Optimal Homotopy Asymptotic Method Research Articles

Tangent hyperbolic nanofluid flow through a vertical cone: Unraveling thermal conductivity and Darcy–Forchheimer effects

Purpose: This paper demonstrates the way tangent hyperbolic nanofluid flow through a vertical cone is influenced by varying viscosity and varying thermal conductivity. This study also seeks to illustrate the impact of convective boundary conditions on a fluid. The mathematical modeling also takes the Darcy–Forchheimer effect into account. Methodology: Using the appropriate similarity transformation, the fluid problem is reduced into a set of nonlinear ordinary differential equations. These systems of differential equations are evaluated numerically by applying the Optimal Homotopy Asymptotic Method. Findings: The nature of emergent parameters is examined in relation to the temperature distribution, nanoparticle concentration profile, and velocity profile. An increase in variable viscosity corresponds to a decrease in fluid velocity, while enhanced thermal conductivity results in elevated fluid temperature. The skin friction coefficient, Sherwood, and Nusselt numbers are numerically examined for active concerned parameters. These findings can be put into practice in a variety of fields such as polymer cooling systems and medication. Originality: Existing literature has yet to explore the combination of tangent hyperbolic nanofluids with varying viscosity and thermal conductivity under convective boundary conditions.

Numerical solutions of thin film flows of non-Newtonian fluids via optimal homotopy asymptotic approach

The objective of this research is to recover new solutions in the lifting and drainage cases of thin film flows involving non-Newtonian fluid models namely Pseudo-Plastic (PP) and Oldroyd 6-Constant (O6C). Both of the considered fluids exhibit numerous uses in industry when coupled with thin film phenomena. Some of the industrial applications include decorative and optical coatings, prevention of metallic corrosion and lithography of various diodes, sensors and detectors. For solution purpose, a modified version of Optimal Homotopy Asymptotic Method (OHAM) is proposed in which Daftardar–Jafari polynomials will replace the classical OHAM polynomials in nonlinear problems and provide better results in terms of accuracy. The paper includes a comprehensive application of modified algorithm in the case of thin film phenomena. To validate the obtained series solutions, the paper employs a rigorous assessment of convergence and validity by computing the residual errors in each scenario. For showing the effectiveness of modified algorithm, numerical comparison of classical and modified OHAMs is also presented in this study. Furthermore, the study conducts an in-depth graphical analysis to assess the impact of fluid parameters on velocity profiles both in lifting and drainage scenarios. The results of this investigation demonstrate that the proposed modification of OHAM ensures better accuracy of solutions than the classical OHAM. Consequently, this method can be effectively utilized for tackling more advanced situations.

A comparative analysis of Noyes-Field model for the non-linear Belousov–Zhabotinsky reaction using two reliable techniques

Within the domain of nonlinear dynamics, the Belousov-Zhabotinsky reaction system has consistently captivated researchers, sustaining its position as a vibrant and dynamic area of exploration. As an enduring subject of study, the Belousov-Zhabotinsky system provides continuous opportunities to unravel the foundational tenets of nonlinear dynamics within intricate systems. In our quest to deepen comprehension of this complex system, we introduce an innovative methodology for addressing the time fractional Belousov-Zhabotinsky system. This novel approach leverages both the Natural Transform Iterative Method (NTIM) and the Optimal Homotopy Asymptotic Method (OHAM), aiming to contribute novel insights and methodologies to the ongoing discourse surrounding this intriguing and influential area of research. We obtained a series solution to test the accuracy of the suggested approach. The proposed technique has the advantage of requiring few calculations while producing high-precision results. To help you better understand, 2D and 3D graphical representations are provided to demonstrate the model's behavior and how changing the fractional order derivative in Caputo's sense and time affects the solutions.

A Hybrid Laplace Transform-Optimal Homotopy Asymptotic Method (LT-OHAM) for Solving Integro-Differential Equations of the Second Kind

The new proposed hybrid method between optimal homotopy asymptotic method and Laplace transform namely LT-OHAM is formulated for the first time in our paper. This hybrid method presents significant features of LT-OHAM and its capability of handling IDEs. This formulation is developed to find the solution of IDEs. By using the new presented hybrid method, some applications of IDEs are solved. This hybrid method seems very efficient and easy to solve these types of equations.

