Analytical approximate solution of nonlinear dynamic system containing fractional derivative by modified decomposition method

• Published in: Applied mathematics and computation Vol. 182; no. 1; pp. 544 - 552
• Main Authors: Saha Ray, S., Chaudhuri, K.S., Bera, R.K.
• Format: Journal Article
• Language: English
• Published: New York, NY Elsevier Inc 01.11.2006; Elsevier
• Subjects: Adomian decomposition method; Exact sciences and technology; Fractional derivative; Fractional differential equation; Global analysis, analysis on manifolds; Mathematics; Modified decomposition method; Nonlinear dynamics and nonlinear dynamical systems; Numerical analysis; Numerical analysis. Scientific computation; Ordinary differential equations; Physics; Sciences and techniques of general use; Statistical physics, thermodynamics, and nonlinear dynamical systems; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Fractional differential equation; Adomian decomposition method; Modified decomposition method; Fractional derivative; Damping; Viscoelasticity; Material science; Differential equation; Electromagnetic; Thin plate; Acoustics; Viscoelastic material; Dynamical system; Non linear system; Exact solution; Numerical analysis; Applied mathematics; Electrochemistry; Newtonian fluid; Non linear mechanics; Approximate solution; Analytical solution
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2006.04.016

The fractional derivative has been occurring in many physical problems such as frequency dependent damping behavior of materials, motion of a large thin plate in a Newtonian fluid, creep and relaxation functions for viscoelastic materials, the PI λ D μ controller for the control of dynamical systems etc. Phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry and material science are also described by differential equations of fractional order. The solution of the differential equation containing fractional derivative is much involved. Instead of application of the existing methods, an attempt has been made in the present analysis to obtain the solution of nonlinear dynamic system containing fractional derivative [Ji-zeng Wang et al., Coiflets-based method in the solution of nonlinear dynamic system containing fractional derivative, in: The Fourth International Conference on Nonlinear Mechanics (ICNM-IV), Shanghai, August 2002, pp. 1304–1308] by the relatively new Adomian decomposition method. The results obtained by this method are then graphically represented and then compared with the exact solution. A good agreement of the results is observed.