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This work applies the Aboodh transform with the homotopy perturbation method (HPM) to solve fractional differential equations. As the Aboodh transform is limited to linear equations only, HPM is an effective and dominant method for nonlinear differential equations to obtain approximate solutions. The role of NWSE is important in nonlinear systems that explain how stripes arise in two‐dimensional systems. The ATHPM solution has also been compared with LTDM and VIM methods, and it has been shown that ATHPM has more accuracy with a less absolute error than the exact solution, VIM, and LTDM. The final results perform very well with the exact solution. Maple is used to represent 3‐D surfaces and to find numerical values shown in tables. [ABSTRACT FROM AUTHOR]

The study of steady natural convection Couette flow is vital in designing as well as the optimization of microfluidic devices, geothermal energy systems, cooling of electronic devices and systems, etc., due to several recent applications. The present investigation aims to analyze radiative heat transfer and dissipative energy in the free convection of Couette flow within a vertically positioned channel. Incorporating carbon nanotubes (CNTs), specifically single-wall carbon nanotubes (SWCNTs) and multi-wall carbon nanotubes (MWCNTs), into the base fluid water enhances the flow phenomena. Additionally, the explanation of heat source/sink on the energy phenomenon encounters various properties. Further, suitable similarity variables are employed for the transformation of the governing equations. However, the homotopy perturbation method (HPM), an analytical approach, is used for the solution of the coupled ordinary differential equations. The thermophysical parameters and their impact are depicted through graphs, and the comparative analysis is presented via tables. Explore the combined effects of CNT nanoparticle concentrations on the Couette flow through a vertical channel. The inclusion of radiating heat on free convection of nanofluid enriches the flow phenomena. The adaptation of various thermophysical properties, i.e., in particular, the thermal conductivity, shows its effectiveness on the heat transport phenomenon. Clarify different contributing parameters by using the homotopy analysis method. [ABSTRACT FROM AUTHOR]

Our manuscript is related to use Caputo fractional order derivative (CFOD) to investigate results of non-linear mode in plasma. We establish results for both temporal and spatial approximate solution. For the require results, we use reduction perturbation method (RPM) to find the analytical solution of the dust acoustic shock waves. Further, using the same technique we find the solitary wave potential and compared the solutions obtained with another very useful technique known as Homotopy perturbation method (HPM). The comparison of results for both approaches are more precise and agreed with the exact solution of the problem. Finally, we present graphical representation for different fractional order for both temporal and spatial approximate solution. [ABSTRACT FROM AUTHOR]

In the present article, the method which was obtained from a combination of the conformable fractional double Laplace transform method (CFDLTM) and the homotopy perturbation method (HPM) was successfully applied to solve linear and nonlinear conformable fractional partial differential equations (CFPDEs). We included three examples to help our presented technique. Moreover, the results show that the proposed method is efficient, dependable, and easy to use for certain problems in PDEs compared with existing methods. The solution graphs show close contact between the exact and CFDLTM solutions. The outcome obtained by the conformable fractional double Laplace transform method is symmetrical to the gain using the double Laplace transform. [ABSTRACT FROM AUTHOR]

Our manuscript is related to use Caputo fractional order derivative (CFOD) to investigate results of non-linear mode in plasma. We establish results for both temporal and spatial approximate solution. For the require results, we use reduction perturbation method (RPM) to find the analytical solution of the dust acoustic shock waves. Further, using the same technique we find the solitary wave potential and compared the solutions obtained with another very useful technique known as Homotopy perturbation method (HPM). The comparison of results for both approaches are more precise and agreed with the exact solution of the problem. Finally, we present graphical representation for different fractional order for both temporal and spatial approximate solution.

Abstract In this work, we use two unique methodologies, the homotopy perturbation transform method and Yang transform decomposition method, to solve the fractional nonlinear seventh-order Kaup–Kupershmidt (KK) problem. The physical phenomena that arise in chemistry, physics, and engineering are mathematically explained in this equation, in particular, nonlinear optics, quantum mechanics, plasma physics, fluid dynamics, and so on. The provided methods are used to solve the fractional nonlinear seventh-order KK problem along with the Yang transform and fractional Caputo derivative. The results are significant and necessary for exploring a range of physical processes. This paper uses modern approaches and the fractional operator to develop satisfactory approximations to the offered problem. To solve the fractional KK equation, we first use the Yang transform and fractional Caputo derivative. He’s and Adomian polynomials are useful to manage nonlinear terms. It is shown that the suggested approximate solution converges to the exact one. In these approaches, the results are calculated as convergent series. The key advantage of the recommended approaches is that they provide highly precise results with little computational work. The suggested approach results are compared to the precise solution. By comparing the outcomes with the precise solution using graphs and tables we can verify the efficacy of the offered strategies. Also, the outcomes of the suggested methods at various fractional orders are examined, demonstrating that the findings get more accurate as the value moves from fractional order to integer order. Moreover, the offered methods are innovative, simple, and quite accurate, demonstrating that they are effective for resolving differential equations.

