Linear delta expansion technique for the solution of anharmonic oscillations

Abstract

The linear delta expansion technique has been developed for solving the differential equation of motion for symmetric and asymmetric anharmonic oscillators. We have also demonstrated the sophistication and simplicity of this new perturbation technique.We have presented the linear delta expansion (LDE) technique [1] for the resolution of oscillatory non-linear problems. This is a powerful technique which has been originally introduced to deal with problems of strong coupling in quantum field theory. This method has also been applied to a wide class of problems [2-9]. In the original formulation of LDE technique, the Lagrangian density L δ which is not exactly solvable, is interpolated with a solvable Lagrangian L 0 . The full Lagrangian was written as L δ = L 0 + δL. For δ = 0, one obtains solvable Lagrangian L 0 . Here δL is treated as a perturbation and δ is used to keep track of the perturbative order.In the theory of harmonics, there is an important phenomenon which should be pointed out because of its practical importance in demonstrating non-linear effects. The thermal expansion of a crystalline solid can be understood from the non-linear force acting between the atoms. The restoring force between a pair of atoms is asymmetric.For large amplitude of vibration, the restoring force often contains additional terms involving higher power of displacement x. In such case, the restoring force is expressed aswhere s, a 2 , a 3 , a 4 , a 5 etc. are constants. For this case, the oscillator becomes nonlinear and the vibration contains a fundamental frequency and higher harmonics. The oscillations of a non-linear oscillator are called anharmonic oscillations. The differential equation of motion for this type of oscillation is 117

Numerical Solution of Nonlinear Fredholm Integro-Differential Equations Using Spectral Homotopy Analysis Method

“…Further, Liao [6] introduced a nonzero auxiliary parameter to solve this limitation. Unlike the special cases of HAM such as Lyapunove's artificial small parameter method [8], Adomian decomposition method [9][10][11][12], and theexpansion method [13], this method need not a small perturbation parameter. In the HAM the perturbation techniques [14] need not be converted a nonlinear problem to infinite number of linear problems.…”