RESEARCH ARTICLE

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

  • M. R. AKBARI 1 ,
  • D. D. GANJI 2 ,
  • M. NIMAFAR , 3 ,
  • A. R. AHMADI 3
Expand
  • 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

Received date: 15 Jul 2014

Accepted date: 19 Aug 2014

Published date: 19 Dec 2014

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg

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.

Cite this article

M. R. AKBARI , D. D. GANJI , M. NIMAFAR , A. R. AHMADI . Significant progress in solution of nonlinear equations at displacement of structure and heat transfer extended surface by new AGM approach[J]. Frontiers of Mechanical Engineering, 2014 , 9(4) : 390 -401 . DOI: 10.1007/s11465-014-0313-y

1
Chajes A. Structural Analysis. 2nd ed. Englewood Cliff: Prentice-Hall Inc., 1999

2
Bhatt P, Nelson H M. Marshall & Nelson's Structures. 3rd ed. New York: Longman Scientific & Technical, 1990

3
Chajes A. Principles of Structural Stability Theory. Englewood Cliff: Prentice-Hall Inc., 1974

4
Kern D Q, Kraus D A. Extended Surface Heat Transfer. New York: McGraw-Hill, 1972

5
Kraus A, Aziz A, Welty J. Extended Surface Heat Transfer. New York: Wiley, 2001

6
Ganji D D,Sadighi A. Application of He's homotopy-perturbation method to nonlinear coupled systems of reaction-diffusion equations.International Journal of Nonlinear Sciences and Numerical Simulation, 2006, 7(4): 411-418

7
Ganji D D. The application of He's homotopy perturbation method to nonlinear equations arising in heat transfer. Physics Letters A, 2006, 355(4-5): 337-341

8
Ganji D D, Rajabi A. Assessment of homotopy-perturbation and perturbation methods in heat radiation equations. International Communications in Heat and Mass Transfer, 2006, 33(3): 391-400

DOI

9
Gorji M, Ganji D D, Soleimani S. New application of He's homotopy perturbation method. International Journal of Nonlinear Sciences and Numerical Simulation, 2007, 8(3): 319-328

DOI

10
Ganji D D, Rafei M, Sadighi A, Ganji Z Z. A comparative comparison of He's method with perturbation and numerical methods for nonlinear vibrations equations. International Journal of Nonlinear Dynamics in Engineering and Sciences, 2009, 1(1): 1-20

11
Golbabai A, Ahmadian D. Homotopy Pade method for solving linear and nonlinear integral equations. International Journal of Nonlinear Dynamics in Engineering and Sciences, 2009, 1(1): 59-66

12
Zhou J K. Differential Transformation and Its Applications for Electrical Circuits. Wuhan: Huazhong University Press, 1986

13
He J H. Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, 1999, 178(3-4): 257-262

DOI

14
Abbasbandy S. A numerical solution of Blasius equation by Adomian’s decomposition method and comparison with homotopy perturbation method. Chaos, Solitons & Fractals, 2007, 31(1): 257-260

DOI

15
Adomian G. Solving Frontier Problems of Physics, the Decomposition Method. Boston: Kluwer Academic Publishers, 1994

16
Aziz A, Na T Y. Perturbation Method in Heat Transfer. Washington, D C: Hemisphere Publishing Corporation, 1984

17
He J H, Wu X H. Exp-function method for nonlinear wave equations. Chaos, Solitons & Fractals, 2006, 30(3): 700-708

18
WU X H, He J H. Solitary solutions, periodic solutions and compaction-like solutions using the Exp-function method. Computers & Mathematics with Applications, 2007, 54(7-8): 966-986

19
Kutluay S, Esen A. Exp-function method for solving the general improved KdV equation. International Journal of Nonlinear Sciences and Numerical Simulation, 2011, 10(6): 717-726

DOI

20
Chang J R. The exp-function method and generalized solitary solutions. Computers & Mathematics with Applications, 2011, 61(8): 2081-2084

21
He J H. Approximate analytical solution for seepage flow with fractional derivatives in porous media. Computer Methods in Applied Mechanics and Engineering, 1998, 167(1-2): 57-68

DOI

22
He J H. Approximate solution for nonlinear differential equations with convolution product nonlinearities. Computer Methods in Applied Mechanics and Engineering, 1998, 167(1-2): 69-73

DOI

23
He J H. Variational iteration method—a kind of non-linear analytical technique: Some examples. International Journal of Non-Linear Mechanics, 1999, 34(4): 699-708

DOI

24
Akbari M R, Ganji D D, Majidian A, Ahmadi A R. Solving nonlinear differential equations of Vanderpol, Rayleigh and Duffing by AGM. Frontiers of Mechanical Engineering, 2014, 9(2): 177-190

25
Ganji D D, Akbari M R, Goltabar A R. Dynamic vibration analysis for non-linear partial differential equation of the beam-columns with shear deformation and rotary inertia by AGM. Development and Applications of Oceanic Engineering, 2014, 3: 22-31

26
Akbari M R, Ganji D D, Ahmadi A R, Hashemikachapi S H. Analyzing the nonlinear vibrational wave differential equation for the simplified model of Tower Cranes by algebraic method. Frontiers of Mechanical Engineering, 2014, 9(1): 58-70

Outlines

/