BAO Si-yuan, DENG Zi-chen. The Approximate Analytical Solution Sequence for Fractional Oscillation Equations Based on the Fractional Variational Iteration Method[J]. Applied Mathematics and Mechanics, 2015, 36(1): 48-60. doi: 10.3879/j.issn.1000-0887.2015.01.004
 Citation: BAO Si-yuan, DENG Zi-chen. The Approximate Analytical Solution Sequence for Fractional Oscillation Equations Based on the Fractional Variational Iteration Method[J]. Applied Mathematics and Mechanics, 2015, 36(1): 48-60.

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

##### doi: 10.3879/j.issn.1000-0887.2015.01.004
Funds:  The National Natural Science Foundation of China(11202146)
• Rev Recd Date: 2014-11-27
• Publish Date: 2015-01-15
• 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.
•  [1] Kilbas A A, Srivastava H M, Trujillo J J.Theory and Applications of Fractional Differential Equations [M]. Amsterdam: Elsevier, 2006. [2] 陈文, 孙洪广, 李西成. 力学与工程问题的分数阶导数建模[M]. 北京: 科学出版社, 2010.(CHEN Wen, SUN Hong-guang, LI Xi-cheng.Modeling Using the Fractional Derivative in Mechanics and Engineering Problems [M]. Beijing: Science Press, 2010.(in Chinese)) [3] 徐明瑜, 谭文长. 中间过程、临界现象——分数阶算子理论、方法、进展及其在现代力学中的应用[J]. 中国科学(G辑: 物理学、力学、天文学), 2006,36(3): 225-238.(XU Ming-yu, TAN Wen-chang. Intermediate processes and critical phenomena—the theory, method, development of fractional operator and its application in modern mechanics[J].Science in China(G Series: Pysica, Mechanica & Astronomica),2006,36(3): 225-238.(in Chinese)) [4] 郭柏灵, 蒲学科, 黄凤辉. 分数阶偏微分方程及其数值解[M]. 北京: 科学出版社, 2011.(GUO Bo-ling, PU Xue-ke, HUANG Feng-hui.Fractional Partial Differential Equations and Their Numerical Solutions [M]. Beijing: Science Press, 2011.(in Chinese)) [5] LIAO Shi-jun. A short review on the homotopy analysis method in fluid mechanics[J].Journal of Hydrodynamics, Series B,2010,22(5): 882-884. [6] Duan J S, Rach R, Buleanu D, Wazwaz A M. A review of the Adomian decomposition method and its applications to fractional differential equations[J].Communications in Fractional Calculus,2012,3(2): 73-99. [7] HE Ji-huan, WU Xu-hong. Variational iteration method: new development and applications[J].Computers & Mathematics With Applications,2007,54(7/8): 881-894. [8] Huang Y-J, Liu H-K. A new modification of the variational iteration method for Van der Pol equations[J].Applied Mathematical Modelling,2013,37(16/17): 8118-8130. [9] GENG Fa-zhan. A modified variational iteration method for solving Riccati differential equations[J].Computers & Mathematics With Applications,2010,60(7): 1868-1872. [10] Ghorbani A. Toward a new analytical method for solving nonlinear fractional differential equations[J].Computer Methods in Applied Mechanics and Engineering,2008,197(49/50): 4173-4179. [11] Abassy T A. Modified variational iteration method (non-homogeneous initial value problem)[J].Mathematical and Computer Modelling,2012,55(3/4): 1222-1232. [12] Altintan D, Ugur O. Solution of initial and boundary value problems by the variational iteration method[J].Journal of Computational and Applied Mathematics, Part B,2014,259: 790-797. [13] 沈淑君, 刘发旺. 解分数阶Bagley-Torvik方程的一种计算有效的数值方法[J]. 厦门大学学报(自然科学版), 2004,43(3): 306-311.(SHEN Shu-jun, LIU Fa-wang. A computat ionally effective numerical method for the fractional order Bagley-Torvik equation[J].Journal of Xiamen University (Natural Science),2004,43(3): 306-311.