Analysis of Unsteady Axisymmetric Squeezing Fluid Flow with Slip and No-Slip Boundaries Using OHAM

In this manuscript, An unsteady axisymmetric flow of nonconducting, Newtonian fluid squeezed between two circular plates is studied with slip and no-slip boundaries. Using similarity transformation, the system of nonlinear partial differential equations is reduced to a single fourth order ordinary differential equation. The resulting boundary value problems are solved by optimal homotopy asymptotic method (OHAM) and fourth order explicit Runge-Kutta method (RK4). It is observed that the results obtained from OHAM are in good agreement with numerical results by means of residuals. Furthermore, the effects of various dimensionless parameters on the velocity profiles are investigated graphically

Homotopy perturbation method-based soliton solutions of the time-fractional (2+1)-dimensional Wu–Zhang system describing long dispersive gravity water waves in the ocean

Physical phenomena and natural disasters, such as tsunamis and floods, are caused due to dispersive water waves and shallow waves caused by earthquakes. In order to analyze and minimize damaging effects of such situations, mathematical models are presented by different researchers. The Wu–Zhang (WZ) system is one such model that describes long dispersive waves. In this regard, the current study focuses on a non-linear (2 + 1)-dimensional time-fractional Wu–Zhang (WZ) system due to its importance in capturing long dispersive gravity water waves in the ocean. A Caputo fractional derivative in the WZ system is considered in this study. For solution purposes, modification of the homotopy perturbation method (HPM) along with the Laplace transform is used to provide improved results in terms of accuracy. For validity and convergence, obtained results are compared with the fractional differential transform method (FDTM), modified variational iteration method (mVIM), and modified Adomian decomposition method (mADM). Analysis of results indicates the effectiveness of the proposed methodology. Furthermore, the effect of fractional parameters on the given model is analyzed numerically and graphically at both integral and fractional orders. Moreover, Caputo, Caputo–Fabrizio, and Atangana–Baleanu approaches of fractional derivatives are applied and compared graphically in the current study. Analysis affirms that the proposed algorithm is a reliable tool and can be used in higher dimensional fractional systems in science and engineering

Modification of Homotopy Perturbation Algorithm Through Least Square Optimizer for Higher Order Integro-Differential Equations

In this manuscript, modification of homotopy perturbationmethod (HPM) is proposed for integro-differential equations by couplingthe least square method (LSM) with HPM. Improved accuracy in a veryfew iterations is the general advantage of this technique. The proposedmethod is applied to different higher order integro-differential equationsof linear and nonlinear nature, and results are compared with exact as wellas available solutions from the literature. Numerical and graphical analysisreveal that the proposed algorithm is reliable for integro-differentialequations and hence can be utilized for more complex problems

New Solutions of Time- and Space-Fractional Black–Scholes European Option Pricing Model via Fractional Extension of He-Aboodh Algorithm

The current study explores the space and time-fractional Black–Scholes European option pricing model that primarily occurs in the financial market. To tackle the complexities associated with solving models in a fractional environment, the Aboodh transform is hybridized with He’s algorithm. This facilitates in improving the efficiency and applicability of the classical homotopy perturbation method (HPM) by ensuring the rapid convergence of the series form solution. Three cases that are time-fractional scenario, space-fractional scenario, and time-space-fractional scenario are observed through graphs and tables. 2D graphical analysis is performed to depict the behaviour of a given option pricing model for varying time, stock price, and fractional parameters. Solutions of the European option pricing model at various fractional orders are also presented as 3D plots. The results obtained through these graphs unfold the interchange between time- and space-fractional derivatives, presenting a comprehensive study of option pricing under fractional dynamics. The competency of the proposed scheme is illustrated via solutions and errors throughout the fractional domain in tabular form. The validity of the He-Aboodh results is exhibited by comparison with existing errors. Analysis shows that the proposed methodology (He-Aboodh algorithm) is a valuable scheme for solving time-space-fractional models arising in business and economics

Constructing and Predicting Solutions for Different Families of Partial Differential Equations: A Reliable Algorithm

