Solving Nonlinear Differential Equation Governing on the Rigid Beams on Viscoelastic Foundation by AGM

在现在的纸，在粘弹性的基础上在一根僵硬横梁上管理的一个震动的微分方程被调查了。在这个颤动的系统上管理的非线性的微分方程被一条简单、创新的途径解决，它被称为 Akbari-Ganjis 方法(AGM ) 。AGM 是一个很合适的计算过程并且为解决各种各样的非线性的微分方程是可用的。而且，用哪个的 AGM 解决一套代数学的方程，没有任何数学操作，复杂非线性的方程能容易被解决。另外，为三个周期每周期失去的抑制比率和精力被调查了。而且，比较被数字方法(Runk45 ) 和 AGM 在获得的结果之间做了。结果显示出 AGM 的高精确性。结果也证明由增加颤动(A) 的起始的振幅的数量，抑制比率的价值将被增加，并且精力由增加周期的数字每周期减少输了。AGM 是为解决微分方程的一条可靠、精确的途径，这被结束。在另一方面，说 AGM 能处于大多数状况直接解决线性、非线性的微分方程更好。这意味着没有任何无尺寸的过程，最后的答案能被获得。因此， AGM 能在非线性的科学被看作重要进步。

Significant progress in solution of nonlinear equations at displacement of structure and heat transfer extended surface by new AGM approach

In this paper, we aim to promote the capability of solving two complicated nonlinear differential equa- tions： 1） Static analysis of the structure with variable cross section areas and materials with slope-deflection method; 2） the problem of one dimensional heat transfer with a logarithmic various surface A （x） and a logarithmic various heat generation G（x） with a simple and innovative approach entitled ＂Akbari-Ganji＇s method＂ （AGM）. Comparisons are made between AGM and numerical method, the results of which reveal that this method is very effective and simple and can be applied for other nonlinear problems. It is significant that there are some valuable advantages in this method and also most of the differential equations sets can be answered in this manner while in other methods there is no guarantee to obtain the good results up to now. Brief excellences of this method compared to other approaches are as follows： 1） Differential equations can be solved directly by this method; 2） without any dimensionless procedure, equation（s） can be solved; 3） it is not necessary to convert variables into new ones. According to the aforementioned assertions which are proved in this case study, the process of solving nonlinear equation（s） is very easy and convenient in comparison to other methods.

Scrutiny of non-linear differential equations Euler-Bernoulli beam with large rotational deviation by AGM

The kinematic assumptions upon which the Euler-Bernoulli beam theory is founded allow it to be extended to more advanced analysis. Simple superposition allows for three-dimensional transverse loading. Using alternative constitutive equations can allow for viscoelastic or plastic beam deformation. Euler-Bernoulli beam theory can also be extended to the analysis of curved beams, beam buckling, composite beams and geometrically nonlinear beam deflection. In this study, solving the nonlinear differential equation governing the calculation of the large rotation deviation of the beam （or column） has been discussed. Previously to calculate the rotational deviation of the beam, the assumption is made that the angular deviation of the beam is small. By considering the small slope in the linearization of the governing differential equation, the solving is easy. The result of this simplifica- tion in some cases will lead to an excessive error. In this paper nonlinear differential equations governing on this system are solved analytically by Akbari-Ganji＇s method （AGM）. Moreover, in AGM by solving a set of algebraic equations, complicated nonlinear equations can easily be solved and without any mathematical operations such as integration solving. The solution of the problem can be obtained very simply and easily. Furthermore, to enhance the accuracy of the results, the Taylor expansion is notneeded in most cases via AGM manner. Also, comparisons are made between AGM and numerical method （Runge- Kutta 4th）. The results reveal that this method is very effective and simple, and can be applied for other nonlinear problems.