Homotopy perturbation and analysis methods have been widely used to obtain both approximate and exact so- lutions to nonlinear problems. In general, these two methods are based on the Taylor series with respect to an embedding parameter. Many researchers have compared the two methods and raised more concerns on the homo- topy perturbation method (HPM) because the homotopy analysis method (HAM) contains a convergence-control parameter ~: For this reason, in this article, a more general form of HPM is introduced as the -homotopy per- turbation method (-HPM), which contains a control parameter : The introduction of parameter in this new modication gives a better way to adjust and control the convergence region and the rate of the series solution. We conrm through the given examples in this study that the HPM is a special case of the -HPM. The error and convergence analysis of this proposed method are also presented

In this study, a new modification of the homotopy analysis method (HAM) called the Tarig homotopy analysis method (THAM) is proposed to the linear and nonlinear wave-like equations. The numerical simulations have been implemented to show that the THAM is very efficient and accurate. The obtained results revealed that the THAM is simple to implement and more convenient for non-linear problems arising in different fields of science.

In this paper, the series solutions of a non-linear delay integral equations are considered by a modified approach of homotopy analysis method (MAHAM). We split the function into infinite sums. The outcomes of the illustrated examples are included to confirm the accuracy and efficiency of the MAHAM. The exact solution can be obtained using special values of the convergence parameter.

We propose a new modification of homotopy perturbation method (HPM) called the δ-homotopy perturbation transform method (δ-HPTM). This modification consists of the Laplace transform method, HPM, and a control parameter δ. This control convergence parameter δ in this new modification helps in adjusting and controlling the convergence region of the series solution and overcome some limitations of HPM and HPTM. The δ-HPTM and q-homotopy analysis transform method (q-HATM) are considered to study the generalized time-fractional perturbed $(3+1)$ -dimensional Zakharov–Kuznetsov equation with Caputo fractional time derivative. This equation describes nonlinear dust-ion-acoustic waves in the magnetized two-ion-temperature dusty plasmas. The selection of an appropriate value of δ in δ-HPTM and the auxiliary parameters n and ħ in q-HATM gives a guaranteed convergence of series solution, but the difference between the two techniques is that the embedding parameter p in δ-HPTM varies from zero to nonzero δ, whereas the embedding parameter q in q-HATM varies from zero to $\frac{1}{n}, n\geq{1}$ . We examine the effect of fractional order on the considered problem and present the error estimate when compared with exact solution. The outcomes reveal complete reliability and efficiency of the proposed algorithm for solving various types of physical models arising in sciences and engineering. Furthermore, we present the convergence and error analysis of the two methods.

In this paper, modified q-homotopy analysis method (mq-HAM) is proposed for solving high-order non-linear partial differential equations. This method improves the convergence of the series solution and overcomes the computing difficulty encountered in the q-HAM, so it is more accurate than nHAM which proposed in Hassan and El-Tawil, Saberi-Nik and Golchaman. The second- and third-order cases are solved as illustrative examples of the proposed method.

In this paper, the time-fractional Fisher’s equation (TFFE) is considered to exam the analytical solution using the Laplace q-Homotopy analysis method (Lq-HAM)”. The Lq-HAM is a combined form of q-homotopy analysis method (q-HAM) and Laplace transform. The aim of utilizing the Laplace transform is to outdo the shortage that is mainly caused by unfulfilled conditions in the other analytical methods. The results show that the analytical solution converges very rapidly to the exact solution.

In this paper, comparative study of q-homotopy analysis method (q-HAM) with the Liao's optimal homotopy analysis method (OHAM) is proposed.We solved two examples, first example is a system of Volterra integro-differential equations and the second one is a nonlinear integro-differential equation.The results show that the q-HAM was more accuracy than the OHAM.

In this work, the q-homotopy analysis transform method (shortly q-HATM) which is a combined form of q-homotopy analysis method and Laplace transform method is employed to find numerical solution to the new modified coupled Korteweg–de Vries system. This method allows us to fine-tune the convergence region along with rate of convergence of the obtained series solution by allowing the auxiliary parameters n and ħ to vary. The obtained solution by the proposed method is presented in a refined convergent series form. The numerical results show that only few terms are sufficient to obtain an approximate solution which is accurate, efficient, and reliable. Furthermore, the graphical depictions of the obtained approximate solution of the system are presented.

In this paper, the solutions of (1 and 2)- dimensional non-linear first kind Fredholm integral equations are studied by combine the q-homotopy analysis method (q-HAM) [2-11] and the regularization method [16,17]. The utilization of this technique depends on converting the first kind Fredholm Integral Equations to the second kind of equations by applying the regularization method. Then q-HAM is employed to the resulting second kind of equations to obtain a solution. Some illustrative examples are given to demonstrate the validity and applicability of this technique.

In this paper, new powerful modification of homotopy analysis technique (NMHAM) was submitted to create an approximate solution of nonhomogeneous nonlinear ordinary and partial differential equations. The NMHAM is a combination of the new technique of homotopy analysis method(NHAM) [4] and the new technique of homotopy analysis method(nHAM) [7].Three illustrative examples are employed to illustrate the accuracy and computational proficiency of this approach. The outcomes uncover that the NMHAM is more accurate than the NHAM and nHAM.

In this paper, a new procedure of the q-homotopy analysis technique (NTqHAM) was submitted for solving non-linear initial value problems. The NTq-HAM contains just a single convergence control parameter α. To show the dependability and proficiency of the technique, this approach is applied to solve two non-linear IVPs, and the outcomes uncover that the NTq-HAM is more general of the He’s homotopy perturbation technique (HPM) [27] and the He’s HPM is only special case of the NTq-HAM when α = 1.

In this paper, an approximate solution for the one-dimensional hyperbolic telegraph equation by using the q-homotopy analysis method (q-HAM) is proposed.The results shows that the convergence of the qhomotopy analysis method is more accurate than the convergence of the homotopy analysis method (HAM).

In this paper, the q-homotopy analysis method is applied to solve linear and nonlinear fractional initial-value problems (fIVPs). The fractional derivatives are described by Caputo’s sense. Exact and/or approximate analytical solutions of the fIVPs are obtained. The results of applying this procedure to the studied cases show the high accuracy and efficiency of theapproach.

In this study the application of a newly developed efficient method namely, optimal q-homotopy analysis method (Oq-HAM) has been illustrated for solving second order initial and boundary value problems. The Oq-HAM is a flexible method and can be applied to solve different types of problems. Moreover, it can easily be implemented in symbolic soft computing tools, e.g. MATHEMATICA.

In this paper, an optimal q-homotopy analysis method (Oq-HAM) is proposed. We present some examples to show the reliability and efficiency of the method. It is compared with the one-step optimal homotopy analysis method. The results reveal that the Oq-HAM has more accuracy to determine the convergence-control parameter than the one-step optimal HAM.

The convergence of q- homotopy analysis method (q-HAM) is studied in the present paper. It is proven that under certain conditions the solution of the equation:

A modified q-homotopy analysis method (mq-HAM) was proposed for solving nth-order nonlinear differential equations. This method improves the convergence of the series solution in the nHAM which was proposed in (see Hassan and El-Tawil 2011, 2012). The proposed method provides an approximate solution by rewriting the nth-order nonlinear differential equation in the form of n first-order differential equations. The solution of these n differential equations is obtained as a power series solution. This scheme is tested on two nonlinear exactly solvable differential equations. The results demonstrate the reliability and efficiency of the algorithm developed.