Homotopy Analysis Method Solutions Research Articles

Collisional breakage population balance equation: An analytical approach

This work presents a unique semi-analytical approach based on the homotopy analysis method (HAM), called accelerated HAM, recently proposed in (Hussain et al., “Semi-analytical methods for solving non-linear differential equations: A review.”, JMAA, 2023), to solve the collisional breakage population balance model, which is an integro-partial differential equation. We compare our findings with those obtained using the Adomian decomposition method, a well-known technique for solving various forms of differential equations. By decomposing the non-linear operator, we investigate how to utilize the convergence control parameter to expedite the convergence of the HAM solution towards its precise value in accelerated HAM. The other objective of the article is to examine the theoretical convergence analysis of the two proposed methods. Additionally, we conduct theoretical research on the error estimates for both the techniques. To validate our schemes, several numerical examples are considered and the numerical simulations demonstrate that the suggested techniques provide accurate estimates for the solution and moments of the collisional breakage equation.

Magnetohydrodynamic study of Micropolar fluid flow in the porous walled channel with variable viscosity and thermal conductivity: HAM Solution

This article examines the production of entropy and the heat transfer rate in the passage of micropolar fluid through a channel. The micropolar fluid takes place through the porous walled channel under the presence of an external magnetic field acting in the perpendicular direction of the flow. In this work, the thermal conductivity and viscosity of the fluid is considered variable and are the function of the temperature. Here, we consider two-dimensional creeping flow with two different forms of temperature boundary conditions, namely, Newtonian heating (NH) boundary constraints and specified surface temperature (SST) boundary conditions. The flow of the micropolar fluid in the proposed problem is governed by the coupled non-linear PDEs. These PDEs are first transformed into the ODEs by the use of some suitable similarity transformations and then obtained ODEs are further solved by using the semi-analytic Homotopy Analysis Method (HAM). This article shows the effect of various flow parameters on the fluid’s velocity, heat transfer rate, temperature of the fluid, and entropy production for both types of thermal boundary conditions. Through this article, it is concluded that the entropy production is minimal for Newtonian heating (NH) boundary conditions as compared to specified surface temperature (SST) boundary conditions. This result suggests that the NH boundary condition is more suitable for electrical appliances as compared to the SST boundary condition. This type of model is applicable in nuclear reactors, engineering appliances, heat reservoirs, etc.

Semi-analytical Approach to Nonlinear Partial Differential Equations Using Homotopy Analysis Technique (HAM)

This work considers a novel semi-analytical method named the homotopy analysis method (HAM) to study the nonlinear gas dynamic equation. The obtained HAM solution is validated by comparing it with the exact available solution and compared with the (Adomian decomposition method) ADM solution and numerical solution to test the efficiency of the proposed method. The efficiency of the proposed approach can be demonstrated numerically and graphically, and it is found to be in excellent agreement with the current approach.

Semi-analytical methods for solving non-linear differential equations: A review

This article develops a new semi-analytical technique based on the homotopy analysis approach for solving linear or non-linear differential equations and the results are compared to the well-known approaches such as the Adomian decomposition method (ADM), homotopy perturbation method (HPM), homotopy analysis method (HAM), and optimized decomposition method (ODM). We discuss the decomposition of the non-linear operator to expedite the HAM solution's convergence to its precise values by using the convergence control parameter. The theoretical convergence analysis and the error estimates are studied. Numerical illustrations show that our proposed scheme improves the accuracy of the non-linear problems discussed in the recently published articles [30] and [31] to an excellent extent and also indicate rapid convergence.

