分数阶振子方程基于变分迭代的近似解析解序列

在粘弹性介质中的阻尼振动中引入分数阶微分算子，建立分数阶非线性振动方程.使用了分数阶变分迭代法（FVIM），推导了Lagrange乘子的若干种形式.对线性分数阶阻尼方程，分别对齐次方程和正弦激励力的非齐次方程应用FVIM得到近似解析解序列.以含激励的Bagley-Torvik方程为例，给出不同分数阶次的位移变化曲线.研究了振子运动与方程中分数阶导数阶次的关系，这可由不同分数阶次下记忆性的强弱来解释.计算方法上，与常规的FVIM相比，引入小参数的改进变分迭代法能够大大扩展问题的收敛区段.最后，以一个含分数导数的Van der Pol方程为例说明了FVIM方法解决非线性分数阶微分问题的有效性和便利性.

The Approximate Analytical Solution Sequence for Fractional Oscillation Equations Based on the Fractional Variational Iteration Method

The fractional calculus was introduced to describe the damped oscillator in viscoelastic medium and the Caputo-type fractional nonlinear oscillation equations were established. The fractional variational iteration method (FVIM) was modified with a small parameter and the Lagrange multiplier was derived. For the linear fractional oscillation equations, both the homogeneous equations and the sinusoidal force-excited nonhomogeneous equations were analyzed with the FVIM to obtain the approximate analytical solution sequence. The varying curves of the displacement for different values of the fractional order were given in the case of the Bagley-Torvik equation. The relationship between oscillator motion and fractional derivative was also studied according to the extent of memorability for different fractional orders. Compared with the ordinary variational iteration method, the proposed FVIM modified with a small parameter expands the interval of convergence significantly for the solution. In the end, the Van der Pol equation with fractional derivative as an example illustrates the method’s effectiveness and convenience to solve non-linear fractional differential problems.