Numerical Investigation with Convergence and Stability Analyses of Integro-Differential Equations of Second Kind

In this paper, integro-differential equations are solved by using an efficient numerical technique, namely, Multistage Optimal Homotopy Asymptotic method. The existence and uniqueness of solutions are established by the application of Lipschitz condition. Convergence of approximate solutions along with stability are also carried out. Some examples are solved to highlight the vital characteristics of the applied numerical scheme. Error estimation and comparison of derived results with existing exact solutions and those results which already available in the literature through graphical illustrations and tables reveal that Multistage Optimal Homotopy Asymptotic algorithm is more efficient and fruitful.

Thermal performance investigation of porous fins with convection and radiation under the influence of magnetic field using optimal homotopy asymptotic method

A study on enhancement of heat transfer in thermal systems by convective-radiative porous fin with temperature-invariant thermal conductivity is presented in this paper using optimal homotopy asymptotic method. The efficacy of the method is displayed through the verification of the results with the previous studies. Also, significance of various parameters of the nonlinear model on the heat transfer enhancement of thermal systems using the solutions presented by the method are discussed. The graphical representation of the thermal behaviour of the extended surfaces is presented for pictorial discussion. The results illustrate that the augmentations of the conductive-radiative, conductive-convective, porosity and magnetic field cause the extended surface temperature to reduce as a result of increased rate of heat flow via the passive device. The graphical illustrations show that the efficiency and effectiveness of the fin is high at low values of the radiative-conductive, convective-conductive, porosity and magnetic field parameters. This study will assist in proper thermal analysis of fins for effective thermal managements of engineering systems.

Application of the Variational Iteration Method for the time-fractional Kaup-Kupershmidt Equation and the Boussinesq-Burger equation

The variational iteration method is used to deal with linear and nonlinear differential equations. The main characteristics of the method lie in its flexibility and ability to accurately and easily solve nonlinear equations. In this work, a general framework is presented for a variational iteration method for the analytical treatment of partial differential equations in fluid mechanics. The Caputo sense is used to describe fractional derivatives. The time-fractional Kaup-Kupershmidt (KK) equation is investigated, as it is the solution of the system of partial differential equations via the Boussinesq-Burger equation. By comparing the results that are obtained by the variational iteration method with those obtained by the two-dimensional Legendre multiwavelet, the optimal homotopy asymptotic method (OHAM), the q-homotopy analysis transform method, the Laplace Adomian Decomposition Method, and the homotopy perturbation method, the first method proved to be very effective and convenient. The main methodology in this work is anticipated to be applied to various fractional calculus, linear, and nonlinear problems.

Analysis of functionally graded rotating disks via analytical approximation methods

Analytical solutions for functionally graded (FG) rotating disks are limited due to the functional representation of the cross-sectional geometry, the density, and the modulus of elasticity which lead to a governing differental equation with variable coefficients. Analytical solutions for such disks are limited according to the functional form chosen for the variation of the property. Hence, analytical approximation approaches gain importance for such problems. In this study, two analytical approximation approaches are proposed for the analysis FG rotating disks. The methods used are the Improved Adomian decomposition method (IADM) and the optimal homotopy asymptotic method (OHAM). The resulting solution is an analytical solution in power series form. It is assumed that the disks considered are subjected to constant angular velocity. Material properties such as modulus of elasticity, density and yield strength are assumed to vary along the radial direction according to the power law function. Stresses are obtained for different values of the gradient parameters to investigate the effects of the parameters. Von Mises yield criterion is used to determine elastic limit angular velocities. The results from this study are in very good agreement with previous results available in the literature. In addition, it is observed that OHAM converges to results much faster than IADM.