This research is a ‌combination of the homotopy ‌perturbation method with ‌Elzaki transform method ‌and Elzaki inverse to ‌solve some nonlinear ‌partial differential ‌equations. Where the method Elzaki ‌transform is not able to ‌deal with nonlinear ‌equations and find a solution by applying ‌the homotopy perturbation ‌method to solve nonlinear ‌differential equations ‌problems. The method ‌used has proven to be an ‌effective and easy way to ‌solve the nonlinear from kind ‌Newell-Whitehead-Segel ‌partial differential ‌equations, which are ‌classified and belong to ‌the category of ‌homogeneous partial ‌differential equations of ‌the second degree‌. The Elzaki transform method and Homotopy pertubation were shown to be very potent and successful integral transform methods for solving some non-linear equations when this method was compared to other well-known methods for handling the problems under consideration.

This study examines Titania nanofluids that conduct electricity and are combined with various base fluids, focusing on various aspects. A magnetic field analyzes a continuous flow of nanofluid over a rotating disk that is positioned at an angle with a three-dimensional geometry assumption. The continuity, momentum, and energy balance equations are established, transformed using similarity variables, and solved through Akbari–Ganji Method (AGM) and Homotopy Perturbation Method (HPM) semi-analytical methods. The effects of several parameters, such as Magnetic parameter (M), Hall parameter (m), Porosity parameter (γ), Radiation parameter (Rd), and Thickness of liquid (δ), are examined through a graphical representation of state variables, including skin friction and Nusselt number. Additionally, the Response Surface Method (RSM) method has been utilized to simultaneously display the effects of porosity parameters and various factors across the disk. The findings showed that the magnetic, porosity, Hall, and thickness parameters had a significant impact on the behavior of the nanofluid. Specifically, the magnetic field had a strong influence on the velocity and temperature distribution of the fluid as it flowed over the rotating disk, while changes in porosity, Hall, and thickness parameters also affected these state variables. [ABSTRACT FROM AUTHOR]

In this article, we investigate the existence and uniqueness of solutions to a generalized nonlinear functional integral equation (G‐NLFIE) associated with certain epidemic models of infectious diseases, defined within the Banach space C[0,1]$$ C\left[0,1\right] $$. Our existence results include several specific cases of nonlinear functional integral equations that commonly occur in nonlinear sciences. We then introduce an iterative algorithm that combines Adomian's decomposition method (ADM) with the modified homotopy perturbation method (mHPM) to approximate solutions to the G‐NLFIE. The paper addresses the convergence properties and error analysis of this method. Finally, we present numerical examples to demonstrate the effectiveness and efficiency of our proposed approach. [ABSTRACT FROM AUTHOR]

This work focuses on the heat transfer analysis of the magnetohydrodynamic peristaltic flow of blood through the gap between two coaxial flexible tubes of different wavelengths. In this model, the non-Newtonian biviscosity fluid is assumed to be blood, which is flowing through the annulus region between the inclined tubes. The governing equations for the considered problem are simplified under the assumptions of a zero Reynolds number and long wavelength approximations. An analytical expression for the temperature is obtained in its closed form, while a semi-analytical solution for the axial velocity of the moving fluid is determined using the homotopy perturbation method. Expressions for various flow variables such as shear stress, volumetric flow rate, pressure rise, and frictional force at the surface of inner and outer tubes are also obtained. In this work, we discussed the impact of various flow parameters like the Hartmann number, Grashof number, heat source, upper limit apparent viscosity coefficient, amplitude ratios of inner and outer tubes, radius ratio, and wavelength ratio on the above flow variables. The streamline contour plots are also drawn for the realization of the flow pattern of the blood inside the endoscope. The comparison of shear stresses for the peristaltic and rigid endoscopes and the validation of the present result with the previously established results are also discussed. A noteworthy observation drawn from the present model is that the pressure gradient enhances for increasing values of wavelength ratio when the wavelength of the inner tube is less than the wavelength of the outer tube, whereas the pressure gradient gets suppressed for increasing values of wavelength ratio when the wavelength of the inner tube is greater than the wavelength of the outer tube. From the present analysis, it is also found that the shear stress τrz of blood is the least for a peristaltic endoscope as compared to the rigid one. Therefore, this study may be applicable to medical practitioners for laparoscopic purposes, as a peristaltic endoscope may be a more appropriate device due to its flexibility than a rigid endoscope. [ABSTRACT FROM AUTHOR]

To understand the dynamics of neurodegenerative disease, we consider a fractional-order yeast prion mathematical model. We characterize the system dynamics by locating the steady states and perform their local stability analysis both theoretically and numerically. We employ the homotopy perturbation method to determine approximate solutions for the proposed model. By varying the fractional orders, we generate plots of these solutions over time to examine the dynamics of the Sup35 monomer and Sup35 prion populations. Our simulations reveal that the dynamical behaviors of both populations can undergo alterations depending upon the chosen fractional orders and the value of the threshold parameter. [ABSTRACT FROM AUTHOR]

