The **first** **integral** **method** was **first** proposed for solving Burger-KdV equation [1] which is based on the ring theory of commutative algebra. This **method** was further developed by the same author in [2-10] and some other mathematicians [2,11,12,13]. The present paper investi- gates for the **first** time the applicability and effectiveness of the **first** **integral** **method** on the Maccari’s. We consid- er Maccari’s system:

Abstract The **first** **integral** **method** is an efficient **method** for obtaining exact solutions of some nonlinear partial differential equations. This **method** can be applied to non integrable equations as well as to integrable ones. In this paper, the **first** **integral** **method** is used to construct exact solutions of the 2D Ginzburg-Landau equation.

The **first** **integral** **method** a general formula, is successful for solving a lot of nonlinear equation, and establishing travelling wave solutions, which is based on the ring theory of commutative algebra, and used to solve compli- cated and tedious algebra calculation. We can also apply them to some other nonlinear partial differential equa- tions.

The generalized Zakharov equation is a coupled equation which is a classic nonlinear mathematic model in plasma. A series of new exact explicit solutions of the system are obtained, by means of the **first** **integral** **method**, in the form of trigonometric and exponential functions. The results show the **first** **integral** **method** is an efficient way to solve the coupled nonlinear equations and get rich explicit analytical solutions.

Recently, most of researchers go on the way of finding exact solutions for nonlinear differential equations. They are very important in studying nonlinear phenomena that are described by differential equations because they have a major role in physics and applied sciences such as fluid dynamics, solid state physics, mechanics, biology and mathematical finance. Some researchers used different methods for solving these equations and finding exact solutions for them such as tanh-sech **method** 14, 16 , extended tanh **method** 1, 7, 18 , the generalized hyperbolic function **method** 8 , sine-cosine **method** 17 , the transformed rational function **method** 13 and F-expansion **method** 9 . In such methods, they use to transform isolated wave solutions so as to change nonlinear partial differential equation to ordinary differential equation easily solved by these methods. The **first** **integral** **method** that was suggested by Feng 6 is one of the methods are used to find the exact solutions for different nonlinear partial differential equations as 3, 4, 15 .

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With the availability of symbolic computation packages like Maple or Mathematica, the search for obtaining exact solutions of nonlinear partial diﬀerential equations (PDEs) has become more and more stimulating for mathematicians and scientists. Having exact so- lutions of nonlinear PDEs makes it possible to study nonlinear physical phenomena thor- oughly and facilitates testing the numerical solvers as well as aiding the stability analysis of solutions. In recent years, many approaches to solve nonlinear PDEs such as the ex- tended tanh function **method** [–], the modiﬁed extended tanh function **method** [, ], the exp-function **method** [–], the Weierstrass elliptic function **method** [], the Laplace decomposition **method** [, ] and so on have been employed.

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In this present work we have presented a number of solitary wave solutions to the Broer-Kaup equations and approximate long water wave equations. The ﬁrst **integral** **method** is a very powerful **method** for ﬁnding exact solutions of the nonlinear diﬀerential equations. From our results, we can see that the technique used in this paper is very eﬀective and can be steadily applied to nonlinear problems.

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′ = (9) By using the division theorem for two variables in complex domain C x y [ ] , which is based on the Hilbert- Nullstellensatz theorem [4], we can obtain a **first** **integral** to Equation (9) which can applied to Equation (7) to obtain a **first**-order ordinary differential equation. An exact solution to Equation (5) is then obtained by solving the equations that are obtained from applying Equation (9) into Equation (7). Now, we wish to quickly recall the division theory.

In this early assignment, we focus to enhance the convergence level of the complete elliptic **integral** of the **first** kind K k by transforming the value of modulus k into an appropriate modulus functions to produce transformation functions. From the literature review that we have conducted, there are two well known examples of such modulus transformation, namely

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bounded sets. We advise the reader to pay particular attention to its reading as it introduces notations which are ubiquitous in the rest of the paper. Section 2 is where the exact definition and basic properties of tubular neighborhoods are given. In Section 3, we prove the key result for the tubular Runge’s theorem (Proposition 3.1), essentially relying on a well-applied Nullstellensatz. In Section 4, we reprove Bombieri’s theorem for curves with Bilu’s idea, as it is not yet published to our knowledge (although this is exactly the principle behind Runge’s **method** in [Bilu and Parent 2011] for example). Finally, we prove and discuss our tubular Runge’s theorem (Theorem 5.1) in Section 5.