CONVECTIVE CATTANEO-CHRISTOV HEAT FLUX AND HEAT GENERATION EFFECT ON MHD WILLIAMSON FLUID FLOW OVER AN EXPONENTIALLY STRETCHING SURFACE WITH THERMAL RADIATION

The two-dimensional steady incompressible MHD boundary layer flow of convective Williamson fluid flow of the Cattaneo-Christov heat flux type over an exponentially stretching surface in the presence of heat generation and thermal radiation is considered. The nonlinear governing partial differential equations are reduced to ordinary differential equations by using a similarity transformation. The Homotopy Analysis Method (HAM) is applied to solve the reduced equations and the effects of importanceof fluid parameters are explained through graphs. Also, the HAM solutions were compared with the numerical solutions and good agreement was observed

Using Homotopy Perturbation and Analysis Methods for Solving Different-dimensions Fractional Analytical Equations

The aim of the research, we extended the one-dimensional to multi-dimensional, we applied the homotopy perturbation and analysis methods to solve Volterra integral equations and to obtain approximate analytical solutions of systems of the second kind multi-dimensional Volterra integral equations. We proved the convergence of the homotopy analysis method (HAM). The HAM solutions contained an auxiliary parameter that provides a convenient way of controlling the convergence region of series solutions. It is shown that the solutions obtained by the homotopy-perturbation method (HPM) are only special cases of the HAM solutions. Several examples are given to illustrate the efficiency and implementation of the method. The results indicate that this method is efficient for the linear and non - linear models with the dissipative terms.

Entropy Generation in Peristaltic Transport of Hybrid Nanofluids with Thermal Conductivity Variations and Electromagnetic Effects.

Entropy generation in peristaltic transport of hybrid nanofluid possessing temperature-dependent thermal conductivity through a two-dimensional vertical channel is studied in this paper. The hybrid nanofluid consists of multi-walled carbon nanotubes mixed with zinc oxide suspended in engine oil. Flow is affected by a uniform external magnetic field, hence generating Lorentz force, Hall and heating effects. Given the vertical orientation of the channel, the analysis accounts for mixed convection. To study heat transfer in the current flow configuration, the model considers phenomena such as viscous dissipation, heat generation or absorption, and thermal radiation. The mathematical modeling process employs the lubrication approach and Galilean transformation for enhanced accuracy. The slip condition for the velocity and convective conditions for the temperature are considered at the boundaries. The study analyzes entropy generation using the Homotopy Analysis Method (HAM) and includes convergence curves for HAM solutions. Results are presented using graphs and bar charts. The analysis shows that higher Brinkman and thermal radiation parameters result in higher temperatures, while higher thermal conductivity parameters lead to reduced entropy generation and temperature profile. Additionally, higher Hall parameter values decrease entropy generation, while an increased Hartman number improves entropy generation.

Radiation mechanism on dissipative ternary hybrid nanoliquid flow through rotating disk encountered by Hall currents: HAM solution

The energy transition to enhance heat transport during the flow of ternary hybrid nanofluid comprised of Copper (Cu), Iron oxide (Fe3O4), Silicon dioxide (SiO2) nanoparticles and Polymer as base liquid over a revolving disk surface has been reported. From the several existing techniques, adding nanoparticles to standard working fluids is an effective approach that might dramatically improve the rate of heat transfer. The Hall and radiation impacts are also considered. A mathematical model is developed by assuming the flow as incompressible and purely radial in a cylindrical coordinate system and appropriate similarity variables are introduced for problem simplification and dimensionless analysis. The solution of complex generated PDEs was calculated using semi analytical method known as Homotopy Analysis Method (HAM) after translating them into corresponding ODEs. It has been observed that the consequences of thermal radiation, dissipation term, unsteadiness and rotation parameters boost the energy transmission rate and enhances the heat transfer rate of tri hybrid nanofluids. However, Cu-Fe3O4 nanoparticles in the base fluid remarkably magnifies the energy transmission rate. The unitary ternary hybrid nanofluid (Cu, Fe3O4, SiO2) has higher velocity profiles as compared to unitary nano fluid (Cu) and hybrid nano-fluid (Cu-Fe3O4).