This paper introduces a groundbreaking method, Homotopy-based Fourier transform, integrating Fourier transform and Homotopy perturbation for refined nonlinear problem-solving. The modification enhances solution technique efficiency, notably accelerating convergence, particularly in solving the Korteweg–de Vries equation. Demonstrating versatility, the method effectively addresses ordinary and partial differential equations, showcasing its applicability across diverse mathematical scenarios. Moreover, the approach is extended to nonlinear dynamical systems, illustrating its robustness in handling complex dynamic behaviors. This method proves especially suitable for highly nonlinear differential equations, offering an efficient and effective tool for scientists and engineers dealing with intricate mathematical models. By significantly improving convergence rates, the Homotopy-based Fourier transform stands out as a valuable asset in unraveling the complexities of nonlinear systems across various scientific and engineering applications. [ABSTRACT FROM AUTHOR]

In this work, we use two unique methodologies, the homotopy perturbation transform method and Yang transform decomposition method, to solve the fractional nonlinear seventh-order Kaup–Kupershmidt (KK) problem. The physical phenomena that arise in chemistry, physics, and engineering are mathematically explained in this equation, in particular, nonlinear optics, quantum mechanics, plasma physics, fluid dynamics, and so on. The provided methods are used to solve the fractional nonlinear seventh-order KK problem along with the Yang transform and fractional Caputo derivative. The results are significant and necessary for exploring a range of physical processes. This paper uses modern approaches and the fractional operator to develop satisfactory approximations to the offered problem. To solve the fractional KK equation, we first use the Yang transform and fractional Caputo derivative. He's and Adomian polynomials are useful to manage nonlinear terms. It is shown that the suggested approximate solution converges to the exact one. In these approaches, the results are calculated as convergent series. The key advantage of the recommended approaches is that they provide highly precise results with little computational work. The suggested approach results are compared to the precise solution. By comparing the outcomes with the precise solution using graphs and tables we can verify the efficacy of the offered strategies. Also, the outcomes of the suggested methods at various fractional orders are examined, demonstrating that the findings get more accurate as the value moves from fractional order to integer order. Moreover, the offered methods are innovative, simple, and quite accurate, demonstrating that they are effective for resolving differential equations. [ABSTRACT FROM AUTHOR]

In this paper, a delay - induced pest control model is proposed. We have introduced a time delay in healthy trees and whitefly population in the infected tree density of the proposed system of equations to reduce the probability of healthy trees becoming infected, as well as the level of infection. We have analyzed the impact of time delay on the stability of the equilibrium and establish requirements to verify its asymptotic stability over all delays. The solutions of this system of non-linear ordinary differential equations(ODEs) and delay differential equations(DDEs) are presented by using homotopy perturbation method(HPM). Numerical simulation is also obtained for the same model of both ODE and DDE by using MATLAB software. The proposed method works very well and is easy to use, as shown by these findings. Our aim is to control the spread of whitefly such that harvest is not affected. [ABSTRACT FROM AUTHOR]

The main focus of this paper is to introduce and analyze numerical methods for solving nonlinear Fredholm integro-differential equations. Two specific methods, namely homotopy perturbation and variational iteration methods, are implemented in this study. To assess the accuracy of the proposed schemes, extensive computational testing is conducted. The convergence analysis is performed using contracting mapping, which is a common method for proving the convergence of iterative algorithms. Additionally, the paper explores the error bound of an approximate solution generated from the partial sum of the series. This provides insights into the quality of the approximation.Comparison studies between the proposed methods are also carried out to evaluate their performance. The accuracy of the methods is assessed through numerical examples and compared against existing solutions. To further analyze the errors, l∞ and l2 norms are used. These norms quantify the differences between the approximate and actual solutions, providing a measure of accuracy. Finally, the efficiency of the approach is evaluated, considering factors such as computational complexity and execution time. The results of the numerical experiments are compared with previous studies and analytical solutions to validate their reliability and compatibility. [ABSTRACT FROM AUTHOR]

The goal of this paper is to utilize the homotopy perturbation method (HPM) and Laplace transform to provide an approximate analytical expression to the non-linear time-dependent reaction diffusion equation arising in a mathematical model of an immobilized enzyme system with external mass transfer resistance. This mathematical model is a non-steady, non-linear reaction diffusion equation based on Michaelis–Menten kinetics. Approximate analytical expressions are also provided for various geometries of the enzyme catalytic pellets, namely, planar, cylindrical, and spherical. Obtained semi-analytical expressions are proven to fit for all the parameters appearing in the system and for all the geometries of enzyme catalytic pellets. When comparing the numerical and approximate analytical solutions, satisfactory results are obtained. Also, approximate analytical expressions of the effectiveness factor (EF) of the immobilized system are presented, and the effect of parameters on the EF is also analyzed. [ABSTRACT FROM AUTHOR]

