Search Results (83)

Nonlinear mixed integro-differential equations (NM-IDEs) of the third kind present a complex challenge during solving initial value problems (IVPs), particularly after converting them from standard forms. In this work, we address the existence and uniqueness of a type of NM-IDEs employing Picard’s method. Additionally, we estimate the solution using the homotopy analysis method (HAM) and analyze the convergence of the approach. To demonstrate the credibility of the theoretical results, various applications are given, and the numerical results are displayed in a group of figures and tables to highlight that solving IVPs by first converting them to NM-IDEs and using the HAM is a computationally efficient approach. Full article

In this study, the optimal q-Homotopy Analysis Method (optimal q-HAM) has been used to investigate fractional Abel differential equations. This article is designed as a case study, where several forms of Abel equations, containing Bernoulli and Riccati equations, are given with ordinary derivatives and fractional derivatives in the Caputo sense to present the application of the method. The optimal q-HAM is an improved version of the Homotopy Analysis Method (HAM) and its modification q-HAM and focuses on finding the optimal value of the convergence parameters for a better approximation. Numerical applications are given where optimal values of the convergence control parameters are found. Additionally, the correspondence of the approximate solutions obtained for these optimal values and the exact or numerical solutions are shown with figures and tables. The results show that the optimal q-HAM improves the convergence of the approximate solutions obtained with the q-HAM. Approximate solutions obtained with the fractional Differential Transform Method, q-HAM and predictor–corrector method are also used to highlight the superiority of the optimal q-HAM. Analysis of the results from various methods points out that optimal q-HAM is a strong tool for the analysis of the approximate analytical solution in Abel-type differential equations. This approach can be used to analyze other fractional differential equations arising in mathematical investigations. Full article

In this current work, we assume the mathematical modelling of non-Newtonian time-dependent hybrid nanoparticles via a cylindrical stenosis artery. In this work, blood is used as a base fluid, and the nanoparticles (copper and aluminum oxide) of cylindrical shape are inserted inside the artery to combine with blood to form hybrid nanofluid (HNF). The homotopy analysis method (HAM) is deployed for the solution of nonlinear resulting equations. For the validation of this current work, the results of the existing work have been compared with our proposed model results. A comparison of key profiles like velocity, temperature, wall shear stress, and flow rate is also performed at a specific critical height of the stenosis. It is also observed that the thermal conductance of hybrid nanofluids is greater than that of nanofluids. Including the hybrid nanoparticles (copper and aluminum oxide) inside the blood enhances the blood axial velocity. These simulations are applicable to the magnetic targeting treatment of stenosed artery disorders and the diffusion of nanodrugs. Full article

Hybrid nanofluids have many real-world applications. Research has shown that mixed nanofluids facilitate heat transfer better than nanofluids with one type of nanoparticle. New applications for this type of material include microfluidics, dynamic sealing, and heat dissipation. In this study, we began by placing copper into H₂O to prepare a Cu-H₂O nanofluid. Next, Cu-H₂O was combined with Al₂O₃ to create a Cu-Al₂O₃-H₂O hybrid nanofluid. In this article, we present an analytical study of the estimated flows and heat transfer of incompressible three-dimensional magnetohydrodynamic hybrid nanofluids in the boundary layer. The application of similarity transformations converts the interconnected governing partial differential equations of the problem into a set of ordinary differential equations. Utilizing the homotopy analysis method (HAM), a uniformly effective series solution was obtained for the entire spatial region of 0 < η < ∞. The errors in the HAM calculation are smaller than 1 × 10⁻⁹ when compared to the results from the references. The volume fractions of the hybrid nanofluid and magnetic fields have significant impacts on the velocity and temperature profiles. The appearance of magnetic fields can alter the properties of hybrid nanofluids, thereby altering the local reduced friction coefficient and Nusselt numbers. As the volume fractions of nanoparticles increase, the effective viscosity of the hybrid nanofluid typically increases, resulting in an increase in the local skin friction coefficient. The increased interaction between the nanoparticles in the hybrid nanofluid leads to a decrease in the Nusselt number distribution. Full article

