Showing papers on "Homotopy analysis method published in 2022"

On Solutions of Fractional-Order Gas Dynamics Equation by Effective Techniques

Abstract: In this work, the novel iterative transformation technique and homotopy perturbation transformation technique are used to calculate the fractional-order gas dynamics equation. In this technique, the novel iteration method and homotopy perturbation method are combined with the Elzaki transformation. The current methods are implemented with four examples to show the efficacy and validation of the techniques. The approximate solutions obtained by the given techniques show that the methods are accurate and easy to apply to other linear and nonlinear problems.

The Numerical Investigation of a Fractional-Order Multi-Dimensional Model of Navier-Stokes Equation via Novel Techniques

Abstract: In this study, numerical results of a fractional-order multi-dimensional model of the Navier–Stokes equations will be achieved via adoption of two analytical methods, i.e., the Adomian decomposition transform method and the q-Homotopy analysis transform method. The Caputo–Fabrizio operator will be used to define the fractional derivative. The proposed methods will be implemented to provide the series form results of the given models. The series form results of proposed techniques will be validated with the exact results available in the literature. The proposed techniques will be investigated to be efficient, straightforward, and reliable for application to many other scientific and engineering problems.

Novel Analysis of the Fractional-Order System of Non-Linear Partial Differential Equations with the Exponential-Decay Kernel

Abstract: This article presents a homotopy perturbation transform method and a variational iterative transform method for analyzing the fractional-order non-linear system of the unsteady flow of a polytropic gas. In this method, the Yang transform is combined with the homotopy perturbation transformation method and the variational iterative transformation method in the sense of Caputo–Fabrizio. A numerical simulation was carried out to verify that the suggested methodologies are accurate and reliable, and the results are revealed using graphs and tables. Comparing the analytical and actual solutions demonstrates that the proposed approaches are effective and efficient in investigating complicated non-linear models. Furthermore, the proposed methodologies control and manipulate the achieved numerical solutions in a very useful way, and this provides us with a simple process to adjust and control the convergence regions of the series solution.

A new numerical investigation of fractional order susceptible-infected-recovered epidemic model of childhood disease

Abstract: The susceptible-infected-recovered (SIR) epidemic model of childhood disease is analyzed in the present framework with the help of q -homotopy analysis transform method ( q -HATM). The considered model consists the system of three differential equations having fractional derivative, and the non-linear system exemplifies the evolution of childhood disease in a population and its influence on the community with susceptible, infected and recovered compartment. The projected method is a mixture of q -homotopy analysis method and Laplace transform. Two distinct explanatory cases are considered, and corresponding simulations have been demonstrated in terms of plots for different value of the order. The present investigation elucidates that the projected both derivative and technique play a vital role in the analysis and illustrate the behaviour of diverse mathematical models described with differential equations in human disease.

Multiple slip, Soret and Dufour effects in fluid flow near a vertical stretching sheet in the presence of magnetic nanoparticles

Abstract: This paper presents a mathematical analysis of multiple slip, Soret and Dufour effects in a boundary layer flow of an electrically conducting nanofluid over a vertically stretching sheet. The flow situation has been described mathematically by using partial differential equations. Suitable transformations are utilized to make the model equation convenient for computation. An efficient optimal homotopy analysis method has been implemented successfully to obtain analytic approximations to the unknown functions in the flow problem. The influences of wall slip parameters, porosity of the medium, Buoyancy forces, magnetic field, thermal radiation, Soret and Dufour effects, heat source and chemical reaction parameters are examined in detail. The variations of the dimensionless velocity, temperature and concentration profiles in relation to the emerging parameters are explored intensively. The rates of momentum, heat and mass transfer near the stretching surface are also studied against the pertinent parameters. The study reveals that the increase in velocity slip parameter speeds up the fluid motion and increasing the Soret effect raises concentration of nanoparticles near the stretching sheet. Further, the analytic approximations for the solutions of the present model obtained by implementing the optimal homotopy analysis method are found in a very good agreement with some early works under common assumptions.

A new numerical investigation of fractional order susceptible-infected-recovered epidemic model of childhood disease

Abstract: The susceptible-infected-recovered (SIR) epidemic model of childhood disease is analyzed in the present framework with the help of q-homotopy analysis transform method (q-HATM). The considered model consists the system of three differential equations having fractional derivative, and the non-linear system exemplifies the evolution of childhood disease in a population and its influence on the community with susceptible, infected and recovered compartment. The projected method is a mixture of q-homotopy analysis method and Laplace transform. Two distinct explanatory cases are considered, and corresponding simulations have been demonstrated in terms of plots for different value of the order. The present investigation elucidates that the projected both derivative and technique play a vital role in the analysis and illustrate the behaviour of diverse mathematical models described with differential equations in human disease.

