AUTHOR]

A magneto couple stress nanofluid flow along with double diffusive convection is presented for peristaltic induce flow through symmetric nonuniform channel. A comprehensive mathematical model is scrutinized for couple stress nanofluid magneto nanofluids and corresponding equations of motions are tackled by applying small Reynolds and long wavelength approximation in viewing the scenario of the biological flow. Computational solution is exhibited with the help of graphical illustration for nanoparticle volume fraction, solutal concentration and temperature profiles in MATHEMTICA software. Stream function is also computed numerically by utilizing the analytical expression for nanoparticle volume fraction, solutal concentration and temperature profiles. Whereas pressure gradient profiles are investigated analytically. Impact of various crucial flow parameter on the pressure gradient, pressure rise per wavelength, nanoparticle volume fraction, solutal concentration, temperature and the velocity distribution are exhibited graphically. It has been deduced that temperature profile is significantly rise with Brownian motion, thermophoresis, Dufour effect, also it is revealed that velocity distribution really effected with strong magnetic field and with increasing non-uniformity of the micro channel. The information of current investigation will be instrumental in the development of smart magneto-peristaltic pumps in certain thermal and drug delivery phenomenon. [ABSTRACT FROM AUTHOR]

A mathematical model is explored to establish the electroosmotic flow for Cu-water nanoliquids within a ciliated symmetric micro-channel, the flow is established with aid of ciliary motion and axial pressure gradient. Nanofluid comprise of Cu as a nanofluid particles and water as base fluid. Maxwell-Garnelt model is exploited for viscosity and thermal conductivity of nanoliquid. Magnetic field is applied in the transverse direction and external electric field is enforced in the axial direction. Equations of motion are simplified for nanofluid flow in the microchannel by employing low Reynolds number and long wavelength approximation theory. Crucial exact analytical expression are gathered for electric potential, temperature profile, axial velocity, volume flux, pressure gradient, stream function, and result for pressure rise per wavelength explored numerically. The influence of crucial flow parameters on, flow behaviour, pumping phenomena, and temperature profile are thoroughly investigated. [ABSTRACT FROM AUTHOR]

In this study a novel mathematical model for electroosmotic flow for Cu-water based nanofluid with heat transfer phenomenon is reported for scraped-surface heat exchanger. The flow is initiated due to motion of lower wall of the channel and axial pressure gradient. The flow is modelled with aid of low Reynolds number and lubrication approximation theory. Exact analytical expressions are gathered for axial velocity, and stream functions for various stations of scraped-surface heat exchanger. Physical phenomenon of electro osmotic parameter are investigated on velocity profile, velocity distribution and pressure rise at edge of the blades. It is reported that electro-osmotic parameter mainly works as dragging force, it can be used to control the flow. This controlling mechanism may be helpful in mixing different materials in scraped-surface heat exchanger. Pressure rise at edge of the blades mainly rises below the blades with electro-osmotic, whereas, this profiles is suppressed for region above the blades and between the blades. [ABSTRACT FROM AUTHOR]

Sometimes, an increased reaction is caused by an immune system of the body to a non-toxic agent (e. g. eggs, dust, pollens or some drugs). This is called allergy or hypersensitivity. The regulatory T cells decrease these allergic reactions. Nowadays, it is noticed that immune system has a deep relationship with micro-organism present in the intestine, that can be explained by the example, that some bacteria of intestine increase production of Treg cells by producing butyric acid like fatty acids. This can also understand that sufficiently different types of T cell receptors of Treg cells are needed to stop the inflammatory response produced by intestinal bacteria. In this study, the dynamic relation of T helper cells, intestinal bacteria and Treg cells are illustrated by a conformable mathematical model. Memory effects are figured out and displayed through graphs. Different plots also show the effects of increasing/decreasing amount of Treg induction efficiency on the whole system. [ABSTRACT FROM AUTHOR]

An innovative concept of water-based Cu–Al2O3 hybrid nanofluid has been employed to investigate the behavior of flow and heat transfer inside a rectangular channel whose permeable walls experiences dilation or contraction in height. The transformed set of ordinary differential equations is then solved by a well-known Runge–Kutta–Fehlberg algorithm. The analysis also includes three different shapes of copper nanocomposites, namely, platelet, cylinder and brick- shaped. The impact of various embedded parameters on the flow and heat transfer distributions have been demonstrated through the graphs. All the flow properties, temperature profile and rate of heat transfer at the walls are greatly influenced by the presence of copper nanoparticles. Furthermore, it was observed that the platelet shaped nanocomposites provide a better heat transfer ability as compared to the other shapes of nanoparticles. [ABSTRACT FROM AUTHOR]

