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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]

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 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]

Diabetes is referred to a chronic metabolic disease signalized by elevated levels of blood glucose (also known as blood sugar level), which results over time in serious damage to the heart, blood vessels, eyes, kidneys, and nerves in the body. A mathematical assessment of the diabetes model using the Caputo fractional order derivative operator is given in this research paper. The concept of a Caputo fractional order derivative is a novel class of noninteger order derivative that has many applications in real-life scenarios. The proposed model is represented by a set of fractional ordinary differential equations. The authors employed the Sumudu Transform Homotopy Perturbation Method (STHPM) for finding the series solutions of the model being studied. By giving various numerical values to the respective model parameters, graphical analysis is also performed. It is observed in the numerical discussion that a decrease in both fractional order and leads to decrease in the number of diabetic people. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to inspect the unsteady MHD mixed convective Couette flow of viscous dissipative fluid in a vertical channel filled with porous material. The steady state analytical solutions of temperature, velocity, frictional force and heat transfer rate are obtained using the Homotopy perturbation method (HPM). The implicit finite difference technique is used to solve the transient leading equations numerically. Plots and tables with the dynamic flow parameters and dimensionless variables are displayed. During this computation, it was found that raising the Darcy parameter (Da) and the mixed convection (Gre) parameters both push the fluid flow upwards, but increasing the magnetic field's values had the opposite effect (M). [ABSTRACT FROM AUTHOR]

In this paper, some approximate analytical methods are reviewed, including the variational iteration method and the homotopy perturbation method. Based on the ideas of the two methods, a new method for solving a class of system of Fredholm integral equations of the second kind is proposed. Some examples are given to show the simple solution process and the accuracy of the solution for each method. The emphasis of this review paper focuses on Ji-Huan He's unapproachable ideas and the mathematics beauty, so the references are not exhaustive. [ABSTRACT FROM AUTHOR]

This paper proposes a numerical approach based on the fractional complex transform and the homotopy perturbation method to solving the space-time fractional Benjamin-Bona-Mahony (mBBM) equation with Caputo fractional derivative. Approximated solutions with high accuracy are provided without linearization or complicated computation. Numerical examples are given to illustrate the efficiency of this method. [ABSTRACT FROM AUTHOR]

Homotopy perturbation method (HPM) is applied in this paper for the transient study of some queueing/inventory/healthcare models. Semi-analytic approximate transient probabilities are obtained as power series in the time variable. The numerical implementation of the method using a programming language like MATLAB, is easy. However, the number of terms that is required in the above power series, to get an accurate approximation, grows with time. This obstacle is overcame by adopting a step-by-step HPM method. Implementation of the method is discussed in the case of the classic M/M/1 queue, the (s, S) inventory model with random replenishment time but without backlogs, the (s, S) inventory model with positive service and replenishment times, allowing infinite backlogs, and the Markov chain model for the patient handoff process in emergency departments. Numerical experimentation has been carried out to evaluate the efficiency of the method and to understand the challenges involved in its implementation. [ABSTRACT FROM AUTHOR]

This paper introduces a novel numerical approach for tackling the nonlinear fractional Phi-four equation by employing the Homotopy perturbation method (HPM) and the Adomian decomposition method (ADM), augmented by the Shehu transform. These established techniques are adept at addressing nonlinear difierential equations. The equation's complexity is reduced by applying the Shehu Transform, rendering it amenable to solutions via HPM and ADM. The eficacy of this approach is underscored by conclusive results, attesting to its proficiency in solving the equation. With extensive ramifications spanning physics and engineering domains like fluid dynamics, heat transfer, and mechanics, the proposed method emerges as a precise and eficient tool for resolving nonlinear fractional difierential equations pervasive in scientific and engineering contexts. Its potential extends to analogous equations, warranting further investigation to unravel its complete capabilities. [ABSTRACT FROM AUTHOR]

This paper presents a comparative study of two popular analytical methods, namely the Homotopy Perturbation Transform Method (HPTM) and the Adomian Decomposition Transform Method (ADTM), to solve two important fractional partial differential equations, namely the fractional heat transfer and porous media equations. The HPTM uses a perturbation approach to construct an approximate solution, while the ADTM decomposes the solution into a series of functions using the Adomian polynomials. The results obtained by the HPTM and ADTM are compared with the exact solutions, and the performance of both methods is evaluated in terms of accuracy and convergence rate. The numerical results show that both methods are efficient in solving the fractional heat transfer and porous media equations, and the HPTM exhibits slightly better accuracy and convergence rate than the ADTM. Overall, the study provides a valuable insight into the application of the HPTM and ADTM in solving fractional differential equations and highlights their potential for solving complex mathematical models in physics and engineering. [ABSTRACT FROM AUTHOR]