Porosity reconstruction based on Biot elastic model of porous media by homotopy perturbation method

Abstract: Fluid-saturated porous media are two-phase media, which are composed of solid and liquid phases. Biot theory for fluid-saturated porous media holds that underground media are composed of porous elastic solid and compressible viscous fluid filled with pore space. Compared with the single-phase media theory, the fluid-saturated porous media theory can describe the subsurface media more precisely, and the elastic wave equations in the fluid-saturated porous media contain more parameters used to describe the formation properties. Therefore, fluid-saturated porous media theory is widely used in geophysical exploration, seismic engineering, and other fields. This paper considers the porosity reconstruction problem of the elastic wave equations in the fluid-saturated porous media based on Biot theory, which is a typically nonlinear ill-posed inverse problem from mathematical viewpoint. The proposed method is a novel iteration regularization scheme based on the homotopy perturbation technique. To verify the validity and applicability, numerical experiments of two-dimensional and three-dimensional porosity models have been carried out. Numerical results illustrate that this method can overcome the numerical instability and are robust to data noise in the reconstruction procedure. Furthermore, compared with the classical regularized Gauss-Newton method, the homotopy perturbation method greatly widens the convergence region while keeping the fast convergence rate.

Homotopy perturbation method for fractal Duffing oscillator with arbitrary conditions

Abstract: A nonlinear vibration system in a fractal space can be effectively modeled using the fractal derivatives, and the homotopy perturbation method is employed to solve fractal Duffing oscillator with arbitrary initial conditions. A detailed solving process is given, and it can be easily followed for applications to other nonlinear vibration problems.

Numerical investigation of fractional-order Kersten–Krasil’shchik coupled KdV–mKdV system with Atangana–Baleanu derivative

Abstract: Abstract In this article, we present a fractional Kersten–Krasil’shchik coupled KdV-mKdV nonlinear model associated with newly introduced Atangana–Baleanu derivative of fractional order which uses Mittag-Leffler function as a nonsingular and nonlocal kernel. We investigate the nonlinear behavior of multi-component plasma. For this effective approach, named homotopy perturbation, transformation approach is suggested. This scheme of nonlinear model generally occurs as a characterization of waves in traffic flow, multi-component plasmas, electrodynamics, electromagnetism, shallow water waves, elastic media, etc. The main objective of this study is to provide a new class of methods, which requires not using small variables for finding estimated solution of fractional coupled frameworks and unrealistic factors and eliminate linearization. Analytical simulation represents that the suggested method is effective, accurate, and straightforward to use to a wide range of physical frameworks. This analysis indicates that analytical simulation obtained by the homotopy perturbation transform method is very efficient and precise for evaluation of the nonlinear behavior of the scheme. This result also suggests that the homotopy perturbation transform method is much simpler and easier, more convenient and effective than other available mathematical techniques.

Homotopic simulation for heat transport phenomenon of the Burgers nanofluids flow over a stretching cylinder with thermal convective and zero mass flux conditions

Abstract: Abstract This study is focused to elaborate on the effect of heat source/sink on the flow of non-Newtonian Burger nanofluid toward the stretching sheet and cylinder. The current flow analysis is designed in the form of higher order nonlinear partial differential equations along with convective heat and zero mass flux conditions. Suitable similarity transformations are used for the conversion of higher order nonlinear partial differential equations into the nonlinear ordinary differential equations. For the computation of graphical and tabular results, the most powerful analytical technique, known as the homotopy analysis method, is applied to the resulting higher order nonlinear ordinary differential equations. The consequence of distinct flow parameters on the Burger nanofluid velocity, temperature, and concentration profiles are determined and debated in a graphical form. The key outcomes of this study are that the Burger nanofluid parameter and Deborah number have reduced the velocity of the Burger nanofluid for both the stretching sheet and cylinder. Also, it is attained that the Burger nanofluid temperature is elevated with the intensifying of thermal Biot number for both stretching sheet and cylinder. The Burger nanofluid concentration becomes higher with the escalating values of Brownian motion parameter and Lewis number for both stretching sheet and cylinder. The Nusselt number of the Burger nanofluid upsurges due to the increment of thermal Biot number for both stretching sheet and cylinder. Also, the different industrial and engineering applications of this study were obtained. The presented model can be used for a variety of industrial and engineering applications such as biotechnology, electrical engineering, cooling of devices, nuclear reactors, mechanical engineering, pharmaceutical science, bioscience, medicine, cancer treatment, industrial-grid engines, automobiles, and many others.