Hydrodynamic analysis of the magnetic field dependent viscous fluid flow and thermosolutal convection between rotating channels

According to research, exposing a person to a magnetic field enhances blood flow and minimizes their risk of suffering a heart attack. Ferrohydrodynamics is the study of fluid motion mechanics that is affected by strong magnetic polarisation forces (FHD). Ferrofluids may transmit heat in a variety of ways by using magnetic fluids. This behaviour is demonstrated by liquid-cooled speakers, which utilise less ferrofluid to prevent heat from reaching the speaker coil. This modification boosts the coil’s ability to expand, which enables the loudspeaker to create high-fidelity sound. It is investigated how the fluid dynamics of spinning, squeezing plates are affected by thermosolutal convection and a magnetic field dependent (MFD) viscosity. Standard differential equations are used to represent the equations of the modified form of Navier Stokes, Maxwell’s, and thermosolutal convection. The magnetic field, modified velocity field equations, and thermosolutal convection equations all yield suitable answers. Additionally computed and thoroughly detailed are the MHD torque and fluid pressure that are imparted to the top plate. To create a technique with quick and certain convergence, the resulting equations for uniform plates are solved using the Homotopy Analysis Method (HAM) with appropriate starting estimates and auxiliary parameters. The validity and reliability of the HAM outcomes are shown by comparing the HAM solutions with the BVP4c numerical solver programme. It has been found that a magnetic Reynolds number lowers the temperature of the fluid as well as the tangential and axial components of the velocity field. Additionally, when the fluid’s MFD viscosity rises, the axial and azimuthal components of the magnetic field behave in opposition to one another. This study has applications in the development of new aircraft take-off gear, magnetorheological airbags for automobiles, heating and cooling systems, bio-prosthetics, and biosensor systems.

Solution of the Quadratic Integral Equation by Homotopy Analysis Method

In the present paper, we derive an approximate solution of the quadratic integral equation by using the homotopy analysis method (HAM). This approach provides a solution in the form of a rapidly converging series, and it includes an auxiliary parameter that controls the series solution's convergence. We compare the HAM solution with the exact solution graphically. Additionally, an absolute error comparison between the exact and HAM solutions is performed. The findings indicate that HAM is a very straightforward and attractive approach for computation.

Heat Transfer Analysis over the Boundary Layer Stagnation-Point Flow of Couple Stress Fluid over an Exponentially Stretching Sheet

The present study provides a comprehensive discussion for the problem of boundary layer flow and heat transfer analysis of stagnation point flow of couple stress fluid over an exponentially stretching surface. The governing equations of couple stress fluid model are assumed under boundary layer approach. The nonlinear partial differential equations are simplified by using similar similarity transformations. The analytical solutions of abridged equations are computed with the help of homotopy analysis method (HAM). The convergence of the HAM solutions have been deliberated by plotting <i>ћ </i>- curves and also through homotopy pade approximation. The physical features of pertinent parameters have been discussed through graphs.

Application of HAM to laminar boundary layer flow over a wedge with an external magnetic field

AbstractThe incompressible laminar along with injection or suction over a wedge for a boundary layer flow is studied. Falkner–Skan transformations reduce the governing partial differential equations (PDEs) to two nonlinear coupled PDEs and are solved by the homotopy analysis method (HAM). The variation of a dimensionless temperature and velocity profiles and for , , and to the nonidentical values of the parameter for injection/suction has been shown in the graph. The results hence obtained show that the flow field is determined by the existence of the applied magnetic field. The finite difference method is applied to the reduced PDEs and the velocity and temperature profiles are compared with the HAM solutions and depicted graphically. To distinguish singularities in the graph, we have applied Pade for the HAM series solution, which is depicted in graphs for all three cases. We have also estimated the radius of convergence of HAM solutions by Domb–Sykes plot for injection, no suction, and suction respectively. The important observation made by us through HAM and numerical solution is the existence of flow separation for injection, which is not shown by previous authors.

