A Nonlinear Model Arising in the Buckling Analysis and its New Analytic Approximate Solution

Abstract

An analytical nonlinear buckling model where the rod is assumed to be an inextensible column and prismatic is studied. The dimensionless parameters reduce the constitutive equation to a nonlinear ordinary differential equation which is solved using the Adomian decomposition method (ADM) through Green’s function technique. The nonlinear terms can be easily handled by the use of Adomian polynomials. The ADM technique allows us to obtain an approximate solution in a series form. Results are presented graphically to study the efficiency and accuracy of the method. To the author’s knowledge, the current paper represents a new approach to the solution of the buckling of the rod problem. The fact that ADM solves nonlinear problems without using perturbations and small parameters can be judged as a lucid benefit of this technique over the other methods

Vibration analysis of a uniform pre-twisted rotating Euler–Bernoulli beam using the modified Adomian decomposition method

“…We can find the coefficients C m from the recurrent equations (27) and get a solution for C j ð Þ from equation (23). The series solution is C j ð Þ = P ' m = 0 C m j m and, of course, all the coefficients cannot be determined and the solutions have to be approximated by the truncated series P nÀ1 m = 0 C m j m .…”

“…where, K Rl = k RL =EI YY etc., the initial term F j ð Þ in equation (23) is the function of C 0 , C 1 and from the recurrence relation of equation (27b) the coefficients C m m ! 8 ð Þ are the function of C 0 , C 1 and l, where…”

“…Both the Adomian modified decomposition and Adomian decomposition methods have been applied as an innovative eigenvalue solver for the free vibration of the Euler-Bernoulli beam under various supporting conditions [21,22]. It has been shown [23] that the accuracy of solution using the AMDM may be improved by incorporating the Green's function technique. Other, more recent work, has been carried out for the rotating cantilever beam problem and related nonlinear oscillations in structural engineering [24] while the Adomian decomposition and Adomian modified decomposition methods have been applied to a non-Newtonian flow problem [25] and 3D flow in a long porous slider [26].…”

Solving Systems of Non-Linear Volterra Integral Equations by Combined Sumudu Transform-Adomian Decomposition Method

“…In the beginning of the 1980's, Adomian [5,6] proposed a new and fruitful method, so-called the Adomian decomposition method (ADM), for solving linear and non-linear (algebraic, differential, partial differential, integral, etc.) equations [7][8][9][10][11][12][13][15][16][17]. It has been shown that this method yields a rapid convergence of the solutions series.…”

A power series solution for rotating nonuniform Euler–Bernoulli cantilever beams

“…(2008a, 2008b) have applied both the Adomian modified decomposition and Adomian decomposition methods as an innovative eigenvalue solver for the free vibration of the Euler–Bernoulli beam under various supporting conditions. It has been shown by Khan and Al-Hayani (2013) that the accuracy of solution using the AMDM may be improved by incorporating Green functions. Recently, much work has been carried out using the homotopy perturbation transform method (Akbarzade and Khan, 2012; Vázquez-Leal et al., 2013) which, in addition to being used to solve the problem presented here, can be used for many problems involving nonlinear differential equations.…”