A study of heat and mass transfer flow of a variable viscosity couple stress fluid between inclined plates

In this paper, the Poiseuille flow of non-isothermal couple stress fluid of Reynolds model between two heated parallel inclined plates has been studied. The strongly nonlinear coupled ordinary differential equations have been investigated using the Asymptotic Homotopy Perturbation method (AHPM) and the Optimal Homotopy Asymptotic Method with DJ Polynomials (OHAM-DJ). Employing both these techniques, the approximate solutions for velocity profile, temperature distributions, shear stress, average velocity and volumetric flow rate have been achieved. The results obtained using both these methods are in the form of infinite series, hence the results can be obtained easily. Comparison of various results, obtained through these methods, is carried out. Here, it is worth to mention that the results gained by both these approaches are in excellent resemblance. The effect of different parameters on the flow problem has been studied numerically as well as presented graphically.

Study of heat transfer in MHD viscoelastic fluid of second grade over a stretching porous sheet with electromagnetic effects and nonuniform source/sink

This work is devoted to the study of non-Newtonian fluid of second grade. This model assumes that the flow and heat transfer in MHD viscoelastic fluid is subject to convective boundary conditions in the presence of electromagnetic field and nonuniform source/sink. The flow is considered over a porous sheet. The governing partial differential equations are converted as ordinary differential equations using suitable similarity transformations. The resulting coupled of nonlinear differential equations are successfully solved by means of Optimal Homotopy Asymptotic Method (OHAM). The effects of different physical parameters such as: the second-grade viscoelastic parameter, thermal radiation parameter or Eckert number are analyzed. Several closed form analytical approximate solutions for dimensionless stream function and temperature are presented for some particular cases. Our procedure is based upon an original construction of the solutions using a moderate number of the convergence-control parameters. These parameters lead to very high precision, comparing our approximate solutions with numerical solutions. Let us note that every nonlinear differential equation is reduced to two linear differential equations. The values of the convergence-control parameters are determined using rigorous mathematical procedures. OHAM is an iterative approach which rapidly converges after only one approximation, is effective, explicit, simple and easy to apply. These remarks are in fact the true power of our procedure to solving nonlinear problems without the presence of small or large parameters in the governing equations or boundary/initial conditions.

Analytical analysis of silver-water, silver-blood base nanofluid flow over fluctuating disk with the influence of viscous dissipation over fluctuating disk

This work considers the analytical analysis of silver-water, silver-blood base nanofluid flow over fluctuating disk with the influence of viscous dissipation over fluctuating disk. The primary goal of this study is an effort to improve the heat transfer ratio, which is a core part of the engineering and industrial sectors. Following a continuity check, the problem is modeled using the conservation rules of momentum and energy. Nonlinear PDEs are produced through modeling, which are then transformed into ODEs using a similarity transformation and thermophysical characteristics. The resultant ODEs are resolved using the Optimal Homotopy Asymptotic Method (HAM). The outcomes of this method are compared to authenticate the outcomes of the obtained results. The Mathematica software is used to run HAM methods, and graphical interpretations are given to highlight the influence of dissimilar contributing factors on the velocity profile and temperature distribution. Nusselt’s number, and the skin friction are examined through graphical representation. Convergence of the problem is checked with the help of graphs and tables by using dual solution of the problem.

Partial Slip Effects for Thermally Radiative Convective Nanofluid Flow

The partial slip effects for radiative convective nanofluid flow over a stretching sheet in porous medium are analytically explored in this work. The Navier–Stokes equations, the momentum and the energy equations are converted into a set of non-linear ODEs by the similarity transformation. Using the modified optimal homotopy asymptotic method (OHAM), the resulting non-linear ODEs are analytically approximately solved. The impact of various parameters, such as: the velocity exponential factor n, the wall thickness parameter γ, the dimensionless velocity slip parameter δ1, the Prandtl number Pr, the radiation parameter R, and the dimensionless temperature jump parameter δ2, on the behaviour of the mass and heat transfer is presented. The influence of these parameters is tabular and graphically presented. An excellent agreement between the approximate analytical solution and the corresponding numerical solution is highlighted. The results obtained confirm that modified OHAM is a useful and competitive mathematical tool to explore a large class of non-linear problems with applications in various fields of science and engineering.