This research delves into the comprehensive study of the mathematical model governing the propagation of tsunami waves along ocean coastlines in imprecise or uncertain environment. The model is based on the widely used shallow-water assumption, which can be represented by a system of non-linear fuzzy partial differential equations. Utilizing fuzzy numbers to represent the problem's uncertainty may have distinct advantages from a certain perspective. As such, this work employs the homotopy perturbation method (HPM) to obtain fuzzy approximate analytical solutions for the governing model under various coastal slopes and ocean depths. The addition of uncertainty to the model may add complexity to the methodology. Therefore, a double parametric approach has been integrated into the HPM procedure. This approach allows us to analyze the impact of coastal slope and sea depth on tsunami wave characteristics, specifically wave velocity and run-up height, in uncertain environments. Furthermore, the numerical results obtained for fuzzy tsunami wave velocity and wave height resemble the real behavior of tsunamis. These can be seen through the given fuzzy plots, both in 2D and 3D forms. Finally, to validate the obtained solutions, a comparison has been made between the special case of the present fuzzy solutions and the existing precise (crisp) solutions. [ABSTRACT FROM AUTHOR]

This paper considers the movement of an incompressible non-Newtonian Walters' B liquid within a rotating annulus. The flow lies within porous media and is magnetized by a uniform normal magnetic field. Mass transmission is included in the arrangements of homogeneous and heterogeneous chemical reactions (HHCR) with a thermal diffusion effect. Heat diffusion is exposed with a temperature- dependent heat source (THS) and an exponential radial-dependent heat source (ERHS), besides Joule heating and viscous dissipations. The mathematical construction is shown by momentum, energy and concentration of chemical materials equations along with time-periodic boundary conditions. The innovation of this study stems from considering mass transfer utilizing HHCR, with thermal diffusion effects in the flow within a rotating annulus, trying to understand the effect of these reactions on the flow and related applications. The principal regulating the structure of nonlinear partial differential equations is transferred by ordinary ones utilizing appropriate similarity analysis. These equations are analyzed by the homotopy perturbation method. Subsequently, the analytical solutions of the main profiles are obtained, and a graphical construction is established to clarify the effects of the relevant physical elements. It is observed that the velocity waves decrease with annulus width enlargement and increase with the wave phase amplitude. With an increase in all significant parameters, heat transmission gets better. Furthermore, as the strength of HHCR is increased, the concentration of molecules decreases. These outcomes are significant in several applications like cancer therapy, biochemical manufacturing purposes through reactors, where biochemical interactions can be regulated by accurate control of various parameters. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study a class of forest disease and insect infection models with time delay. The exact solutions are successfully obtained by using the homotopy perturbation method. [ABSTRACT FROM AUTHOR]

The present work describes the fractional view analysis of Newell-Whitehead-Segal equations, using an innovative technique. The work is carried with the help of the Caputo operator of fractional derivative. The analytical solutions of some numerical examples are presented to confirm the reliability of the proposed method. The derived results are very consistent with the actual solutions to the problems. A graphical representation has been done for the solution of the problems at various fractional-order derivatives. Moreover, the solution in series form has the desired rate of convergence and provides the closed-form solutions. It is noted that the procedure can be modified in other directions for fractional order problems. [ABSTRACT FROM AUTHOR]

This work presents experimental and numerical studies of heat transfer during cooling fluid flow in a group of five minichannels 1 mm deep. The main purpose was to determine the heat transfer coefficient on the contact surface between the fluid and the heated wall of the selected minichannel at subcooled boiling. The temperature distribution on the outer surface of the heated plate was measured by means of an infrared camera. Thermal and flow parameters were monitored by an appropriate data-acquisition system. The test section was placed horizontally with fluid flowing above the heated wall. The HFE-649, HFE-7100 and HFE-7200 working fluids were examined in the experiments. Simcenter STAR-CCM+ software was used for numerical analysis of heat transfer in the test section. Furthermore, a simplified two-dimensional (2D) model was proposed that designates subcooled boiling heat transfer during fluid flow in a central minichannel. The heat-transfer process in the heated plate and the working fluid was described using indicated partial differential equations with appropriate boundary conditions. The solution to the proposed system of equations led to the solving of two more inverse Cauchy-type problems. The classical Trefftz method (TM) and the homotopy perturbation method (HPM) combined with the TM allowed for obtaining temperature distributions in the heater and the fluid and consequently, the heat transfer coefficient at the heater–fluid interface from the Robin boundary condition. Comparison of the results from numerical simulation due to Simcenter STAR-CCM+ showed similar temperature distributions at the heated surface. The calculated heat transfer coefficients, by HPM and Simcenter STAR-CCM+, were validated using the 1D approach. Furthermore, the results from simulations in Simcenter STAR-CCM+ in the form of local temperatures of the heater were confronted with experimental data for comparison. Similar results were achieved. [ABSTRACT FROM AUTHOR]

This paper addresses the semi-analytical modeling and stability analysis of coupled slosh-tank dynamics within a multi-body system framework using the Homotopy Perturbation Method (HPM). The sloshing motion of the liquid inside the tank is represented using an equivalent pendulum model, which allows for a more accurate depiction of the dynamics involved. Nonlinear equations of motion are derived using the Lagrangian approach to account for lateral and longitudinal excitations, explicitly focusing on compressive oscillations. Our model investigates the influence of critical parameters, including viscous damping, amplitude, and excitation frequency. These parameters are examined at two specific points, one within and one outside the stability regions, to understand their effects on the overall system behavior. The study demonstrates that viscous damping is particularly significant in moving points from unstable to stable regions compared to other principal parameters. Simulations are conducted to visualize stability phenomena through stability diagrams, phase portraits, and time histories of sloshing amplitude. The results obtained using HPM are compared to those from the numerical Runge-Kutta method, validating the analytical approach. This comparison highlights the effectiveness of HPM in accurately capturing the dynamics and stability characteristics of coupled slosh-tank systems, offering valuable insights into the design and control of such systems. [ABSTRACT FROM AUTHOR]

