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In this paper, we solve the nonlinear equations by using a classical **method** and a powerful **method**. A powerful **method** known as **homotopy** **continuation** **method** (HCM) is used to solve the problem of classical **method**. We use Newton-HCM to solve the divergence problem that the classical Newton’s **method** always faces. The divergence problem occurs when a bad initial guess is used. The problem with Newton’s **method** happens when the derivative of given function at initial point equal to zero. The division by zero makes the scheme become nonsense. Thus, an approach used to solve this mathematical problem by using Newton-HCM. The results are implemented by mathematical software known as Mathematica 7.0. The results obtained indicate the ability of Newton-HCM to solve this mathematical problem.

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Recently, nonlinear systems of equations appear in numerical applications frequently. Normally, we will use Newton's **method** and quasi-Newton **method** to solve the nonlinear systems of equations. In this study, we employ another **method**, **homotopy** **continuation** **method** to use in solving nonlinear systems of equations. So, this study will illustrate the performance of the three methods for solving nonlinear systems of equations.

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In the present paper, we propose a new **method** called **Homotopy** perturbation Sumudu transform **method** (HPSTM) for solving the nonlinear equations. It is worth mentioning that the proposed **method** is an elegant combination Of the Sumudu transformation, the **Homotopy** perturbation **method** and He’s Polynomials and is mainly due to Ghorbani [40, 41]. The use of He’s polynomials in the nonlinear term was first introduced by Ghorbani [40, 41]. The Proposed algorithm provides the solution in a rapid convergent series which may lead to the solution in a closed form. The advantage of this **method** is its capability of combining two powerful methods for obtaining exact and numerical solutions for nonlinear dispersive equations. This paper considers the effectiveness of the **Homotopy** Perturbation Sumudu transform **method** (HPSTM) in solving nonlinear dispersive equations.

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Most of engineering problems, especially heat transfer and fluid flow equations are nonlinear. Therefore, some of them are solved using computational fluid dynamic (numerical) **method** and the other using analytical perurbation **method** [1-3]. In the numerical **method**, stability and convergence should be considered to avoid divergent or inappropriate results. In analytical perturbation **method**, the small parameter should be exerted in the equation [4]. Thus, finding the small parameter and exerting it into the equation are the problems of this **method**. The perturbation **method** is one of the well-known methods to solve the nonlinear equations which was studied by a large number of researchers such as Bellman [5] and Cole [6]. Actually, these scientists paid more attention to the mathematical aspects of the subject which included a loss of physical verification. This loss in the physical verification of the subject was recovered by Nayfeh [7] and Van Dyke [8].

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In most cases, the **homotopy** analysis **method** leads to a very rapidly convergent series solution, and usually only a few iterations are suffi- cient to arrive at a very accurate solution, specially when the auxiliary parameter ~ i and the base functions are properly chosen.

Inspired and motivated by the ongoing research in this area, we introduce a new ap- proximate **method**, namely **homotopy** analysis sumudu transform **method** (HASTM) for solving the nonlinear equations in this article. It is worth mentioning that the proposed **method** is an elegant combination of sumudu transform **method** and **homotopy** analysis **method**. It provides the solutions in terms of convergent series with easily computable components in a direct way without using linearization, perturbation or restrictive as- sumptions. The advantage of this **method** is its capability of combining two powerful methods for obtaining exact and approximate analytical solutions for nonlinear equations. This paper considers the eﬀectiveness of the **homotopy** analysis sumudu transform **method** (HASTM) in solving linear and nonlinear Fokker-Planck equations.

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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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The **homotopy** perturbation **method** (HPM) was first proposed by the Chinese mathematician Ji-Huan He [1-3]. Unlike classical techniques, the **homotopy** perturbation **method** leads to an analytical approximate and exact solutions of the nonlinear equations easily and elegantly without transforming the equation or linearizing the problem and with high accuracy, minimal calculation and avoidance of physically unrealistic assumptions. As a numerical tool, the **method** provide us with numerical solution without discretization of the given equation, and therefore, it is not effected by computation round-off errors and one is not faced with necessity of large computer memory and time. This technique has been employed to solve a large variety of linear and nonlinear problems [4- 10]. In the present study, **homotopy** perturbation **method** has been applied to solve the Kuramoto– Sivashinsky equations. The numerical results are compared with the exact solutions. It is shown that the errors are very small.

