In this paper, comparison of **homotopy** **perturbation** **method** (HPM) and **homotopy** **perturbation** transform **method** (HPTM) is made, revealing that **homotopy** **perturbation** transform **method** is very fast convergent to the solu- tion of the partial differential equation. For illustration and more explanation of the idea, some examples are provided.

It is interesting to study partial differential equations since they appear in many physical phenomena which in general are very hard to solve [2]. Only a few have known exact solutions. In recent years, we have seen an increase in the study of new analytical methods or approximate analytical methods. Among others are **homotopy** analysis **method** [3], **homotopy** **perturbation** **method** [4], variational iteration **method** [5], differential transformation **method** [6], adomian decomposition **method** [7], sumudu transform [8], ELzaki transform [9, 1], etc.

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In this article, the **homotopy** **perturbation** **method** (HPM) has been successfully applied to solve the dynamics of a coupled spring system with cubic nonlinearity. Our results suggest that it is an efficient **method** for obtaining solutions of nonlinear differential equations governing the motion of such problems. They also confirm the simplicity and efficiency of the **method** for solving any nonlinear ordinary differential equation or systems of nonlinear ordinary differential equations. It is also observed that the HPM is a promising **method** for solving other linear and nonlinear partial differential equations. Acknowledgements: This research is supported by Islamic University, Kushtia, Bangladesh. We like to thank all honourable members of the Academic Committee of the Department of Mathematics for providing us with logistic supports in carrying out the research. We also gratefully thank the anonymous reviewers for their valuable comments which helped us improve the manuscript significantly.

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Abstract: Mathematical modeling of real-life problems usually results in functional equations, such as ordinary or partial differential equations, integral and integral-differential equations etc. The theory of integral equation is one of the major topics of applied mathematics. In this paper a new **Homotopy** **Perturbation** **Method** (HPM) is introduced to obtain exact solutions of the systems of integral equations-differential and is provided examples for the accuracy of this **method**. This paper presents an introduction to new **method** of HPM, then introduces the system of integral - differential linear equations and also introduces applications and literature. In second section we will introduce categorizations of averaging integral - differential and several methods to solve this kind of achievement. The third section introduces a new **method** of HPM. Fourth section determines quarter of integral - differential equations by using HPM. Therefore, we provide Conclusion and some examples that illustrate the effectiveness and convenience of the proposed **method**.

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The **homotopy** pertirabation **method** is one of the important methods to find the approximate solutions for non- linear partial differential equations in mathematical physics. The **homotopy** **perturbation** **method**, which was originally proposed by J. H. He [2]-[5] in 1999, has been proved by many authors to be a powerful mathematical tool for solving various kinds of problems [2]-[5]. This **method** introduces an efficient approach for a wide va- riety of scientific and enginering applications. We should point out that this **method** can give the approximte or exact solutions without the computation of the Adomian polynomial, discretization, linearization, transformation or **perturbation**. To illustrate, consider the following nonlinear differential equation [2]-[5]:

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In this paper, we present a comparative study between the modified Sumudu decomposition **method** (MSDM) and **homotopy** **perturbation** **method** (HPM). The study outlines the important features of the two methods. The analysis will be explained by discussing the nonhomogeneous Kortewege-de Vries (KdV) problems.

He’s **Homotopy** **perturbation** **method** suggested in this article is an efficient **method** for obtaining the most accurate solution of a highly nonlinear partial differential equation with initial condition. Therefore, this **method** is a powerful mathematical technique to solve the wide classes of nonlinear partial differential equations in the form of analytical expressions. The HPM has got many important advantages and it does not require small parameters in the equation, so that the limitations of the traditional perturbations can be eliminated. Also the calculations in the HPM are simple and straightforward. The reliability of the **method** and the reduction in the size of the computational domain gives this **method** a wider applicability.

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In the recent years, the application of the **homotopy** **perturbation** **method** (HPM) [1, 7] in nonlinear problems has been developed by scientists and engineers, because this **method** continuously deforms the difficult problem under study into a simple problem which is easy to solve. The **homotopy** **perturbation** **method** [6], proposed first by He in 1998 and was further developed and improved by He [7, 8, 11]. The **method** yields a very rapid convergence of the solution series in the most cases. Usually, one iteration leads to high accuracy of the solution. Although goal of He’s **homotopy** pertur- bation **method** was to find a technique to unify linear and nonlinear, ordinary or partial differential equations for solving initial and boundary value problems. Most **perturbation** methods assume a small parameter exists, but most nonlinear problems have no small parameter at all. A review of recently developed nonlinear analysis methods can be found in [9]. Recently, the applications of **homotopy** **perturbation** theory among scientists were appeared [1-6], which has become a powerful mathematical tool, when it is successfully coupled with the **perturbation** theory [7, 10, 11].

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Abstract In this paper, the **homotopy** **perturbation** **method** (HPM) is considered for finding approximate solutions of two-dimensional viscous flow. This technique provides a sequence of functions which converges to the exact solution of the problem. The HPM does not need a small parameters in the equations, but; the **perturbation** **method** depends on small parameter assumption and the obtained results. In most cases, it ends up with a non-physical result, so **homotopy** **perturbation** **method** overcomes completely the above shortcomings. HPM is very convenient and effective and the solutions is compared with the exact solution.