Chemical reaction impact in dual diffusive non-Newtonian liquid featuring variable fluid thermo-solutal attributes

Abstract: This study focusses on two-dimensional rate type Maxwell material flow configured by a vertical moving surface. Heat source/sink and variable conductivity attributes are included in energy expression. Mass transport analysis is scrutinized through variable diffusivity and chemical reaction. Thermo-solutal buoyancy forces are introduced to capture mixed convection effects. The fundamental laws of fluid mechanics are utilized to model the problem. The implementation of order ensued in PDEs of governed physical problem. Apposite transformations are introduced to transfigure these PDEs into ordinary differential expressions. Computations of significant physical quantities are achieved through homotopy scheme. The convergence is ensured through tabular and graphical outcomes. The description of sundry variables is elucidated via pictorial forms.

Numerical investigation of fractional-order Kersten–Krasil’shchik coupled KdV–mKdV system with Atangana–Baleanu derivative

Abstract: Abstract In this article, we present a fractional Kersten–Krasil’shchik coupled KdV-mKdV nonlinear model associated with newly introduced Atangana–Baleanu derivative of fractional order which uses Mittag-Leffler function as a nonsingular and nonlocal kernel. We investigate the nonlinear behavior of multi-component plasma. For this effective approach, named homotopy perturbation, transformation approach is suggested. This scheme of nonlinear model generally occurs as a characterization of waves in traffic flow, multi-component plasmas, electrodynamics, electromagnetism, shallow water waves, elastic media, etc. The main objective of this study is to provide a new class of methods, which requires not using small variables for finding estimated solution of fractional coupled frameworks and unrealistic factors and eliminate linearization. Analytical simulation represents that the suggested method is effective, accurate, and straightforward to use to a wide range of physical frameworks. This analysis indicates that analytical simulation obtained by the homotopy perturbation transform method is very efficient and precise for evaluation of the nonlinear behavior of the scheme. This result also suggests that the homotopy perturbation transform method is much simpler and easier, more convenient and effective than other available mathematical techniques.

On the Existence and Uniqueness Analysis of Fractional Blood Glucose-Insulin Minimal Model

Abstract: In our research work, we suggest the modified minimal model of fractional order and analyze it using the homotopy decomposition method (HDM). The minimal model is quite a useful mathematical model which describes the behavior of glucose-insulin metabolism. The original model was given in the 80s and has been updated over a different period. In this modified model, we add a one-factor diet which plays an important role in the blood-glucose analysis. We obtained the numerical results by using the homotopy decomposition method. HDM is extremely useful, significant, and very simple. We also discuss the existence and uniqueness of the fractional model.

Numerical Investigation of Nonlinear Shock Wave Equations with Fractional Order in Propagating Disturbance

Abstract: The symmetry design of the system contains integer partial differential equations and fractional-order partial differential equations with fractional derivative. In this paper, we develop a scheme to examine fractional-order shock wave equations and wave equations occurring in the motion of gases in the Caputo sense. This scheme is formulated using the Mohand transform (MT) and the homotopy perturbation method (HPM), altogether called Mohand homotopy perturbation transform (MHPT). Our main finding in this paper is the handling of the recurrence relation that produces the series solutions after only a few iterations. This approach presents the approximate and precise solutions in the form of convergent results with certain countable elements, without any discretization or slight perturbation theory. The numerical findings and solution graphs attained using the MHPT confirm that this approach is significant and reliable.

An Application of Homotopy Perturbation Method to Fractional-Order Thin Film Flow of the Johnson–Segalman Fluid Model

Abstract: Thin film flow is an important theme in fluid mechanics and has many industrial applications. These flows can be observed in oil refinement process, laser cutting, and nuclear reactors. In this theoretical study, we explore thin film flow of non-Newtonian Johnson–Segalman fluid on a vertical belt in fractional space in lifting and drainage scenarios. Modelled fractional-order boundary value problems are solved numerically using the homotopy perturbation method along with Caputo definition of fractional derivative. In this study, instantaneous and average velocities and volumetric flux are computed in lifting and drainage cases. Validity and convergence of homotopy-based solutions are confirmed by finding residual errors in each case. Moreover, the consequences of different fractional and fluid parameters are graphically studied on the velocity profile. Analysis shows that fractional parameters have opposite effects of the fluid velocity.

The Analysis of the Fractional-Order Navier-Stokes Equations by a Novel Approach

Abstract: This article introduces modified semianalytical methods, namely, the Shehu decomposition method and q-homotopy analysis transform method, a combination of decomposition method, the q-homotopy analysis method, and the Shehu transform method to provide an approximate method analytical solution to fractional-order Navier-Stokes equations. Navier-Stokes equations are widely applied as models for spatial effects in biology, ecology, and applied sciences. A good agreement between the exact and obtained solutions shows the accuracy and efficiency of the present techniques. These results reveal that the suggested methods are straightforward and effective for engineering sciences models.