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Abstract. Laplace **Homotopy** Perturbation **method** is projected in a modified form to solve highly nonlinear differential equations.This approach simplifies the equations and avoid the computations of Laplace Transforms of complicated functions, which in turn save the computation time of the computational tools. The results are compared with results obtained by using reduced differential transform **method**.

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Abstract. In this paper, a combined form of natural transform with **homotopy** analysis **method** is proposed to solve nonlinear fractional partial differential equations. This **method** is called the fractional **homotopy** analysis natural transform **method** (FHANTM). The FHANTM can easily be applied to many problems and is capable of reducing the size of computational work. The fractional derivative is described in the Caputo sense. The results show that the FHANTM is an appropriate **method** for solving nonlinear fractional partial differentia equation.

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In this paper effect of **continuation** power flow in analyzing voltage stability is shown. It is very useful for evaluation of the critical point of the PV curve in static voltage stability assessment. PV- curves are constructed to calculate load ability margins and identify the weakest bus in the system. However, drawing the PV- curve is time consuming in large- scale power system. As a result, the open source power system analysis toolbox (PSAT) MATLAB software package is used for analysis. The results presented in this paper clearly show how CPF technique can be used to increase system load ability in power systems limits.

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are compared with other methods for a wide range of test problems, and are shown to significantly reduce the number of function evaluations for the difficult.. For the easier problems th[r]

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This leads us to ask what is feasible to address programmatically, both for Save the Children and other organizations looking to support women in continuing contraceptive use. For instance, when considering factors contributing to early **method** removal, nearly one-third of women in this sample who removed their IUD or implant prior to 24 months of **method** use indicated that the reason for removal was socio-cultural reasons, more specifically that their husband or a family member requested that the contraceptive **method** be removed. Examination of household dynamics in Pakistan points to the influence of both the husband and mother-in-law in decision making around contraceptive use. Perceived support from a woman’s in laws has a high association with intentions to use methods of contraception. 23 Women are also ten times more likely to use FP if their husband approves. 24 There is therefore a recognition, based on

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Partial differential equations which arise in real-world physical problems are often too complicated to be solved exactly. And even if an exact solution is obtainable, the required calculations may be too complicated to be practical, or it might be diffucult to interpret the outcome. Very recently, some promising approximate analytical solutions are proposed, such as Exp-function **method** [1–2], Adomian decomposition **method** [3–7], variational iteration **method** [8–10] and **Homotopy**-perturbation **method** [11–16]. Other methods are reviewed

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such as the **method** of averaging, the renormalization **method**, the **method** of multiple scales, the Lindstedt-Poincare **method**, the **method** of matched asymptotic expansions, plus their variants were developed to make our results as accurate as possible [12, 13]. The main drawback of perturbation methods is the prerequi- site of a small parameter, which is hard to fulfill. That is why sometimes a small parameter may have to be artificially introduced into the equation. The solutions therefore are not entirely valid. They may be for weakly nonlinear problems, but the validity mostly does not cover strongly nonlinear problems.

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In this paper, the **homotopy** perturbation α-integral transform **method** is successfully applied for solving nonlinear time fractional gas dynamics equation.Therefore, this **method** is very powerful and efficient technique for solving different kinds of linear and nonlinear fractional differential equations arising in different fields of science and engineering. Inconclusion, this **method** is a nice refinement in existing numerical techniques and might find the wide applications.

A multitude of generalizations and variants of Banach’s contractive condition have been given after Banach’s theorem see, e.g., Rhoades 13 and, recently, Agarwal and O’Regan 11 have given a **homotopy** result thus generalizing a fixed point theorem of Hardy and Rogers 14 under the following generalized contractive condition: there exists a ∈ 0, 1 such that for all x, y ∈ X