(in Chinese)) [14] 廖少锴, 张卫. 非线性分数阶微分振子的动力学研究[J]. 振动工程学报, 2007,20(5): 459-467.(LIAO Shao-kai, ZHANG wei. Dynamics of nonlinear fractional differential oscillator[J].Journal of Vibration Engineering,2007,20(5): 459-467.(in Chinese)) [15] 曹军义, 谢航, 蒋庄德. 分数阶阻尼Duffing系统的非线性动力学特性[J]. 西安交通大学学报, 2009,43(3): 50-54.(CAO Jun-yi, XIE Hang, JIANG Zhuang-de. Nonlinear dynamics of Duffing system with fractional order damping[J].Journal of Xi’an Jiaotong University,2009,43(3): 50-54.(in Chinese)) [16] 王振滨, 曹广益, 朱新坚. 分数阶系统状态空间描述的数值算法[J]. 控制理论与应用, 2005,22(1): 101-105, 109.(WANG Zhen-bin, CAO Guang-yi, ZHU Xin-jian. A numerical algorithm for the state-space representation of fractional order systems[J].Control Theory & Applications,2005,22(1): 101-105, 109.(in Chinese)) [17] LI Chang-pin, DENG Wei-hua. Remarks on fractional derivatives[J].Applied Mathematics and Computation,2007,187(2): 777-784. [18] 申永军, 杨绍普, 邢海军. 分数阶Duffing振子的超谐共振[J]. 力学学报. 2012,44(4): 762-768.(SHEN Yong-jun, YANG Shao-pu, XING Hai-jun. Super-harmonic resonance of fractional-order Duffing oscillator[J].Chinese Journal of Theoretical and Applied Mechanics,2012,44(4): 762-768.(in Chinese)) [19] Plfalvi A. Efficient solution of a vibration equation involving fractional derivatives[J].International Journal of Non-Linear Mechanics,2010,45(2): 169-175. [20] Wang Z H, Wang X. General solution of the Bagley-Torvik equation with fractional-order derivative[J].Communications in Nonlinear Science and Numerical Simulation,2010,15(5): 1279-1285. [21] 刘艳芹. 一类分数阶非线性振子的特性研究[J]. 计算机工程与应用, 2012,48(16): 30-32.(LIU Yan-qin. Study on properties of fractional nonlinear oscillator equations[J].Computer Engineering and Applications,2012,48(16): 30-32.(in Chinese)) [22] 鲍四元, 邓子辰. 分数阶Fornberg-Whitham方程及其改进方程的变分迭代解[J]. 应用数学和力学, 2013,34(12): 1236-1246.(BAO Si-yuan, DENG Zi-chen. Variational iteration solutions for fractional Fornberg-Whitham equation and its modified equation[J].Applied Mathematics and Mechanics,2013,34(12): 1236-1246.(in Chinese)) [23] WU Guo-cheng, Baleanu D. New applications of the variational iteration method-from differential equations to q-fractional difference equations[J].Advances in Difference Equations,2013,21: 1-16. [24] Tatari M, Dehghan M. On the convergence of He’s variational iteration method[J].Journal of Computational and Applied Mathematics,2007,207(1): 121-128. [25] WEN Zhi-wu, YI Jie, LIU Hong-liang. Convergence analysis of variational iteration method for caputo fractional differential equations[C]// Communications in Computer and Information Science,AsiaSim 2012, 2012: 296-307. [26] Khuri S A, Sayfy A. Variational iteration method: Green’s functions and fixed point iterations perspective[J].Applied Mathematics Letters,2014,32: 28-34. [27] Barari A, Omidvar M, Ghotbi A R, Ganji D D. Application of homotopy perturbation method and variational iteration method to nonlinear oscillator differential equations[J].Acta Applicandae Mathematicae,2008,104(2): 161-171. [28] Merdan M, Gokdogan A, Yildirim A. On numerical solution to fractional non-linear oscillatory equations[J].Meccanica,2013,48(5): 1201-1213.

### Catalog

###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142