Coating of pseudo-plastic material in reverse roll-coating: Mathematical analysis

This paper describes a theoretical framework and computational methods of a thin layer coating of a non-Newtonian polymeric material while it moves through a tiny space among two inverted rollers. Order of magnitude is accustomed to clarify the nondimensional forms of the governing equations. Semi-analytical solutions of pressure gradient, velocity profile and rate of the flow are acquired via optimal homotopy asymptotic method (OHAM). The graphical representation depicts the physical quantities of the effects of velocity profile ratio k and Weissenberg number We. It is observed that by increasing the values of k and We, velocity profile decreases while pressure distribution increases.

Application of the Optimal Homotopy Asymptotic Approach for Solving Two-Point Fuzzy Ordinary Differential Equations of Fractional Order Arising in Physics

This work focuses on solving and analyzing two-point fuzzy boundary value problems in the form of fractional ordinary differential equations (FFOBVPs) using a new version of the approximation analytical approach. FFOBVPs are useful in describing complex scientific phenomena that include heritable characteristics and uncertainty, and obtaining exact or close analytical solutions for these equations can be challenging, especially in the case of nonlinear problems. To address these difficulties, the optimal homotopy asymptotic method (OHAM) was studied and extended in a new form to solve FFOBVPs. The OHAM is known for its ability to solve both linear and nonlinear fractional models and provides a straightforward methodology that uses multiple convergence control parameters to optimally manage the convergence of approximate series solutions. The new form of the OHAM presented in this work incorporates the concepts of fuzzy sets theory and some fractional calculus principles to include fuzzy analysis in the method. The steps of fuzzification and defuzzification are used to transform the fuzzy problem into a crisp problem that can be solved using the OHAM. The method is demonstrated by solving and analyzing linear and nonlinear FFOBVPs at different values of fractional derivatives. The results obtained using the new form of the fuzzy OHAM are analyzed and compared to those found in the literature to demonstrate the method’s efficiency and high accuracy in the fuzzy domain. Overall, this work presents a feasible and efficient approach for solving FFOBVPs using a new form of the OHAM with fuzzy analysis.

Elastic Stress Analysis of Hyperbolic Rotating Disks via Optimal Homotopy Asymptotic Method

This study, it is aimed to use an effective approximation method, called Optimal Homotopy Asymptotic Method (OHAM) for elastic stress analysis. This study is carried out with the idea that utilizable of the considered method in many areas and having advantages will add a new perspective to the rotating disc problems that can provide convenience and practicality. The considered disk is an annular disk with both free ends and is subjected to centrifugal force. The thickness variation is assumed as hyperbolic. Deformations occurred on the disk, and radial and tangential stresses are calculated with the proposed method. The approximate solutions are in very good agreement with the exact solution. It is observed that the results converge to exact solutions faster than other approximate methods, like the Improved Adomian Decomposition method, in the literature. The results of this study show that OHAM is a very practical method with fast results for rotating disk problems. The study also shows the advantages of OHAM being directly applicable to differential equations of rotating disks without any transform function. Results by the comparison of approximate results and exact solutions indicated that OHAM can effectively be used in the analysis of rotating variable thickness disks.

Computational Investigations of an Internally Heated Convective-Radiative Porous Fin subjected to Magnetic Field: Comparative Methods and Parametric Studies

The concern of this paper is to compare the computational efficiencies and accuracies of three approximate analytical methods; namely, homotopy analysis method (HAM), optimal homotopy asymptotic method (OHAM) and differential transform method (DTM) for the nonlinear thermal performance analysis of a convective-radiative porous fin with temperature-dependent internal heat generation under the influence of magnetic field. To establish the computational accuracies of the three methods, the results of the three series solutions are compared with the results of the developed exact analytical and numerical methods. Also, the symbolic solutions developed in this work are used to explore the impacts of the controlling parameters on the performance of the passive device. It is established that as the coductive-convective, conductive-radiative and magnetic field parameters increase, the fin temperature distribution decreases and hence, the fin thermal efficiency is improved. An increase in temperature distribution in the fin is noticed as the nonlinear thermal conductivity parameter increases. It is envisaged that the present study will give a good insight into the nonlinear analysis of extended surfaces which will aid proper design in thermal systems.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited.