This paper focuses on novel approaches to finding solitary wave (SW) solutions for the modified Degasperis-Procesi and fractionally modified Camassa-Holm equations. The study presents two innovative methodologies: the Yang transformation decomposition technique and the homotopy perturbation transformation method. These methods use the Caputo sense fractional order derivative, the Yang transformation, the adomian decomposition technique, and the homotopy perturbation method. The inquiry effectively solves the fractional Camassa-Holm and Degasperis-Procesi equations, which also provides a detailed numerical and graphical comparison of the solutions found. The results, which include accurate solutions, derived solutions, and absolute error displayed in tabular style, demonstrate the effectiveness of the suggested procedures. These procedures are iterative, which results in several answers. The estimated absolute error attests to the correctness and simplicity of these solutions. Especially in plasma physics, these approaches may be expanded to handle various linear and nonlinear physical issues, including the evolution equations controlling nonlinear waves. [ABSTRACT FROM AUTHOR]

In this paper, the He–Elzaki transform method (HEM) is proposed. The method is formulated by combining He's variation iteration method and the modified Laplace transform, known as the Elzaki integral transform. This method is designed to solve the time-fractional telegraph equation that arises in electromagnetics. The Caputo sense is used to describe fractional derivatives. One of the advantages of this method is that the computation of the Lagrange multiplier is not necessarily required through the convolution theorem or integration in recurrence relations. Additionally, to reduce nonlinear computational terms, He's polynomial is determined using the homotopy perturbation method. The proposed method is applied to several examples of nonlinear fractional telegraph equations. The results obtained from these examples demonstrate that the proposed method is an efficient technique that facilitates the process of solving time-fractional differential equations. [ABSTRACT FROM AUTHOR]

Introduction: In this paper, we analyzed the study of a mathematical model of non-linear differential equation on the effect of HIV/AIDS disease among unaware HIV/AIDS-infected population. Material and methods: Population was divided into four categories, including HIV-negative individuals, unaware HIV-positive cases, aware HIV-positive, and AIDS patients. The model was investigated numerically and analytically using fourth-order Runge-Kutta approach and homotopy perturbation method (HPM). Results: We have discussed the parameter variation graphically. Conclusions: Determining the dynamics of HIV prevalence and investigating the effect of each parameter on the governing equation can be simple with analytical solution. [ABSTRACT FROM AUTHOR]

This paper presents a symplectic homotopy perturbation method for Hamiltonian systems with uncertainties. The effects of uncertainties cannot be ignored, but the research on symplectic algorithms of Hamiltonian systems with uncertainties is still weak. Besides, the nonlinear dynamic systems with uncertainties are far more complicated than deterministic linear systems. In this paper, taking two kinds of uncertainties into consideration, the dynamic responses of stochastic and interval Hamiltonian systems, especially uncertain nonlinear Hamiltonian systems, are investigated based on the homotopy perturbation method, respectively. By introducing an embedding parameter, a series of homotopy perturbation equations are deduced, and the uncertain mathematical characteristics of the dynamic responses of stochastic and interval Hamiltonian systems can be obtained by the symplectic algorithms, respectively. Then, the comparative study of calculation results by the symplectic homotopy perturbation method of stochastic and interval Hamiltonian systems is carried out and their compatible relationship is discussed. Eventually, three numerical examples are used to demonstrate the validity and engineering applicability in structural dynamic systems of the proposed method. The numerical results show the superiority of the proposed method in accuracy, stability and symplectic conservative compared with the non-symplectic Runge–Kutta method. [ABSTRACT FROM AUTHOR]

Considering the prevailing situations, the mathematical modeling and dynamics of novel coronavirus (2019-nCoV) particularly in India are studied in this paper. The goal of this work is to create an effective SEIRS model to study about the epidemic. Four different SEIRS models are considered and solved in this paper using an efficient homotopy perturbation method. A clear picture of disease spreading can be obtained from the solutions derived using this method. We parametrized the model by considering the number of infection cases from 1 April 2020 to 30 June 2020. Finally, numerical analysis and graphical representations are provided to interpret the spread of virus. [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 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]

This study develops a numerical strategy for finding the approximate solution of the nonlinear foam drainage (NFD) equation with a time-fractional derivative. In this paper, we formulate the idea of the Laplace homotopy perturbation transform method (LHPTM) using Laplace transform and the homotopy perturbation method. This approach is free from the heavy calculation of integration and the convolution theorem for the recurrence relation and obtains the solution in the form of a series. Two-dimensional and three-dimensional graphical models are described at various fractional orders. This paper puts forward a practical application to indicate the performance of the proposed method and reveals that all the outputs are in excellent agreement with the exact solutions. [ABSTRACT FROM AUTHOR]

This paper studies a hyperbolic tangent oscillator arising in a nonlinear packaging system. The high-order nonlinearity makes it difficult to study the system analytically. A modified homotopy perturbation method is proposed, where the initial guess is an approximate solution of the well-known Duffing oscillator; the solution accuracy can be improved by either improvement of the initial guess or continuous iteration. This paper gives a general approach to high-order nonlinear oscillators. [ABSTRACT FROM AUTHOR]