A study using mathematical modeling has been conducted to analyze how both man-made and natural sources of contaminants affect various layers of an aquifer-aquitard system. The xy-, yz-, and zx-plane have been used to depict the locations where the natural sources of contaminant occur on the xz- and yz-plane, and where the man-made sources occur, on the xy-plane. It is assumed that the sources occurring in different planes are constant, while the velocity of groundwater flow has been considered only along the x-axis. A three-dimensional advection dispersion equation (ADE) has been used to accurately model the flow of groundwater and contaminants through a porous medium. Three distinct sources exert their influence on three separate planes throughout the entire duration of this study, thus making it possible to model these sources using initial conditions. This study presents a profile of contaminant concentration in space and time when constant sources are located on different planes. Some physical assumptions have been considered to make the model relatable to real-world phenomena. Often, finding stability conditions for numerical solutions becomes difficult, so an unconditionally stable solution is more appreciable. The homotopy analysis method (HAM), a method known for its unconditional stability, has been used to solve a three-dimensional mathematical model (ADE) along with its initial conditions. Man-made sources show more impact than equal-strength natural sources in the aquifer-aquitard system. Full article

The homotopy analysis method (HAM) is a useful method to derive analytical approximate solutions of black holes in modified gravity theories. In this paper, we study the Einstein–Weyl gravity coupled with Maxwell field and obtain analytical approximation solutions for charged black holes by using the HAM. It is found that the analytical approximate solutions are sufficiently accurate in the entire spacetime outside the black hole’s event horizon and also consistent with numerical ones for charged black holes in the Einstein–Maxwell–Weyl gravity. Full article

This investigation focuses on the impact of Stefan blowing on the flow of hybrid nanoliquids over a moving slender needle with magnetohydrodynamics (MHD), thermal radiation, and entropy generation. To facilitate analysis, suitable transformations are applied to convert the governing partial differential equations into a set of ordinary differential equations, which are then solved analytically using Homotopy Analysis Method (HAM) in Mathematica. This study investigates how varying the values of Stefan blowing, magnetic field, and thermal radiation parameters impact the profiles of velocity, temperature, and concentration. Additionally, the study analyzes the outcomes of the local skin friction, local Nusselt number, and local Sherwood number. Increasing the magnetic field reduces the velocity profile. The temperature profile is enhanced by a rise in the thermal radiation parameter. Also, the results reveal that an increase in the Stefan blowing number leads to higher profiles of velocity. Full article

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. Full article

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. Full article

In this research, three numerical methods, namely the variational iteration method, the Adomian decomposition method, and the homotopy analysis method are considered to achieve an approximate solution for a third-order time-fractional partial differential Equation (TFPDE). The equation is obtained from the classical (FW) equation by replacing the integer-order time derivative with the Caputo fractional derivative of order $\mathsf{\eta }=(0,1]$ with variable coefficients. We consider homogeneous boundary conditions to find the approximate solutions for the bounded space variable $l<\mathit{\chi }<L$ and $l,L\in \mathbb{R}$. To confirm the effectiveness of the proposed methods of non-integer order $\mathsf{\eta }$, the computation of two test problems was presented. A comparison is made between the obtained results of the (VIM), (ADM), and (HAM) through tables and graphs. The numerical results demonstrate the effectiveness of the three numerical methods. Full article