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Block-Pulse functions have been used by many researchers for various problems such as solving differential equations [12], **integral** equations [6], population balance equations [5]. Recently, Maleknejad have used BPFs for solving Fredholm **integral** equations system [9]. Babolian presented a direct **method** to solve Volterra **integral** equation of the **first** kind using operational matrix with BPFs [2] . Numerical solution for a system of **first** kind Volterra **integral** equations is presented in [8]. In this paper, we present Block-Pulse Functions(BPFs)

Abstract: To determine the photon emission or absorption probability for a diatomic system in the context of the semiclassical approximation it is necessary to calculate the characteristic canonical oscillatory **integral** which has one or more saddle points. Integrals like that appear in a whole range of physical problems, e.g. the atom-atom and atom-surface scattering and various optical phenomena. A uniform approximation of the **integral**, based on the stationary phase **method** is proposed, where the **integral** with several saddle points is replaced by a sum of integrals each having only one or at most two real saddle points and is easily soluble. In this way we formally reduce the codimension in canonical integrals of "elementary catastrophes" with codimensions greater than 1. The validity of the proposed **method** was tested on examples of integrals with three saddle points ("cusp" catastrophe) and four saddle points ("swallow-tail" catastrophe).

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[16] Abualnaja, K.M. and Khader, M.M. (2016) A Computational Solution of the Mul- ti-Term Nonlinear ODEs with Variable Coefficients Using the **Integral**-Collocation- Approach Based on Legender Polynomials. Journal of Progressive Research in Ma- thematics , 9, 1406-1410.

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Abstract. The innovative design of the next international Material Testing Reactor, the Jules Horowitz Reactor (JHR), induced the development of a new neutron and photon calculation formular HORUS3D/P&N, based on deterministic and stochastic codes and the European nuclear data library JEFF3.1.1. A new **integral** experiment, named the AMMON experiment, was designed in order to make the experimental validation of HORUS3D. The objectives of this experimental program are to calibrate the biases and uncertainties associated with the HORUS3D/N&P calculations for JHR safety and design calculations, but also the validation of some specific nuclear data (concerning mainly hafnium and beryllium isotopes). The experiment began in 2010 and is currently performed in the EOLE zero-power critical mock-up at CEA Cadarache. This paper deals with the **first** feedback of the AMMON experiments with 3D Monte Carlo TRIPOLI4©/JEFF3.1.1 calculations.

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Based on the numerical simulation using ANSYS finite element program through the use of ANSYS Parametric Design Language (APDL), successful fracture mechanics analysis is conducted to study the plain strain plate contained crack. In ANSYS, there are several techniques can be used to calculate the stress intensity factor (SIF) such as Displacement Extrapolation **Method** (DEM), J-**Integral**, Interaction **Integral** **Method** and Energy Release Rate **Method**. The last three methods can be utilized to extract the SIF as long as the analysis remain within the elastic limit. On the other hand, such methods are not necessary to implement singular element around the crack tip except for the case of displacement extrapolation **method**.

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After a new soil fills the outcropped volume, “true” stiffness matrix of this soil is added to the “artificial” stiffness matrix mentioned above. If a new soil is more or less comparable to the old one in terms of stiffness, “artificial” matrix does not spoil “true” stiffness matrix, and the results in terms of **integral** impedances are reasonable. However, if hard soil is substituted with soft soil, the “Cheshire Cat Effect” leads to the unacceptable errors in terms of **integral** impedances.

In some cases an analytical solution can not be found for **integral** equation, therefore numerical methods have been applied. In this paper we have worked out a computational **method** for the ap- proximate solution of the linear Fredholm fuzzy **integral** equations of the second kind. the pre- sented course in this study is a **method** for com- puting unknown Taylor coeﬃcients of the solution functions. Consider that to get the best approxi- mating solution of the present fuzzy equation, the truncation limit N must be chosen large enough. An interesting feature of this **method** is ﬁnding the analytical solution for given equation, if the exact solution was a polynomial of degree N or less than N . The analyzed examples illustrate the ability and reliability of the present **method**.

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In this paper, **first** we propose a new **method** to approximate the solution of two-dimensional linear fuzzy Fredholm **integral** equations of the second kind based on the fuzzy wavelet like operator. Then, we discuss and investigate the convergence and error analysis of the proposed **method**. Finally, to show the accuracy of the proposed **method**, we present two numerical examples.

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This The main methods of geometric function theory are the ideas of parametric **method**, area **method**, variational **method**, **method** of **integral** representations. These methods have appeared in different times, and a cause for their design were different extreme problems, which include L. Bieberbach problem of coefficients holomorphic univalent in the unit circle on the class S , as well as the practical task of constructing conformal mappings of simply connected and multiply connected domains. In the paper the application of the **method** of parametric representations to the problem of finding the range of the Schwartz derivative for classes S

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This paper presents a comparison between variational iteration **method** (VIM) and modified varia- tional iteration **method** (MVIM) for approximate solution a system of Volterra **integral** equation of the **first** kind. We convert a system of Volterra **integral** equations to a system of Volterra integro- differential equations that use VIM and MVIM to approximate solution of this system and hence obtain an approximation for system of Volterra **integral** equations. Some examples are given to show the pertinent features of this methods.