efficiently solved by means of the perturbation technique. However, if a ~ 1, then a problem becomes considerably difficult. If a solution to the boundary problem (1)~(3) is sought in the form Ä . . mx . Ttjt u= > Λ, sin — sin — , >=1,3,5,... Z Z then one obtains an infinite set of non-linear algebraic equations with the unknowns Af It can be solved using a numerical approximation, but sometimes fundamental difficulties occur even for a relatively low number of equations. In reference [6] the authors suggest introducing artificially a "small parameter μ" standing by non-diagonal terms, then constructing a solution as a series of the introduced parameter and finally taking μ 1 (the method of the solution extension with respect to the parameter). In some cases the Pade approximants can be applied. Although the method seems to work effectively for some problems, it needs further investigation as well as verification of the obtained results [7], An idea of searching another parameter of an asymptotic series, which allows to avoid problems with small denominators seems to be more reliable. In the works [8, 9] the so called small δ method has been proposed. A small artificial δ parameter is introduced in the power exponent of a non-linear term. According to that approach the boundary condition (3) can be formulated in the following manner
[1]
Wanda Szemplińska-Stupnicka,et al.
The Behavior of Nonlinear Vibrating Systems
,
1990
.
[2]
Kimball A. Milton,et al.
A new perturbative approach to nonlinear partial differential equations
,
1991
.
[3]
Ji-Huan He,et al.
Α Review on Some New Recently Developed Nonlinear Analytical Techniques
,
2000
.
[4]
J. Bajkowski,et al.
The 12 subharmonic resonance and its transition to chaotic motion in a non-linear oscillator
,
1986
.
[5]
C. Bender,et al.
A new perturbative approach to nonlinear problems
,
1989
.
[6]
J. Awrejcewicz,et al.
Asymptotic approaches in nonlinear dynamics : new trends and applications
,
1998
.
[7]
Jan Awrejcewicz,et al.
Strange nonlinear behaviour governed by a set of four averaged amplitude equations
,
1996
.