Static analysis of composite beams on variable stiffness elastic foundations by the Homotopy Analysis Method

New analytical solutions for the static deflection of anisotropic composite beams resting on variable stiffness elastic foundations are obtained by the means of the Homotopy Analysis Method (HAM). The method provides a closed-form series solution for the problem described by a non-homogeneous system of coupled ordinary differential equations with constant coefficients and one variable coefficient reflecting variable stiffness elastic foundation. Analytical solutions are obtained based on two different algorithms, namely conventional HAM and iterative HAM (iHAM). To investigate the computational efficiency and convergence of HAM solutions, the preliminary studies are performed for a composite beam without elastic foundation under the action of transverse uniformly distributed loads considering three different types of stacking sequence which provide different levels and types of anisotropy. It is shown that applying the iterative approach results in better convergence of the solution compared with conventional HAM for the same level of accuracy. Then, analytical solutions are developed for composite beams on elastic foundations. New analytical results based on HAM are presented for the static deflection of composite beams resting on variable stiffness elastic foundations. Results are compared to those reported in the literature and those obtained by the Chebyshev Collocation Method in order to verify the validity and accuracy of the method. Numerical experiments reveal the accuracy and efficiency of the Homotopy Analysis Method in static beam problems.

Characteristics of thermophoresis and Brownian motion on radiative reactive micropolar fluid flow towards continuously moving flat plate: HAM solution

The present objective of this study is to develop a mathematical model and rheological aspects by combining the micropolar fluid model to simulate the reactive flow from a continuously moving flat plate. The current flow model is formulated with employment Thermophoretic diffusion, Brownian motion and chemically reactive species. The focal micropolar based flow equations are transmitted into a non-dimensional form via similarity transformation for which Homotopy Analysis Method (HAM) is employed to obtain approximate analytical solutions. The role of numerous parameters on dimensionless flow phenomena are observed in very effective way utilizing graphical presentation. Various order of approximations for the convergence of HAM has been tabulated. Proper validation is conducted to show a strong relationship between our results with related published ones in literature with some limiting conditions. Furthermore, streamlines and isotherms are also plotted to know the pattern of the fluid flow. More so, the analysis shows that enhancement in Thermal Grashof number, solutal Grashof number and microrotation parameter rises the fluid spin rotation. Temperatures are suppressed with higher thermophoresis and Brownian parameters. Concentrations are elevated with higher thermophoresis parameter. Finally, the observations revealed that the presence of thermophoresis and Brownian motion is more effective to improve heat transportation phenomenon.

Flow dynamics of a time-dependent non-Newtonian and non-isothermal fluid between coaxial squeezing disks

The goal of this research is to investigate the behaviours of porosity and squeezing phenomena in the presence of time-dependent heat flow that affect the flow rate and improve the system’s heating/cooling mechanism, reduce non-Newtonian fluid turbulence and scale-up flow tracers. Squeezing discs in the presence of no-slip velocity and convective surface boundary conditions induces a laminar, unstable and incompressible non-Newtonian fluid. The convective form of the momentum, concentration and energy equations are modelled for smooth discs to evaluate and offer an analytical and numerical examination of the flow for heat and mass transfer, which are further transformed to a highly non-linear system of ordinary differential equation using similarity transformations. In the case of smooth disks, the self-similar equations are solved using Homotopy Analysis Method (HAM) with appropriate initial guesses and auxiliary parameters to produce an algorithm with an accelerated and assured convergence. The comparison of HAM solutions with numerical solver programme BVP4 c proves the validity and correctness of HAM results. It is found that increasing or bypassing the Hartman number reduces the capillary region, making the Lorentz force effect more visible for small values of non-Newtonian parameter. The concentration rate at the bottom disc rises rapidly as the thermal diffusivity rises. In addition, because the rate of outflow from the flow domain increases, the suction/injection parameter lowers the radial velocity. Additionally, as the non-Newtonian parameter is increased, skin friction and heat/mass flux rise. In the suction/injection situation, all physical characteristics have the opposite effect on flow field profiles.

Variable Viscosity and Thermal Conductivity Effects on Entropy Generation in Nanofluid Flow in an Inclined Channel: HAM Solution

This work investigates the entropy generation in nanofluid flow in a sloping channel with Navier slip and asymmetric wall temperatures. The viscosity and thermal conductivity are assumed to be dependent on temperature. The equations governing the flow, temperature are nonlinear and are solved using Homotopy Analysis Method (HAM) after non-dimensionalisation. Comparisons with existing literature have been produced and are found to be in excellent agreement, for special case of the current formulation. The impact of pertinent flow and fluid parameters on entropy generation, Bejan number, Nusselt number and skin friction is addressed, developed and displayed graphically.