The paper is devoted to a new extension in Gegenbauer wavelet method (GWM) to investigate the transfer of heat and MHD boundary-layer flow of ferrofluids beside a flat plate with velocity slip. A homogenous model study is conducted in which we assumed the heat transfer and forced convective flow of ferrofluids along a flat plate with a uniform wall heat flux. In the direction of transverse to plate, a magnetic field is imposed. Three various magnetic nanoparticle types including Mn-ZnFe2O4, CoFe2O4, Fe3O4 are incorporated inside the base fluid. Two types of base fluids (water and kerosene) with bad thermal conductivity as compared to nanoparticles of solid magnetic have been assumed. The mathematical model is tackled via modified Gegenbauer wavelet method (MGWM). A simulation is accomplished for individual ferrofluid mixture by assuming the prevailing impacts of uniform and slip heat fluxes. The variation of heat transfers and skin friction were also observed at the surface of the plate and we analyzed the better heat transfer for every mixture. Kerosene-based magnetite (Fe3O4) delivers the better rate of heat transfer at wall due to its association with the kerosene-based Mn-Zn and cobalt ferrites. The slip velocity and magnetic field effects on the temperature, dimensionless velocity, rate of heat transfer and skin friction are examined for various magnetic nanoparticles inside the kerosene oil and water. We observed that the primary influence of magnetic field reduces the dimensionless surface temperature and accelerates the dimensionless velocity as compared to the hydrodynamic case, thus enhancing the rate of heat transfer and skin friction ferrofluids. Moreover, a detailed evaluation of outcomes obtained by MGWM, already published work and numerical RK-4 were found to be in excellent agreement. The error and convergence analysis are presented. Comparison of results, graphical plots, error and convergence analysis reveal the appropriateness of proposed method. The proposed algorithm can be extended for other nonlinear problems. [ABSTRACT FROM AUTHOR]

Fractional integral and differential inequalities provide the bounds for the uniqueness of solutions of the fractional differential equations. In this paper we compute some new Hadamard and the Fejéer-Hadamard fractional inequalities for harmonically convex functions via Caputo k-fractional derivatives. Also results for Caputo fractional derivatives have been induced. [ABSTRACT FROM AUTHOR]

This paper aims to explore the flow of water saturated with copper nanoparticles of different shapes between parallel Riga plates. The plates are placed horizontally in the coordinate axis. Influence of the linear thermal radiation is also taken into account. The equations governing the flow have been transformed into a nondimensional form by employing a set of similarity transformations. The obtained system is solved analytically (variation-of-parameters method) and numerically (Runge-Kutta scheme). Under certain conditions, a special case of the model is also explored. Furthermore, influences of the physical quantities on velocity and thermal fields are discussed with the graphical aid over the domain of interest. The quantities of engineering and practical interest (skin friction coefficient and local rate of heat transfer) are also explored graphically. [ABSTRACT FROM AUTHOR]

The combined effects of heat transfer and chemical reaction are studied for the flow through a semi-infinite asymmetric channel with orthogonally deformable porous walls. The similarity transforms have been used to reduce the conservation laws to a corresponding system of nonlinear ordinary differential equations. The resulting equations are solved, both analytically and numerically, by using Homotopy Analysis Method (HAM) and the fourth-order Runge-Kutta (RK-4) method, respectively. The convergence of the analytical solution is assured through the so-called total squared residual error analysis. The optimal values of auxiliary parameters are obtained by minimizing the total squared residual error. [ABSTRACT FROM AUTHOR]

This article is devoted to establish a novel analytical solution scheme for the fractional diffusion equations. Caputo's formulation followed by the variation of parameters method has been employed to obtain the analytical solutions. Following the derived analytical scheme, solution of the fractional diffusion equation for several initial functions has been obtained. Graphs are plotted to see the physical behavior of obtained solutions. [ABSTRACT FROM AUTHOR]

we use the modified variation of parameters method for finding the analytical solution of a system of third-order nonlinear boundary value problems associated with obstacle, unilateral, and contact problems. The results are calculated in terms of convergent series with easily computable components. The suggested technique is applied without any discretization, perturbation, transformation, and restrictive assumptions. Moreover, it is free from round off errors. Some examples are given to illustrate the implementation and efficiency of the modified variation of parameters method. [ABSTRACT FROM AUTHOR]

In this paper, we apply the exp-function method to construct generalized solitary and periodic solutions of good Boussinesq equations which play an important role in mathematical physics, applied and engineering sciences. The suggested algorithm is quite efficient and is practically well suited for use in these problems. Numerical results clearly indicate the reliability and efficiency of the suggested technique. [ABSTRACT FROM AUTHOR]

In this paper, we use the Exp-function method to construct the generalized solitary and periodic solution of the Kuramoto–Sivashinsky and Boussinesq equations. These equations play very important role in mathematical physics and engineering sciences. The suggested algorithm is quite efficient and is practically well suited for use in these problems. The results show the reliability and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

This paper is based on finding the exact solutions for Burger's equation, Zakharov-Kuznetsov (ZK) equation and Kortewegde vries (KdV) equation by utilizing exponential function method that depends on the series of exponential functions. The exponential function method utilizes the homogeneous balancing principle to find the solutions of nonlinear equations. This method is simple, wide-reaching and helpful for finding the exact solution of nonlinear conformable PDEs. [ABSTRACT FROM AUTHOR]

The analysis of Homotopy Perturbation Method (HPM) for the solution of fractional partial differential equations (FPDEs) is presented. A unified convergence theorem is given. In order to validate the theory, the solution of fractional-order Burger-Poisson (FBP) equation is obtained. Furthermore, this work presents the method to find the solution of FPDEs, while the same partial differential equation (PDE) with ordinary derivative i.e., for α = 1 , is not defined in the given domain. Moreover, HPM is applied to a complicated obstacle boundary value problem (BVP) of fractional order. [ABSTRACT FROM AUTHOR]

In this paper, we use the variation of parameters method to solve a class of eighth-order boundary-value problems. The analytical results are calculated in terms of convergent series. Error estimates are tabulated and are compared with some other well known techniques which reveal the complete reliability of the suggested scheme. Several examples are given to verify the reliability and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]