Homotopycontinuation methods (HCMs) are now widely used to find the roots of polynomial equations as well as transcendental equations. HCM can be used to solve the divergence problem as well as starting value problem. Obviously, the divergence problem of traditional methods occurs when a method cannot be operated at the beginning of iteration for some points, known as bad initial guesses. Meanwhile, the starting value problem occurs when the initial guess is far away from the exact solutions. The starting value problem has been solved using Super Ostrowski homotopycontinuationmethod for the initial guesses between 1000 x 0 1000 .
HomotopyContinuationMethod can be used to generate a good starting value. This method can guarantee to converge to the answer if we choose the auxiliary homotopy function well. HomotopyContinuationMethod does some simple iteration process to obtain our solutions more precisely. The algorithm of this method is clear and easy as well as its convergence speed is fast.
In this paper, we solve the nonlinear equations by using a classical method and a powerful method. A powerful method known as homotopycontinuationmethod (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.
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, homotopycontinuationmethod 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.
HomotopyContinuationMethod (HCM) was introduced to solve the problems of nonlinear optimization and also systems of nonlinear equations (Allgower and George, 1990). This method deforms a simple function into the function of interest by tracing path, computes series of zeros and ends in zero of that function of interest. Since the homotopy methods converge to a solution for any arbitrary chosen initial condition, they are said to be globally convergent.
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.
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 . 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  and Cole . 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  and Van Dyke .
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.
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.
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.
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.
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.
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.
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
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
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.
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