Critical flow over an uneven bottom topography using Forced Korteweg-de Vries (fKdV)

This research examined the critical flow over an uneven bump using forced Korteweg-de Vries (fKdV) model. The forced KdV model containing forcing term which represent an uneven bump is solved using Homotopy Analysis Method (HAM). HAM is a semi-analytic technique whereby its solution contains a series of approximated solution in which it converges immediately to the exact solution. A particular HAM solution is chosen with an appropriate convergence parameter by referring to horizontal line segment. The convergent HAM solution depicts that waves only exhibited over the sloping region and no rise of waves found on flat part of bottom topography.

Numerical Solution of SEIR Model of The MERS-CoV Disease using Homotopy Analysis Method

The spread of MERS-Cov disease which was modelled by Susceptible Exposed Infected Recovered (SEIR) model has been solved by a reliable method so-called Homotopy Analysis Method (HAM). The solution using HAM is done by constructing the zero order deformation equation of SEIR model into a high order equation and selecting the convergence control (ℏ). The closeness of HAM and Fourth order Runge Kutta (RK4) solutions and also the existence of residual error showa benchmark of the success of the HAM. The result shows that the minimum errors of the closeness of HAM and fourth order Runge Kutta (RK4) solutions are 10−17 while the minimum residual error of HAM solutions are 10−18. Therefore, HAM has successfully obtained solution of SEIR model approximately. Overall, HAM can be an alternative method for solving more complex models.

Homotopy analysis method approximate solution for fuzzy pantograph equation

This paper investigates the powerful method namely the homotopy analysis method (HAM), to solve the fuzzy pantograph equation (FPE) in approximate analytic form. HAM yields a convergent infinite series solution to the solution of FPE without the need to reduce the FPE to the first order system or compare it with the exact solution, and this is one of the advantages of this method. For a better approximate solution, the HAM uses a convergence control parameter from the convergence region of the infinite series solution. HAM solution of FPE is obtained by reformulate crisp standard approximation via the properties of the fuzzy set theory.

Homotopy Simulation of Dissipative Micropolar Flow and Heat Transfer from a Two-Dimensional Body with Heat Sink Effect

Non-Newtonian flow from a wedge constitutes a fundamental problem in chemical&lt;br /&gt; engineering systems and is relevant to processing of polymers, coating systems, etc. Motivated by such applications, the homotopy analysis method (HAM) was employed to&lt;br /&gt; obtain semi-analytical solutions for thermal convection boundary layer flow of incompressible micropolar fluid from a two-dimensional body (wedge). Viscous dissipation&lt;br /&gt; and heat sink effects were included. The non-dimensional boundary value problem&lt;br /&gt; emerges as a system of nonlinear coupled ordinary differential equations, by virtue of&lt;br /&gt; suitable coordinate transformations. The so-called Falkner-Skan flow cases are elaborated. Validation of the HAM solutions was achieved with earlier simpler models, as well as with a Nakamura finite difference method for the general model. The micropolar model employed simulates certain polymeric solutions quite accurately, and features rotary motions of micro-elements. Primary and secondary shear stress, wall couple stress, Nusselt number, microrotation velocity, and temperature were computed for the effect of&lt;br /&gt; vortex viscosity parameter (micropolar rheological), Eckert number (viscous dissipation),&lt;br /&gt; Falkner-Skan (pressure gradient) parameter, micro-inertia density, and heat sink parameter. The special cases of Blasius and stagnation flow were also addressed. It was observed from the study that the temperature and thermal boundary layer thickness are both suppressed with increasing wedge parameter and wall heat sink effect, which is beneficial to temperature regulation in polymer coating dynamics. Further, strong reverse spin was generated in the microrotation with increasing vortex viscosity, which resulted in&lt;br /&gt; increase in angular momentum boundary layer thickness. Also, both primary and secondary skin friction components were reduced with increasing wedge parameter. Nusselt number was also enhanced substantially with greater wedge parameter.