This paper estimated the problem of one-dimensional generalized thermoelastic half-space in medium considering two models: the Lord and Shulman (LS) model and the Dual-Phase-Lag (DPL) model. We assumed that the surface of the half-space was free from traction force and under an exponentially varying external heat source at the boundary with time. The technique of homotopy perturbation has been applied to find the approximate solution for the interactions of thermoelasticity with the applied boundary condition. The effect of a heat source that varies with the time and the free traction force are investigated for the temperature, displacement, and stress. The numerical results obtained are presented graphically to show the influence of the new external parameters. The results obtained illustrate the strong impacts on the displacement, temperature, and stress with the variations in the two models as well as the relaxation time parameter. The results show the agreement between the present results and the previous obtained results of the phenomenon and applicable, especially in biology, acoustics, engineering, and geophysics. [ABSTRACT FROM AUTHOR]

This study focuses on an artery with multiple stenoses, emphasizing the electro-osmotic effects. The artery's walls are porous, and slip boundary effects are present. Blood ow problems are better modeled with a slip and porous border. It is examined extensively due to the wide range of applications in the medical field, especially in diagnosing drug delivery and handling cellular irregularities. In this paper, we have visualized the non-Newtonian behavior of blood by using viscoelastic fluids as Williamson's fluid model. A mathematical model for an incompressible fluid is created, and the mathematical issue is then transformed into its dimensionless form by applying limitations in the case of mild multiple stenoses. The partial differential equations for the velocity and temperature profiles can be found when the problem is put into a dimensionless form. Analytical solutions of the resulting system are calculated with the help of the Homotopy Perturbation Method (HPM). The visual representation of analytically obtained solutions is investigated for both symmetric and non-symmetric geometries of stenosis. For varied values of ow rate Q and electro-osmotic parameter m, the streamlines are examined in detail. [ABSTRACT FROM AUTHOR]

In this paper, we examined the approximations to the time-fractional Kawahara equation and modified Kawahara equation, which model the creation of nonlinear water waves in the long wavelength area and the transmission of signals. We implemented two novel techniques, namely the homotopy perturbation transform method and the Elzaki transform decomposition method. The derivative having fractional-order is taken in Caputo sense. The Adomian and He's polynomials make it simple to handle the nonlinear terms. To illustrate the adaptability and effectiveness of derivatives with fractional order to represent the water waves in long wavelength regions, numerical data have been given graphically. A key component of the Kawahara equation is the symmetry pattern, and the symmetrical nature of the solution may be observed in the graphs. The importance of our suggested methods is illustrated by the convergence of analytical solutions to the precise solutions. The techniques currently in use are straightforward and effective for solving fractional-order issues. The offered methods reduced computational time is their main advantage. It will be possible to solve fractional partial differential equations using the study's findings as a tool. [ABSTRACT FROM AUTHOR]

Cardiac catheterization is an invasive diagnostic procedure for treating the cardiovascular diseases. This paper aims to understand the transport phenomena of blood nanofluid through a flexible arterial domain, which has stenosis/dilation with the catheter outer surface layered with nanoparticles. Blood is considered as a micro-polar fluid, governed by nonlinear equations, which are then solved by using the homotopy perturbation method. We consider the flexible nature of the arterial wall as a function of time due to the heart's pumping action. Different slip velocities with the Darcy effects in the constricted domain's abnormal segments represent the dysfunction of a blood vessel. In this paper, we explore the effects of the various fluid parameters and the embedded geometrical parameters on physiological characteristics. We observe that the radial velocity increases with the increasing value of Gr and Br. However, impedance has shown the opposite pattern in the dilation segment. Also, we understand the helical flow distribution in the abnormal segments of the considered blood vessel. These findings explain that the study of nanofluid could be an assuring therapeutic approach against arterial conditions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, He's fractional derivative is adopted to establish fractional evolution equations in a fractal space. He's fractional complex transform is used to convent the fractional evolution equation into its traditional partner, and the homotopy perturbation method is used to solve the equations. Some illustrative examples are presented to show that the proposed technology is very excellent. [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]

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]

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]

In this paper, we investigate the fractional-order arms race model. The model has emerged as an important tool for the investigation of international conflict and arms races. The variational iteration method, the homotopy perturbation method, and the adomian decomposition method are used to solve the mathematical model with Caputo’s fractional derivative. Several numerical computations have been provided to establish the validity and accuracy of the acquired results. It is shown that the fractional-order model can be solved easily using semi-analytical methods. The results obtained by all methods are compared. [ABSTRACT FROM AUTHOR]

Rotavirus infection is also a major cause of death in babies and children. Rotavirus causes severe diarrhea in babies and infants in developed and underdeveloped countries. It has multiple types of transmission: human to human and human to the environment. This paper investigates the time‐fractional order rotavirus epidemic model in an uncertain environment defined in the Caputo–Fabrizio (CF) derivative sense. In the titled model, the initial conditions are considered as fuzzy numbers. A double parametric form (DPF) of fuzzy numbers has been used in the fuzzy fractional rotavirus model. A semi‐analytical approach called the homotopy perturbation Elzaki transform method (HPETM) has been employed to solve the present model with the fuzzy initial condition. In order to derive the existence and uniqueness of the solution of the model, the concept of fixed‐point theory is utilized. Various fuzzy and interval solutions have been computed by considering different values of fractional orders, fuzzy parameters, and parameters involved in the given model. [ABSTRACT FROM AUTHOR]