This article describes the non-linear reaction-diffusion model, which depicts the behaviour of hydrogen peroxide production and glucose oxidation in the chitosan-alginate microsphere. Analytical solutions of glucose, oxygen, gluconic acid and hydrogen peroxide concentrations in planar coordinates under steady-state circumstances are obtained for all reaction-diffusion parameters. The non-linear boundary value problem's approximate analytical expressions are obtained using asymptotic techniques based on the homotopy perturbation and Akbari-Ganji method. The exact and commonly used MATLAB program provided a numerical simulation. It is demonstrated that the resulting analytical expressions strongly agree with the numerical outcomes reported in the literature. Furthermore, determining hydrogen peroxide flux and pH profiles within the microspheres are also discussed. The theoretical findings make it possible to forecast and enhance the efficiency of enzyme kinetics.

In this investigation, a new approach is proposed to study blood flow through a stenotic artery under the impact of electro-osmosis and several other main parameters. The new approach depends on the Chebyshev series, the Akbari-Ganji method, and the new homotopy perturbation method named the Chebyshev Akbari-Ganji homotopy perturbation method (CAGHPM). The validity of the proposed method was ensured by the good agreement of its results with the results obtained by other methods mentioned in previous studies. The influences of the electro-osmotic parameter and the Helmholtz-Smoluchowski velocity, as well as the inclination angle, magnetic field, porosity, and chemical reaction, are discussed in terms of the velocity, temperature, wall shear stress, and concentration of blood flow. The results obtained by using the CAGHPM are more accurate than those obtained by other methods used to solve the current problem. Furthermore, error tables, figures, and convergence analysis theoretically and computationally show the efficiency and effectiveness of the proposed new method.

In this study, a biosensor based on an electrode that has been chemically altered is represented mathematically. The homotopy perturbation method analytically solves the model, and simulation results are verified against analytical solutions for various physical parameter values. The biosensor's mathematical model comprises three layers: an exterior diffusion layer, a dialysis membrane, and an enzyme layer. An effective diffusion coefficient is added to the mathematical model to integrate the diffusion layer with the dialysis membrane. Then, the problem is reduced to the two-layer model, which can effectively replace the three-layer model. The homotopy perturbation method (HPM) is used to solve the non-linear system of chemically modified electrode equations into an approximate analytical formulation. These formulations will be compared with MATLAB-based numerical simulations. Furthermore, an alternative to the current density in steady-state and the biosensor's sensitivity, resistance are explored.

The Fitzhugh–Nagumo equation is an essential governing equation that explains the process of impulse transmission from the nerves. This work suggests a novel generalized arbitrary-order Fitzhugh–Nagumo equation that is based on the recently developed nonsingular fractional operator. The modified Fitzhugh–Nagumo equation has been generalized using fractional Yang–Abdel–Cattani operator to provide a more accurate representation of the equation with memory effects and non-local behavior. Further, we discussed some interesting and new properties of the proposed fractional operator and the Rabotnov exponential fractional kernel with a modified Sumudu integral transform. In order to solve the proposed nonlinear differential equation, the homotopy perturbation method coupled with the modified Sumudu integral transform technique is implemented to examine the result of a newly proposed Yang–Abdel–Cattani fractional Fitzhugh–Nagumo equation, which converges to the exact solution. The nonlinear terms of this equation are handled in terms of He’s polynomials. The existence and uniqueness of the solution have been demonstrated using the Banach space fixed point theorem. The convergence and error analysis of the suggested technique are also discussed. The findings are expressed in terms of generalized Mittag-Leffler functions. The obtained results are plotted with the help of MATLAB R2016a mathematical software. The results in this work are more accurate, and it is proposed that the new Yang–Abdel–Cattani fractional derivative operator is an effective tool for finding the results of any other nonlinear problems arising in science and engineering. All three problems have been computed for different orders to confirm the significance of the fractionalizing concept in the proposed equation. The increase of the fractional order value ℘ to 1 confirms that our proposed time-fractional Fitzhugh–Nagumo equation in the Yang–Abdel–Cattani sense gets close to the analytic solution.

The goal of this paper is to utilize the homotopy perturbation method (HPM) and Laplace transform to provide an approximate analytical expression to the non-linear time-dependent reaction diffusion equation arising in a mathematical model of an immobilized enzyme system with external mass transfer resistance. This mathematical model is a non-steady, non-linear reaction diffusion equation based on Michaelis–Menten kinetics. Approximate analytical expressions are also provided for various geometries of the enzyme catalytic pellets, namely, planar, cylindrical, and spherical. Obtained semi-analytical expressions are proven to fit for all the parameters appearing in the system and for all the geometries of enzyme catalytic pellets. When comparing the numerical and approximate analytical solutions, satisfactory results are obtained. Also, approximate analytical expressions of the effectiveness factor (EF) of the immobilized system are presented, and the effect of parameters on the EF is also analyzed.

