Standard Homotopy Perturbation Method Research Articles

Homotopy Perturbation Method for Semi-Bounded Solution of the System of Cauchy-Type Singular Integral Equations of the First Kind

In this paper, we introduce the standard Homotopy Perturbation Method (HPM) for obtaining semi-bounded solutions of the first kind of system of Cauchy-type singular integral equations (CSIEs) with constant coefficients. We use the Gauss elimination technique to reduce the system of CSIEs to a diagonal triangle system of algebraic equations. We then apply the HPM to solve the resulting equations. By applying the theory of semi-bounded solutions of CSIEs, we can determine the inverse operators for the first kind of CSIEs. We demonstrate that the proposed method is exact for the system of characteristic SIEs, regardless of the choice of initial guesses (in the Holder class of functions). To illustrate the validity and accuracy of our proposed method, we supply and analyse three examples. We compare the results obtained using our method with those obtained using the Chebyshev collocation and Galerkin methods. Our method includes the ability to solve the complex-valued system of CSIEs. Based on the numerical results, we conclude that the HPM is more dominant than the other methods.

Time-fractional lorenz type chaotic systems with asymmetric gaussian uncertainty: series solutions via extended He-Mohand algorithm in fuzzy-caputo sense

In this paper, modeling and analysis of 3D fuzzy-fractional Lorenz type systems is presented. System under-consideration includes classical Lorenz, Chen and Burke-Shaw chaotic systems. Asymmetrical Gaussian fuzzy logic with fractional calculus is applied to model complex systems with intricate patterns. The focus of this study is fuzzy-fractional modeling and simulations. For solution purpose, a hybrid perturbation method is introduced where standard homotopy perturbation method (HPM) is enhanced by incorporating Mohand transform in fuzzy-Caputo sense. This hybrid mechanism provides an efficient way to find solutions in fuzzy-fractional environment. Validity of obtained solutions is checked by computing residual errors, which ultimately confirms the convergence of applied methodology. The dynamical behavior of fuzzy-fractional chaotic models is analyzed through various 2-3D plots to represent the chaotic regions as well unpredictable trajectories at both upper and lower bounds. Fuzzy membership functions of 3D models at different values of fractional derivative are also demonstrated through 2D plots. Analysis reveals that extended hybrid methodology proves to be a valuable tool for researchers dealing with nonlinear chaotic fractional systems with fuzzy characteristics.

An analysis of convergence of hybridized approximation techniques for the analytical solution of partial differential equations

In this paper, we use the hybridization of two well-known semi-analytical methods to obtain the numerical solutions for some special linear and nonlinear partial differential equations that are predominant in most physical science disciplines. The first among these methods is the coupling of Laplace transform and Adomian decomposition method (LADM) while the second method is standard Homotopy perturbation method with Laplace transform method (HPTM). The accuracy and dependability of these proposed techniques is confirmed by applying them to solve linear and nonlinear Kleidon-Gordon equations, linear transverse equation of a vibrating beam, homogeneous and inhomogeneous nonlinear PDEs, advection equation, diffusion-convection and Korteweg-DeVries equation. Thereafter, comparison between the solutions obtained by the methods is presented in tables for convergence analysis. Consequently, the findings from our study showed the two methods can be effective alternative approaches for obtaining solutions to linear and nonlinear PDEs and higher-order initial value problems.

Hypersingular integral equations of the first kind: A modified homotopy perturbation method and its application to vibration and active control

This paper attempts to propose and investigate a modification of the homotopy perturbation method to study hypersingular integral equations of the first kind. Along with considering this matter, of course, the novel method has been compared with the standard homotopy perturbation method. This method can be conveniently fast to get the exact solutions. The validity and reliability of the proposed scheme are discussed. Different examples are included to prove so. According to the results, we further state that new simple homotopy perturbation method is so efficient and promises the exact solution. The modification of the homotopy perturbation method has been discovered to be the significant ideal tool in dealing with the complicated function-theoretic analytical structures within an analytical method.

Periodic solution of the cubic nonlinear Klein–Gordon equation and the stability criteria via the He-multiple-scales method

The current work demonstrated a new technique to improve the accuracy and computational efficiency of the nonlinear partial differential equation based on the homotopy perturbation method (HPM). In this proposal, two different homotopy perturbation expansions, the outer expansion and the inner one, are introduced based on two different homotopy parameters. The multiple-scale homotopy technique (He-multiple-scalas method) is applied as an outer perturbation for the nonlinear Klein–Gordon equation. A highly accurate periodic temporal solution has been derived from three orders of perturbation. The amplitude equation, which is imposed as a uniform condition, is of the fourth-order cubic–quintic nonlinear Schrodinger equation. The standard HPM with another homotopy parameter has been used as an inner perturbation to obtain a spatial solution of the nonlinear Schrodinger equation. The cubic–quintic Landau equation is obtained in the inner perturbation technique. Finally, the approximate solution is derived from the temporal and spatial solutions. Further, two different tools are used to obtain the same stability conditions. One of them is a new tool based on the HPM, by constructing the nonlinear frequency. The method adopted here is important and powerful for solving partial differential nonlinear oscillator systems arising in nonlinear science and engineering.