Non-axisymmetric stagnant-point flows for flat plates in porous media containing spherical Cu-Al₂O₃-H₂O nanoparticles are studied using the homotopy analysis method (HAM). The governing equations are transformed into three coupled non-linear ordinary differential equations through similarity transformations. A large degree of freedom is provided by HAM when selecting auxiliary linear operators. By transforming nonlinear coupled ordinary differential equations with variable coefficients into linear ordinary differential equations with constant coefficients, nonlinear coupled ordinary differential equations can be solved. Over the entire domain, these equations can be solved approximately analytically. The analysis involves a discussion of the impact of many physical parameters generated in the proposed model. The results have shown that skin friction coefficients of Cf_x and Cf_y increase with volume fraction of hybrid nanofluid and the coefficient of permeability increasing. For the axisymmetric case of γ = 0, when volume fraction, φ, φ₁, φ₂ = 0, 5%, 10%, 20%, Cf_x = Cf_y = 1.33634, 1.51918, 1.73905, 2.33449, it can be found that the wall shear stress values increase by 13.68%, 30.14%, and 74.69%, respectively. In response to an increase in hybrid nanofluid volume fractions, local Nusselt numbers Nu_x increase. Nu_x decrease and change clearly with the coefficient of permeability increasing in the range of γ < 0; the values of Nu_x are less affected in the range of γ > 0. Full article

The present study is concerned with studying the dynamical behavior of two space-dimensional nonlinear time-fractional models governing the unsteady-flow of polytropic-gas (in brief, pGas) that occurred in cosmology and astronomy. For this purpose, two efficient hybrid methods so-called optimal homotopy analysis $\mathbb{J}$-transform method (${}_O$HA$\mathbb{J}$TM) and $\mathbb{J}$-variational iteration transform method ($\mathbb{J}$-VITM) have been adopted. The _OHA$\mathbb{J}$TM is the hybrid method, where optimal-homotopy analysis method (${}_O$HAM) is utilized after implementing the properties of $\mathbb{J}$-transform ($\mathbb{J}$T), and in $\mathbb{J}$-VITM is the $\mathbb{J}$-transform-based variational iteration method. Banach’s fixed point approach is adopted to analyze the convergence of these methods. It is demonstrated that $\mathbb{J}$-VITM is $\mathrm{T}$-stable, and the evaluated dynamics of pGas are described in terms of Mittag–Leffler functions. The proposed evaluation confirms that the implemented methods perform better for the referred model equation of pGas. In addition, for a given iteration, the proposed behavior via ${}_O$HA$\mathbb{J}$TM performs better in producing more accurate behavior in comparison to $\mathbb{J}$-VITM and the methods introduced recently. Full article

This paper focuses on the usage of the homotopy analysis method (HAM) to solve the fractional heat conduction equation. In the presented mathematical model, Caputo-type fractional derivatives over time or space are considered. In the HAM, it is not necessary to discretize the considered domain, which is its great advantage. As a result of the method, a continuous function is obtained, which can be used for further analysis. For the first time, for the considered equations, we proved that if the series created in the method converges, then the sum of the series is a solution of the equation. A sufficient condition for this convergence is provided, as well as an estimation of the error of the approximate solution. This paper also presents examples illustrating the accuracy and stability of the proposed algorithm. Full article

A non-Darcy flow with moving boundary conditions in a low-permeability reservoir was solved using the homotopy analysis method (HAM), which was converted into a fixed-boundary mathematical model via similarity transformation. Approximate analytical solutions based on the HAM are guaranteed to be more accurate than exact analytical solutions, with relative errors between 0.0089% and 2.64%. When λ = 0, the pressure drop of the Darcy seepage model could be instantaneously transmitted to infinity. When λ > 0, the pressure drop curve of the non-Darcy seepage model exhibited the characteristics of tight support, which was clearly different from the Darcy seepage model’s formation pressure distribution curve. According to the results of the HAM, a moving boundary is more influenced by threshold pressure gradients with a longer time. When the threshold pressure gradients were smaller, the moving boundaries move more quickly and are more sensitive to external influences. One-dimensional, low-permeability porous media with a non-Darcy flow with moving boundary conditions can be reduced to a Darcy seepage model if the threshold pressure gradient values tend to zero. Full article

The spontaneous scalarization of Schwarzscild-AdS is investigated in the Einstein-scalar-Gauss–Bonnet (ESGB) theory. Firstly, we construct scalarized AdS black holes numerically. Secondly, making use of the homotopy analysis method (HAM), we obtain analytical approximate solutions for scalarized AdS black holes in the ESGB theory. It is found that scalarized AdS black holes constructed numerically are consistent with analytical approximate solutions in the whole space. Full article