论文研究-一类分数阶非线性振子方程的特性研究.pdf

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将分数阶微分算子引入到黏弹性介质中的阻尼振动中建立分数阶非线性振动方程。利用Adomian分解方法借助Mathematica软件的符号计算功能求解了该类分数阶阻尼振动方程的近似解,研究了振子运动与方程中分数阶导数的关系。
32 2012,48(16) Computer Engineering and Applications计算机工程与应用 l2(0)=Jt1-2a 6 9t tion to fractional derivati (1-2a)(1+3a tions. to methods of their solution and some of their 4a2t2+3r(-1+2a),4t°r(+2a)3a2t2+4(-1+2a) applications[M].New York: Academic Press, 1999 r2(+a)/(-1+3a)r2(1+a)r(1+3a)P2(1+a)(-1+4a) [2] Tofighi A The intrinsic damping of the fractional oscilla- 12a2t2+4(-1+3a) 3tr(1+2a) r(+a)(1+20)(-1+4)F(1+)(+4 tor[J]. Physica A: Statistical Mechanics and its Applica- ions,2003,329(1/2):29-34 6t"(1+3ax) 4at(-1+2a)/(-3+4a) r(1+a)(1+2a)(1+4a)r(+a)/(-1+3a)/(-3+5a) [3 Ryabov Y E, Puzenko A Damped oscillation in view of the fractional oscillator equation[J]. Physical Review B 6a4t-4+r(-1+2a)/(-3+4a)2a2t2+r(-1+2a)/(-1+4a) 厂(+a)/(-1+3a)/(-3+5a)r(1+a)7(-1+3a)/(-1+5a) 2002,66(18):184201 6a22+3r(+2a)/(-1+4a),2tr(1+2a)/(1+4a) [4 Momani S, Ibrahim R WAnalytical solutions of a frac (22) r(+a)/(+3a)/(-1+5a)r(+a/(1+3a)/(1+5o) tional oscillator by the decomposition method[J]. Interna tional Journal of Pure and Applied mathematics, 2007 37(1):119-131 利用 Mathematica软件的符号计算功能计算出级 [5] Al-rabtah A, Erturk V S, Momani S Solutions of a frac 数解的其他项,方程(13)的近似解表示为: tional oscillator by using differential transform method[J L()=2- Computers Mathematics with Applications, 2010, 59 7(+a)T(1+2a) (3):1356-1362 a2t17(-1+2 t"r(1+2a) (23) [6] Ma J H, Liu Y Q Exact solutions for a generalized non- F(1+a)(-1+3a)(+a)l(+3a) linear fractional Fokker-Planck equation[J]. Nonlinear Analy 利用 Adomian多项式可以简单有效地得到分数 sis: Real World Applications, 2010, 11(1):515-521 阶非线性振子方程的近似解,取方程(13)的三阶近[7]IiYQ, Ma J H Exact solutions of a generalized 似解,观察振子位移的振动情况如图2,图2给出了振 multi-fractional nonlinear diffusion cquation in radical 子位移u随分数阶导数α和时间t的三维图像,当分 symmetry[]. Communications in Theoretical Physics 4数阶导数a较小时,对振子振动的影响越明显 2009,52(5):857-861 [8] Jiang X Y, Xu M YThe time fractional heat conduc- tion equation in the general orthogonal curvilinear coor dinate and the cylindrical coordinate systems[J]. Physica A: Statistical Mech and its applications, 2010, 389 (17):3368-3374 6 1.0 [9 He J H Homotopy perturbation method a new nonlin- 0.6 car analytical technique [J] Applied Mathematics and 0.4 Computation,2003,135(1):73-79 1.6 0.2 [10 Wazwaz A M.The variational iteration method for ana 00 lytic treatment for linear and no ODEs[J].Applied 图2非线性振子方程(13)的三阶近似解 Mathematics and Computation, 2009, 212(1): 120-134 随分数阶导数α和时间t的振动情况 [11] Adomian G. A review of the decomposition method in applied mathematics[J]Journal of Mathematical Anal 3结果和讨论 sis and Applications, 1988, 135(2): 501-544 本文利用 Adomian分解方法求解了一类黏弹性12] Momani s, Odibat Z, Analytical solutions of a time 介质中的分数阶非线性振动方程。借助 Mathematica fractional Navier-Stokes equation by adomian decompo 软件的符号计算功能求解了该类分数阶阻尼振动方 sition method[ J. Applied Mathematics and Computa- 程的近似解,研究了振子运动与方程中分数阶导数 tion,2006,177(2):488-494 的关系。 [13] Barari A, Omidvar M, Abdoul R, et al. Application of homotopy perturbation method and variational itera- tion method to nonlinear Oscillator Differential Equa 参考文献: tions[J].Acta Applicandae Mathematicae, 2008, 104(2) [1 Podlubny I Fractional differential equations an introduc 161-171

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