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

doi:10.1007/s11465-014-0313-y

Front. Mech. Eng.

2014, Vol. 9

Issue (4) : 390-401 https://doi.org/10.1007/s11465-014-0313-y

RESEARCH ARTICLE

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

M. R. AKBARI¹,D. D. GANJI²,M. NIMAFAR^3,^*(

),A. R. AHMADI³

1. Department of Civil Engineering, University of Tehran, Tehran 47618-18853, Iran
2. Department of Mechanical Engineering, Babol University of Technology, Babol, Iran
3. Department of Mechanical Engineering, Sari Branch, Islamic Azad University, Sari 48148-83494, Iran

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Abstract

In this paper, we aim to promote the capability of solving two complicated nonlinear differential equations: 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.

Keywords AGM extended surface heat transfer slope-deflection method

Corresponding Author(s): M. NIMAFAR

Online First Date: 28 October 2014 Issue Date: 19 December 2014

Cite this article:

M. R. AKBARI,D. D. GANJI,M. NIMAFAR, et al. Significant progress in solution of nonlinear equations at displacement of structure and heat transfer extended surface by new AGM approach[J]. Front. Mech. Eng., 2014, 9(4): 390-401.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-014-0313-y
https://academic.hep.com.cn/fme/EN/Y2014/V9/I4/390

Fig.1 Physical model

Fig.2 Schematic diagram for computing the elastic modulus and diameter along the member length

Fig.3 Displacement of the structure element (y) versus length (x) obtained by AGM

Fig.4 Chart of the first derivative for the obtained solution by AGM

Fig.5 Schematic of the structure for obtaining the bending moment at two ends

Tab.1 Tab.1

Fig.6 Comparison between AGM and numerical method for computing displacement of the structure

Fig.7 Comparison between the first derivative of the obtained solutions by AGM and numerical method

Fig.8 Comparison between AGM and exact solution for computing displacement of the structure

Fig.9 Comparison between the first derivative of the obtained solutions by AGM and exact solution

Fig.10 Differences between the obtained solutions by AGM and numerical method

Fig.11 Differences between the first derivative of the obtained solution between AGM and numerical method

Fig.12 Schematic of the extended surface

Fig.13 Temperature distribution along the length of the extended surface

Fig.14 First derivative of the obtained solution by AGM

Fig.15 Heat transfer rate for obtained solution by AGM

Tab.2 Obtained results according to the numerical solution of Eq. (50) in the specified domain

Fig.16 Comparison between AGM and numerical solution for temperature distribution in the specified domain

Fig.17 Comparison between AGM and numerical solution for u ( t )

Fig.18 Absolute value difference of the obtained solutions between AGM and numerical method

Tab.3

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