The construction of mathematical models for different phenomena, and developing their solutions, are critical issues in science and engineering. Among many, the Buckmaster and Korteweg-de Vries (KdV) models are very important due to their ability of capturing different physical situations such as thin film flows and waves on shallow water surfaces. In this manuscript, a new approach based on the generalized Taylor series and residual function is proposed to predict and analyze Buckmaster and KdV type models. This algorithm estimates convergent series with an easy-to-use way of finding solution components through symbolic computation. The proposed algorithm is tested against the Buckmaster and KdV equations, and the results are compared with available solutions in the literature. At first, proposed algorithm is applied to Buckmaster-type linear and nonlinear equations, and attained the closed-form solutions. In the next phase, the proposed algorithm is applied to highly nonlinear KdV equations (namely, classical, modified, and generalized KdV) and approximate solutions are obtained. Simulations of the test problems clearly reassert the dominance and capability of the proposed methodology in terms of accuracy. Analysis reveals that the projected scheme is reliable, and hence, can be utilized for more complex problems in engineering and the sciences

Radiative Mixed Convection Flow of Casson Nanofluid through Exponentially Permeable Stretching Sheet with Internal Heat Generation

This paper investigates the mixed convection boundary-layer flow of Casson nanofluid with an internal heat source on an exponentially stretched sheet. The Buongiorno model, incorporating thermophoresis and Brownian motion, describes fluid temperature. The modeled system is solved numerically using bvp4c routine to analyze the impact of different fluid parameters on velocity, temperature, and concentration profiles. The analysis reveals that the suction effect, magnetic field, and Casson parameter reduce momentum boundary layer thickness and hence slow fluid motion. Conversely, buoyancy forces increase mass boundary layer thickness which results in accelerating fluid motion. Temperature and concentration profiles show similar trends for Brownian motion, radiation, and thermophoresis

A New and Reliable Modification of Homotopy Perturbation Method

This manuscript introduces a new alteration to the Homotopy Perturbation Method by coupling it with the Laplace Transform. The corresponding Homotopy Perturbation Laplace Method (HPLM) promises better results in terms of accuracy, efficiency, and easy-of-use when compared to other semi-numerical schemes, and is therefore conveniently poised to be used for various problems in science and engineering. The method is tested against standard fifth and sixth order linear and nonlinear ordinary differential equations. For validity, the obtained results are compared with well known analytical and numerical schemes.&nbsp

Slip Analysis at Fluid-Solid Interface in MHD Squeezing Flow of Casson Fluid through Porous Medium

An unsteady squeezing flow of Casson fluid having Magneto Hydro Dynamic effect and passing through porous medium channel with slip at the boundaries has been modelled and analyzed. Similarity transformations are applied to the governing partial differential equations of the Casson model to get a highly non-linear fourth order ordinary differential equation. The obtained equation is then solved analytically using the Homotopy Perturbation Method (HPM) for uniform and non-uniform slip at the boundaries. Five cases of boundary conditions, representing slip at upper wall only, uniform slip at both walls, non-uniform slip where slip at upper wall is greater than that of lower wall, non-uniform slip where slip at lower wall is greater than that of upper wall, and slip at lower wall only are considered and thoroughly investigated. Validation is performed by solving the equation numerically using fourth order explicit Runge Kutta method (ERK4). Both analytical and numerical results show good agreement. Lastly, the effects of various fluid parameters on the velocity profile are investigated for each case graphically. Analysis of these plots show that the positive and negative squeeze numbers have opposite effect on the velocity profile throughout all the cases. It is also observed that various fluid parameters like Casson, MHD, and Permeability have similar effects on the velocity profile in the cases when slip is occurring at the upper wall only, and non-uniform slip at both the boundaries with slip at lower wall is greater than upper wall. Furthermore, similar effects have been observed when slip is uniform at both the boundaries, and in case of non-uniform slip with slip at lower wall is less than the upper wall. Keywords: Squeezing flow, Casson fluid, Porous media, Magneto Hydro Dynamic, Slip paramete

Improved Analysis for Squeezing of Newtonian Material between Two Circular Plates

This article presents a scheme for the analysis of an unsteady axisymmetric flow of incompressible Newtonian material in the form of liquid squeezed between two circular plates. The scheme combines traditional perturbation technique with homotopy using an adaptation of the Laplace Transform. The proposed method is tested against other schemes such as the Regular Perturbation Method (RPM), Homotopy Perturbation Method (HPM), Optimal Homotopy Asymptotic Method (OHAM), and the fourth-order Explicit Runge-Kutta Method (ERK4). Comparison of the solutions along with absolute residual errors confirms that the proposed scheme surpasses HPM, OHAM, RPM, and ERK4 in terms of accuracy. The article also investigates the effect of Reynolds number on the velocity profile and pressure variation graphically