On the multistage homotopy procedure for a fractional Lorenz system

In this article, the multistage homotopy perturbation method (MHPM) is applied for solving differential systems with fractional order derivatives in the Caputo sense. This method is a modification of the standard homotopy perturbation method (HPM). A fractional Lorenz system as an application is presented for which some numerical comparisons between the (MHPM) and (HPM) with the 4th order Runge-Kutta method (RK4). The results reveal that the used pre-mentioned procedure (MHPM) is a reliable and an effective tool for constructing an accurate approximate solution for the fractional Lorenz system.

Optimal solutions for the evolution of a social obesity epidemic model

In this work, a novel modification in the traditional homotopy perturbation method (HPM) is proposed by embedding an auxiliary parameter in the boundary condition. The scheme is used to carry out a mathematical evaluation of the social obesity epidemic model. The incidence of excess weight and obesity in adulthood population and prediction of its behavior in the coming years is analyzed by using a modified algorithm. The proposed method increases the convergence of the approximate analytical solution over the domain of the problem. Furthermore, a convenient way is considered for choosing an optimal value of auxiliary parameters via minimizing the total residual error. The graphical comparison of the obtained results with the standard HPM explicitly reveals the accuracy and efficiency of the developed scheme.

Modified homotopy perturbation method for solving hypersingular integral equations of the first kind.

Modified homotopy perturbation method (HPM) was used to solve the hypersingular integral equations (HSIEs) of the first kind on the interval [−1,1] with the assumption that the kernel of the hypersingular integral is constant on the diagonal of the domain. Existence of inverse of hypersingular integral operator leads to the convergence of HPM in certain cases. Modified HPM and its norm convergence are obtained in Hilbert space. Comparisons between modified HPM, standard HPM, Bernstein polynomials approach Mandal and Bhattacharya (Appl Math Comput 190:1707−1716, 2007), Chebyshev expansion method Mahiub et al. (Int J Pure Appl Math 69(3):265–274, 2011) and reproducing kernel Chen and Zhou (Appl Math Lett 24:636–641, 2011) are made by solving five examples. Theoretical and practical examples revealed that the modified HPM dominates the standard HPM and others. Finally, it is found that the modified HPM is exact, if the solution of the problem is a product of weights and polynomial functions. For rational solution the absolute error decreases very fast by increasing the number of collocation points.

The Multistage Homotopy Perturbation Method for Solving Chaotic and Hyperchaotic Lü System

The multistage homotopy-perturbation method (MHPM) is applied to the nonlinear chaotic and hyperchaotic Lü systems. MHPM is a technique adapted from the standard homotopy-perturbation method (HPM) where the HPM is treated as an algorithm in a sequence of time intervals. To ensure the precision of the technique applied in this work, the results are compared with a fourth-order Runge-Kutta method and the standard HPM. The results show that the MHPM is an efficient and powerful technique in solving both chaotic and hyperchaotic systems.

Application of the homotopy perturbation method for the large deflection problem of a circular plate

A moderate improvement of the standard homotopy perturbation method is given and applied to the large deflection problem of a uniformly loaded circular plate with edge simply supported and fastened. Compared with the general homotopy perturbation method, there are two innovations in our method. On the one hand, the homotopy equations are constructed based on part of equations instead of all equations of the system. On the other hand, a parameter in the system is expanded to finite terms which have more freedom to be selected and it can be used to adjust the accuracy of the approximate solutions. More accurate solutions for the large deflection problem are obtained by the proposed method. The error comparisons with classical perturbation solutions show the efficiency and convenience of the proposed homotopy perturbation method.

Spectral-Homotopy Perturbation Method for Solving Governing MHD Jeffery-Hamel Problem

We present a new modification of the homotopy perturbation method (HPM) for solving nonlinear boundary value problems. The technique is based on the standard homotopy perturbation method and blending of the Chebyshev pseudospectral methods. The implementation of the new approach is demonstrated by solving the MHD Jeffery-Hamel flow and the effect of MHD on the flow has been discussed. Comparisons are made between the proposed technique, the previous studies, the standard homotopy perturbation method, and the numerical solutions to demonstrate the applicability, validity, and high accuracy of the presented approach. The results demonstrate that the new modification is more efficient and converges faster than the standard homotopy perturbation method at small orders. The MATLAB software has been used to solve all the equations in this study.

A New Spectral-Homotopy Perturbation Method and Its Application to Jeffery-Hamel Nanofluid Flow with High Magnetic Field

We present a new modification of the homotopy perturbation method (HPM) for solving nonlinear boundary value problems. The technique is based on the standard homotopy perturbation method, and blending of the Chebyshev pseudospectral methods. The implementation of the new approach is demonstrated by solving the Jeffery-Hamel flow considering the effects of magnetic field and nanoparticle. Comparisons are made between the proposed technique, the standard homotopy perturbation method, and the numerical solutions to demonstrate the applicability, validity, and high accuracy of the present approach. The results demonstrate that the new modification is more efficient and converges faster than the standard homotopy perturbation method.