AbstractThis article investigates the thermal, variable viscosity axial Couette flow between two con- centric circular cylinders, taking into account both the Navier and the dynamic slip boundary conditions. In order to put in evidence the effect of dynamic slip on the behavior of the solution, we have kept the flowing fluid to a simple Newtonian viscous fluid. The nonlinear governing equations for momentum and energy balance are solved under startup condition using the Laplace transform (LT) technique, in conjunction with the Homotopy Perturbation Method (HPM). The first two orders of approximation for temperature and velocity are obtained, as well as the entropy generation in the space occupied by the flow and the volumetric flux. Numerical results and graphical representations of the solution are provided and discussed to depict the effects of the slip parameters on the velocity and the temperature profiles, on the entropy generation rate and on the volumetric flux. [ABSTRACT FROM AUTHOR]

In this paper, inscribed non-Newtonian Williamson fluid relaxation and contraction phenomena due to the effect of magnetohydrodynamics (MHD) is taken into account. This issue is modeled, and precise solutions are handled using a low Reynolds number and long wavelength approximation. Graphical depiction of the consequences of various parameters on temperature and velocity are provided. It becomes clear that the temperature reactions to increases in Biot numbers are decreased. Considering a cylindrical coordinate system, the impact of various physical parameters on the temperature distribution, velocity, the pressure gradient, and frictional force are secured from the flow for non-Newtonian Williamson fluid exposed from MATLAB. Additionally, it has been discovered that the velocity profile at the tube's center increases for shear-thinning a fluid and decreases for thickening a fluid. Furthermore, the peristaltic walls show the opposite behavior. [ABSTRACT FROM AUTHOR]

The current study aims to utilize the homotopy perturbation method (HPM) to solve nonlinear dynamical models, with a particular focus on models related to predicting and controlling pandemics, such as the SIR model. Specifically, we apply this method to solve a six-compartment model for the novel coronavirus (COVID-19), which includes susceptible, exposed, asymptomatic infected, symptomatic infected, and recovered individuals, and the concentration of COVID-19 in the environment is indicated by S (t) , E (t) , A (t) , I (t) , R (t) , and B (t) , respectively. We present the series solution of this model by varying the controlling parameters and representing them graphically. Additionally, we verify the accuracy of the series solution (up to the (n − 1) t h -degree polynomial) that satisfies both the initial conditions and the model, with all coefficients correct at 18 decimal places. Furthermore, we have compared our results with the Runge–Kutta fourth-order method. Based on our findings, we conclude that the homotopy perturbation method is a promising approach to solve nonlinear dynamical models, particularly those associated with pandemics. This method provides valuable insight into how the control of various parameters can affect the model. We suggest that future studies can expand on our work by exploring additional models and assessing the applicability of other analytical methods. [ABSTRACT FROM AUTHOR]

AbstractCarbon nanotubes have gained significant attention due to their unique properties like small dimensions, high thermal conductivity, and unique architecture. These nanotubes have enormous potential in various fields such as nanomedicine, mechanical compound materials, sensor devices, bio-sensors, and many other applications. The present study investigates the impact of adding nanoparticles such as iron oxide (Fe3O4), single-walled carbon nanotubes (SWCNT), and multi-walled carbon nanotubes (MWCNT) on magnetohydrodynamics (MHD) oscillatory water-based flow over a porous medium. The aim is to examine the effect of heat and mass transfer rate and radiative viscous dissipation that interacts with Fe3O4 nanoparticles and carbon nanotubes. The governing equations of the dynamic nanofluid systems are transformed into a set of coupled nonlinear partial differential equations through suitable dimensionless transformations and further converted into ordinary differential equations by taking appropriate solutions for oscillatory-type nanofluid flow. The homotopy perturbation method is applied to solve the governing equations with oscillatory boundary conditions and the solution is also compared with the numerical solver bvp4c. The rate of increase in the temperature and Heat Transfer are computed by the linear slope regression method and results are computed for nanoparticles SWCNT, MWCNT, and Ferro. Also, Skin friction and rate of heat transfer are computed and compared for the nanoparticles with varying Peclet and Eckert numbers. The velocity profile and temperature profile of SWCNT/MWCNT/Ferro nanofluids are being compared. The heat transfer rate is observed to be high for Ferro nanoparticles compared to SWCNT and MWCNT. [ABSTRACT FROM AUTHOR]

In this paper, the mathematical model of the biofiltration of a mixture of n-propanol and toluene is developed based on the gasbiofilm phases. The homotopy perturbation technique is applied to get the approximate analytical expression for the concentration of toluene and n-propanol in non-steady-state conditions for both gasbiofilm interfaces. Our analytical results were validated by direct comparison with numerical simulations generated using the "bvp4c" function in the MATLAB platform. For various parameter values, the comparison was found to be in good agreement. The saturated and unsaturated kinetics also have been investigated analytically. [ABSTRACT FROM AUTHOR]