Analytical Approximations of Well Function by Solving the Governing Differential Equation Representing Unsteady Groundwater Flow in a Confined Aquifer

A solution of the governing equation representing the drawdown in a horizontal confined aquifer, where groundwater flow is unsteady, was first provided by Theis and is famously known as the Theis solution. This solution was given in terms of an exponential integral, also called the well function, for which simple and reliable approximations are preferred due to their practical applications. To that end, several approximations are available in the literature, of which some are based on series approximations for the integral, and others are numerical approximations. This study employs three kinds of homotopy-based methods, namely the homotopy analysis method (HAM), homotopy perturbation method (HPM), and optimal homotopy asymptotic method (OHAM), for analytically solving the governing partial differential equation (PDE). For convenience, the PDE is first converted to a boundary value problem (BVP) using a similarity transformation. Comparing the series approximations obtained using these methods with the Theis solution, it is found that the 10th-order HAM, and just three terms of OHAM-based solutions, provide accurate approximations. On the other hand, the HPM-based solution is found to be accurate only within a small domain. Further, the proposed approximations are compared with several series and numerical approximations available in the literature using the percentage error. The proposed methodology provides accurate approximations of the well function by directly solving the governing differential equation in a general framework and thus can be adapted to other practical situations arising in groundwater flow.

Heat and Mass Transfer Analysis for the Viscous Fluid Flow: Dual Approximate Solutions

The aim of this paper is to investigate effective and accurate dual analytic approximate solutions, while taking into account thermal effects. The heat and mass transfer problem in a viscous fluid flow are analytically explored by using the modified Optimal Homotopy Asymptotic Method (OHAM). By using similarity transformations, the motion equations are reduced to a set of nonlinear ordinary differential equations. Based on the numerical results, it was revealed that there are dual analytic approximate solutions within the mass transfer problem. The variation of the physical parameters (the Prandtl number and the temperature distribution parameter) over the temperature profile is analytically explored and graphically depicted for the first approximate and the corresponding dual solution, respectively. The advantage of the proposed method arises from using only one iteration for obtaining the dual analytical solutions. The presented results are effective, accurate and in good agreement with the corresponding numerical results with relevance for further engineering applications of heat and mass transfer problems.

Optimal Homotopy Asymptotic Method for an Anharmonic Oscillator: Application to the Chen System

The aim of our work is to obtain the analytic solutions for a new nonlinear anharmonic oscillator by means of the Optimal Homotopy Asymptotic Method (OHAM), using only one iteration. The accuracy of the obtained results comes from the comparison with the corresponding numerical ones for specified physical parameters. Moreover, the OHAM method has a greater degree of flexibility than an iterative method as is presented in this paper. Based on these results, the analytically solutions of the Chen system were obtained for a special case (just one analytic first integral). The chaotic behaviors were excluded here. The provided solutions are usefully for many engineering applications.

A new optimal multistep optimal homotopy asymptotic method to solve nonlinear system of two biological species

Abstract Recently solving integro-differential equations have been the focus of attention among many researchers in the field of mathematic and engineering. The aim of current study is to apply the well-known optimal homotopy asymptotic method (OHAM) on a specific and famous model of these equations. It is illustrated that auxiliary functions and the number of Taylor series terms affect the accuracy of the solution. Hence, at first a solution has been found with an acceptable error by OHAM. Then, it has been continued to attain a better solution by Multistep optimal homotopy asymptotic method. All these processes had improved the precision of the solution. Auxiliary polynomials of two, three, and four degrees and different numbers of Taylor series term have been investigated to solve a nonlinear system derived by two biological species ‎living together. Ultimately, appropriate results with auxiliary polynomials of degree four and Taylor series with six terms have been obtained precisely. In addition, the error values decrease significantly compared to the other cases.