A New Modification of The HPM for The Duffing Equation With High Nonlinearity

In this work we introduce a new modification of the homotopy perturbation method for solving nonlinear ordinary differential equations. The technique is based on the blending of the Chebyshev pseudo-spectral methods and the homotopy perturbation method (HPM). The method is tested by solving the strongly nonlinear Duffing equation for undamped oscillators. Comparison is made between the proposed technique, the standard HPM, an earlier modification of the HPM and the numerical solutions to demonstrate the high accuracy, applicability and validity of the present approach.

Solution of the heat equation in the cast‐mould heterogeneous domain using a weighted algorithm based on the homotopy perturbation method

PurposeThe purpose of this paper is to present a weighted algorithm based on the homotopy perturbation method for solving the heat transfer equation in the cast‐mould heterogeneous domain.Design/methodology/approachA weighted algorithm based on the homotopy perturbation method is used to minimize the volume of computations. The authors show that this technique yields the analytical solution of the desired problem in the form of a rapidly convergent series with easily computable components.FindingsThe authors illustrate that the proposed method produces satisfactory results with respect to Adomian decomposition method and standard homotopy perturbation method. The reliability of the method and the reduction in the size of computational domain give this method a wider applicability.Originality/valueThis research presents, for the first time, a new modification of the proposed technique, for aforementioned problems and some interesting results are obtained.

An improvement to the homotopy perturbation method for solving the Hamilton-Jacobi-Bellman equation

In this paper, the piecewise homotopy perturbation method (PHPM) is employed to solve the Hamilton–Jacobi–Bellman (HJB) equation arising in the optimal control problems. The method is a simple modification of the standard homotopy perturbation method (HPM), in which it is treated as an algorithm in a sequence of small intervals (i.e. time step) for finding accurate approximate solutions to the corresponding HJB equation. Applying the PHPM with He's polynomials reveals that the modified homotopy perturbation is more impressive than the standard HPM. Also, the convergence of the algorithm is discussed in detail. The comparison of the PHPM results with the standard HPM, exact solution, Modal series, multiwavelets spectral method, differential transformations and the measure theory method is made. Simulation examples are employed to test the validity of the PHPM.

Modified HPMs Inspired by Homotopy Continuation Methods

Nonlinear differential equations have applications in the modelling area for a broad variety of phenomena and physical processes; having applications for all areas in science and engineering. At the present time, the homotopy perturbation method (HPM) is amply used to solve in an approximate or exact manner such nonlinear differential equations. This method has found wide acceptance for its versatility and ease of use. The origin of the HPM is found in the coupling of homotopy methods with perturbation methods. Homotopy methods are a well established research area with applications, in particular, an applied branch of such methods are the homotopy continuation methods, which are employed on the numerical solution of nonlinear algebraic equation systems. Therefore, this paper presents two modified versions of standard HPM method inspired in homotopy continuation methods. Both modified HPMs deal with nonlinearities distribution of the nonlinear differential equation. Besides, we will use a calcium‐induced calcium released mechanism model as study case to test the proposed techniques. Finally, results will be discussed and possible research lines will be proposed using this work as a starting point.

Application of Multistage Homotopy Perturbation Method to the Chaotic Genesio System

Finding accurate solution of chaotic system by using efficient existing numerical methods is very hard for its complex dynamical behaviors. In this paper, the multistage homotopy‐perturbation method (MHPM) is applied to the Chaotic Genesio system. The MHPM is a simple reliable modification based on an adaptation of the standard homotopy‐perturbation method (HPM). The HPM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions to the Chaotic Genesio system. Numerical comparisons between the MHPM and the classical fourth‐order Runge‐Kutta (RK4) solutions are made. The results reveal that the new technique is a promising tool for the nonlinear chaotic systems of ordinary differential equations.

MODIFIED HOMOTOPY PERTURBATION METHOD FOR SOLVING INTEGRAL EQUATIONS

This paper proposes a modified homotopy perturbation method to solve Fredholm and Volterra integral equations. Comparison of the obtained results with those obtained by the standard homotopy perturbation method shows that the proposed method is very effective and simple.

Application of multistage homotopy-perturbation method in hybrid synchronization of chaotic systems

In this paper, the multistage homotopy-perturbation method (HPM) is applied to realize the synchronization between the Lorenz system and the Chen system using active control. The multistage HPM is only a simple modification of the standard HPM, in which it is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions. The simulation results reveal that the multistage method is an effective tool to give the approximate analytical solutions for chaotic systems.

Application of the multistage homotopy-perturbation method to solve a class of hyperchaotic systems

Due to the complex dynamical behaviors of hyperchaotic system, it is very difficult to gain its valid analytical solution by using many existing methods. In this paper, the multistage homotopy-perturbation method is first employed to solve a class of hyperchaotic systems. The method is only a simple modification of the standard homotopy-perturbation method, in which it is treated as an algorithm in a sequence of small intervals (i.e. time step) for finding accurate approximate solutions to the corresponding hyperchaotic systems. Finally, some numerical comparisons among the multistage homotopy-perturbation method, the standard homotopy-perturbation method and the Runge–Kutta method have been made, which manifest that the modified method is a very accurate and effective algorithm to solve the hyperchaotic systems.