This article depicts an approximate solution of systems of nonlinear fractional biochemical reactions for the Michaelis-Menten enzyme kinetic model arising from the enzymatic reaction process. This present work is concerned with fundamental enzyme kinetics utilised to assess the efficacy of powerful mathematical approaches such as the homotopy perturbation method (HPM), homotopy analysis method (HAM), and homotopy analysis transform method (HATM) to get the approximate solutions of the biochemical reaction model with time-fractional derivatives. The Caputo-type fractional derivatives are explored. The proposed method is implemented to formulate a fractional differential biochemical reaction model to obtain approximate results subject to various settings of the fractional parameters with statistical validation at different stages. The comparison results reveal the complexity of the enzyme process and obtain approximate solutions to the nonlinear fractional differential biochemical reaction model. [ABSTRACT FROM AUTHOR]

A local fractional modification of the Duffing equation is considered, and the homotopy perturbation method is employed to reveal its frequency-amplitude relationship, which is of paramount importance in the optimal design of the energy harvesting devices and chatter detection. Effects of the initial conditions on the periodic property is also discussed. [ABSTRACT FROM AUTHOR]

Purpose: Financial mathematics is one of the most rapidly evolving fields in today's banking and cooperative industries. In the current study, a new fractional differentiation operator with a nonsingular kernel based on the Robotnov fractional exponential function (RFEF) is considered for the Black–Scholes model, which is the most important model in finance. For simulations, homotopy perturbation and the Laplace transform are used and the obtained solutions are expressed in terms of the generalized Mittag-Leffler function (MLF). Design/methodology/approach: The homotopy perturbation method (HPM) with the help of the Laplace transform is presented here to check the behaviours of the solutions of the Black–Scholes model. HPM is well known for its accuracy and simplicity. Findings: In this attempt, the exact solutions to a famous financial market problem, namely, the BS option pricing model, are obtained using homotopy perturbation and the LT method, where the fractional derivative is taken in a new YAC sense. We obtained solutions for each financial market problem in terms of the generalized Mittag-Leffler function. Originality/value: The Black–Scholes model is presented using a new kind of operator, the Yang-Abdel-Aty-Cattani (YAC) operator. That is a new concept. The revised model is solved using a well-known semi-analytic technique, the homotopy perturbation method (HPM), with the help of the Laplace transform. Also, the obtained solutions are compared with the exact solutions to prove the effectiveness of the proposed work. The different characteristics of the solutions are investigated for different values of fractional-order derivatives. [ABSTRACT FROM AUTHOR]

Homotopy perturbation method (HPM) is applied to approximate and simulate the cell-cell adhesion model. [ABSTRACT FROM AUTHOR]

The analysis of MHD UCM fluid flow between a pair of rectangular plates with the top plate moving either towards or apart from the stationary bottom porous plate is presented in this work. By employing similarity transformations, the governing equations of momentum are transformed into non-linear ordinary differential equations. An approximate analytical solution is achieved by adopting the homotopy perturbation technique. The aim of this work is to determine the velocity profile and coefficients of skin friction for various values of different physical parameters. It has been found that as the magnetic parameter enhances, the squeezing flow retards in injection and suction cases. It is also observed that the velocity field declines in the core region with a rise in the porosity parameter. This theoretical study is helpful in the processing of visco-elastic polymers in industry. [ABSTRACT FROM AUTHOR]

The study in this manuscript aims to analyse the impact of thermal radiation on the two-dimensional magnetohydrodynamic flow of upper convected Maxwell (UCM) fluid between parallel plates. The lower plate is porous and stationary, while the top plate is impermeable and moving. The equations that describe the flow are transformed into non-linear ordinary differential equations with boundary conditions by employing similarity transformations. The Homotopy Perturbation Method (HPM) is then employed to approach the obtained non-linear ordinary differential equations and get an approximate analytical solution. The analysis includes plotting the velocity profile for different Reynolds number values and temperature distribution curves for distinct physical parameters such as Reynolds number, Deborah number, magnetic parameter, porosity parameter, radiation parameter, and Prandtl number. In the case of injection, the temporal profile declines with an increase in radiation parameter as the plates move away from each other, and an opposite trend is observed as plates move towards each other. Furthermore, the skin friction coefficient and heat transfer rate are analysed for the impact of these parameters using HPM. The numerical values obtained using HPM are compared using the classical finite difference method. The results show good agreement between the semi-analytical and numerical solutions. [ABSTRACT FROM AUTHOR]