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In this paper, **homotopy** **perturbation** **method** is applied to solve moving boundary and isoperimetric problems. This **method** does not depend upon a small parameter in the equation, **homotopy** is constructed with an imbedding parameter p, which is considered as a “small parameter”. Finally, we use combined **homotopy** **perturbation** **method** and Green’s function **method** for solving second order problems. Some examples are given to illustrate the effectiveness of methods. The results show that these methods provides a powerful mathematical tools for solving problems.

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where a, b, g and f are continuous functions in some region of the plane and g() = . By solving this boundary value problem by the **homotopy** **perturbation** **method**, we obtain an approximate or exact solution u(x, y). Before proceeding further, let us introduce the integral operator S deﬁned in the following form:

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A user friendly algorithm based on new **homotopy** **perturbation**, a new local fractional α - integral transform **method** is proposed to solve nonlinear fractional gas dynamics equation. Further, the same problem is solved by Adomian decomposition **method** The results obtained by the two methods are in agreement and hence this technique may be considered an alternative and efficient **method** for finding approximate solutions of both linear and nonlinear fractional differential equations.This **method** is a combined form of the new integral transform, **homotopy** **perturbation** **method**, and He’s polynomials. The nonlinear terms can be easily handled by the use of He’s polynomials. The numerical solutions obtained by the proposed **method** show that the approach is easy to implement and computationally very attractive.

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Several authors have proposed a variety of the modiﬁed **homotopy** **perturbation** meth- ods. E. Yusufo˘ glu in [32] proposed the improved **homotopy** **perturbation** **method** for solv- ing Fredholm type integro-diﬀerential equations. M. Javidi in [25] proposed the modiﬁed **homotopy** **perturbation** **method** (MHPM) to solve nonlinear Fredholm integral equations. The result reveals that the MHPM is very eﬀective and convenient. In another study, he applied the MHPM for solving the system of linear Fredholm integral equations [24]. A. Golbabai in [15,16] used the MHPM for solving Fredholm integral equations and nonlinear Fredholm integral equations of the ﬁrst kind. In another work, he introduced new iterative methods for nonlinear equations by the MHPM [14].

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[2] Gurhan Gurarslan and Murat Sari 2009 “Numerical solutions of linear and nonlinear diffusion equation by a differential quadrature **method**” Int. J. Nume. Meth. Biomed. Engg. Vol. 27 Pp-69-77. [3] J.H. He, 2003 “ **Homotopy** **perturbation** **method**: A new nonlinear

In this work, the modified **homotopy** **perturbation** **method** was successfully used for solving Fisher’s equation with time-fractional derivative, and the classical HPM has been used for solving Fisher’s equation with space-fractional derivative. The final results obtained from modified HPM and com- pared with the exact solution shown that there is a similarity between the exact and the approximate solutions. Calculations show that the exact solution can be obtained from the third term. That’s why we say that modified HPM is an alternative analytical **method** for solving the nonlinear time-fractional equations.

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Abstract In recent years, numerous approaches have been applied for finding the solutions of functional equations. One of them is the optimal **homotopy** asymptotic **method**. In current paper, this **method** has been applied for obtaining the approximate solution of Fisher equation. The reliability of the **method** will be shown by solving some examples of various kinds and comparing the obtained outcomes with the results of **homotopy** **Perturbation** **method**.

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Abstract. In this paper, the multistage **homotopy** **perturbation** **method** is extended to solve the Chen fractional order systems. The multistage **homotopy** **perturbation** **method** is only a simple modification of the standard **homotopy** **perturbation** **method**, in which it is treated as an algorithm in a sequence of intervals for finding accurate approximate analytical solutions. The fractional derivatives are described in the Caputo sense. The solutions are obtained in the form of rapidly convergent infinite series with easily computable terms. Numerical results reveal that the multistage **method** is a promising tool for the Chen fractional order systems.

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Abstract In the present paper, we study a homo- geneous cosmological model in Friedmann-Robertson- Walker (FRW) space-time by means of the so-called **Homotopy** **Perturbation** **Method** (HPM). First, we brieﬂy recall the main equations of the cosmological model and the basic idea of HPM. Next we consider the test example when the exact solution of the model is known, in order to approbate the HPM in cosmology and present the main steps in solving by this **method**. Finally, we obtain a solution for the spatially ﬂat FRW model of the universe ﬁlled with the dust and quintessence when the exact solution cannot be found. A comparison of our solution with the corresponding numerical solution shows that it is of a high degree of accuracy.

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H u p = − p F u + p E u − F u . (2.2) where F u ( ) is a functional operator with the known solution v 0 . It is clear that when p is equal to zero then H u ( ) , 0 = F u ( ) = 0 , and when p is equal to 1, then H u ( ) ,1 = E u ( ) = 0 . It is worth noting that as the embedding parameter p increases monotonically from zero to unity the zero order solution v 0 continuously deforms into the original problem E u ( ) = 0 . The embedding parameter, p ∈ [ ] 0,1 , is considered as an expand- ing parameter [2]. In the **homotopy** **perturbation** **method** the embedding parameter p is used to get series ex- pansion for solution as:

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