The non-Newtonian characteristic of the fluid flow phenomena has several applications in industries and the production of engineering gadgets. Based upon the role, the present work deals with the flow of Williamson nanofluid past two inclined parallel sheets. Further, the influence of radiating heat due to thermal radiation, Ohmic heating because of the magnetization, and inclined angle on Williamson fluid are presented, which are vital for the heat transfer enhancement. The proposed study became novel due to the scrutiny of entropy for the irreversibility processes within the system that is obtained due to heat transfer. The designed model with dimensional form is developed by converting to dimensionless forms with the help of similarity transformations. The finalized transformed equations are handled analytically using the Homotopy perturbation method (HPM). Heat irreversibility is calculated by inserting the observed values into entropy generation and Bejan number. The suggested method of solution for the momentum profile is verified by comparing it to a previously published numerical result in some limited circumstances, and great agreement is found. The result reveals that the entropy generation is enhanced near the upper wall of the channel for the magnetized particle but thermal radiation attenuates it, however, the fluid temperature boosts up for the enhanced coupling constraint i.e. for the existence of Eckert number. [ABSTRACT FROM AUTHOR]

This study's primary objective is to investigate the Jeffery–Hamel model and entropy generation in the magnetohydrodynamic (MHD) flow of hybrid nanofluid across stretching, shrinking, converging, and diverging channels. Aluminum oxide (Al2O3) and silver (Ag) are nanoparticles, using blood as the base fluid. The controlling nonlinear coupled partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs) with similarity transformations and then solved using the Homotopy Perturbation Method (HPM) and shooting technique (Bvp4c). The HPM is compared to the numerical method (NM), and the results are more accurate and reliable. The effects of velocity, temperature, entropy generation, and the Bejan number on physical parameters like a magnetic field, the Reynolds number, magnetic field, and the Brinkman number are discussed through graphs. The heat transfer and skin friction coefficients are also studied and portrayed as graphs and tables. The influence of a magnetic field on the velocity profile of stretching and shrinking through converging and diverging channels. When increasing the magnetic field, the velocity profile increases. Physically, an increase in the magnetic field produces a higher Lorentz force, which improves fluid resistance and restricts fluid particle flow within the given topology. The temperature profile decreases in the stretching and shrinking of converging channels as magnetic field values rise. As well as the opposite behavior observed in stretching and shrinking for diverging channels. In this model, biomedical applications include studying physiological systems, surgical procedures, nano-pharmacological delivery systems, and nano-mediated treatment of atherosclerosis. [ABSTRACT FROM AUTHOR]

The research aims to scrutinize the significance of MHD, radiation, Joule heating, porous medium, and heat generation impacts on the convective boundary flow. It contains blood as the base fluid, silver (Ag), and aluminum oxide (Al2O3) as the nanoparticles over a curved stretching and shrinking sheet. To investigate the rheology of blood, we used a Williamson fluid model with comparisons for both curved stretching/shrinking sheets for velocity and a comparison between linear and nonlinear thermal radiation for temperature, entropy production, and Bejan number. This model is used to study physiological systems, surgical procedures, and nano-pharmacological delivery systems. The governing nonlinear coupled PDEs are transformed into nonlinear associated ODEs via the similarity transformation in the Maple software solver using the Homotopy Perturbation Method. The Homotopy Perturbation Method (HPM) results are compared with the Numerical Method (NM) results, where HPM gives reliable results. The results and discussion provide graphical results with several parameters for velocity, temperature, entropy production, Bejan number, Nusselt number, and skin friction. As the magnetic field parameter (M) increases, the velocity profile decreases for curved stretching sheet. We observe the opposite behavior of the velocity profile for a curved shrinking sheet. As the volume fractions increase, the temperature profile increases for linear and non-linear thermal radiation. The temperature profile rises due to the thermal conductivity of nanoparticles, which is amplified by estimations of the nanoparticle volume fractions. [ABSTRACT FROM AUTHOR]

Wireless sensor networks (WSNs) have attracted a lot of interest due to their enormous potential for both military and civilian uses. Worm attacks can quickly target WSNs because of the network's weak security. The worm can spread throughout the network by interacting with a single unsafe node. Moreover, the analysis of worm spread in WSNs can benefit from the use of mathematical epidemic models. We suggest a five-compartment model to characterize the mechanisms of worm proliferation with respect to time in WSN. Taking into account the ZZ transform convoluted with the Atangana-Baleanu-Caputo (ABC) fractional derivative operator, we employ it to analyze the characteristics and applications of the ZZ transformation using the Mittag-Leffler kernel. Moreover, we construct a new algorithm for the homotopy perturbation method (HPM) in conjunction with the ZZ transform technique to generate analytical solutions for the worm transmission model. Also, we address the qualitative aspects such as positivity, boundness, worm-free state, endemic state, basic reproduction number (R0) and worm-free equilibrium stability. Furthermore, we prove that the virus rate in sensor nodes is extinct if R01. In addition, we develop analytical findings to evaluate the series of solutions. Furthermore, a detailed statistical analysis is conducted to verify the nonlinear dynamics of the system by verifying the 0−1 test to determine whether uncertainty exists using approximation entropy and the C0 data. An extensive analysis of the vaccination class with respect to the transmitting class as well as the susceptible class is being used to investigate the effects of stepping up precautions on WP in WSN. Moreover, the modeling of the WSN revealed that reducing the fractional-order from 1 requires that the recommended approach be implemented at the highest rate so that there is no long-lasting immunization; instead, nodes remain briefly defensive before becoming vulnerable to future worm attacks.