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ContentsPreface i1.Introduction 11.1 The weighted residual methods 31.1.1 Problem description 31.1.2 Primal methods 41.1.3 Mixed methods 81.2 Application of weighted residual methods 91.2.1 Transient motions 101.2.2 Periodic motions 121.3 Finite difference methods 141.3.1 Explicit methods 151.3.2 Implicit methods 161.4 Asymptotic methods 161.4.1 Perturbation method 161.4.2 Adomian decomposition method 191.4.3 Picard iteration method 22References 222.Harmonic Balance Method and Time Domain Collocation Method 272.1 Time collocation in a period of oscillation 292.2 Relationship between collocation and harmonic balance 312.2.1 Harmonic balance method 312.2.2 High dimensional harmonic balance method 332.2.3 Equivalence between HDHB and collocation 352.3 Initialization of Newton-Raphson method 382.3.1 Initial values for undamped system 392.3.2 Initial values for damped system 412.4 Numerical examples 422.4.1 Undamped Duffing equation422.4.2 Damped Duffing equation 47Appendix A 51Appendix B 51References 533.Dealiasing for Harmonic Balance and Time Domain Collocation Methods 553.1 Governing equations of the airfoil model 563.2 Formulation of the HB method 593.2.1 Numerical approximation of Jacobian matrix 613.2.2 Explicit Jacobian matrix of HB 623.2.3 Mathematical aliasing of HB method 673.3 Formulation of the TDC method 683.3.1 Explicit Jacobian matrix of TDC 723.3.2 Mathematical aliasing of the TDC method 733.4 Reconstruction harmonic balance method 793.5 Numerical examples 803.5.1 RK4 results and spectral analysis 803.5.2 HBEJ vs.HBNJ 843.5.3 Aliasing analysis of the HB and TDC methods 873.5.4 Dealiasing via a marching procedure 93Appendix 96References 1014.Application of Time Domain Collocation in FormationFlying of Satellites 1034.1 TDC searching scheme for periodic relative orbits 1044.2 Initial values for TDC method 1094.2.1 The C-W equations 1104.2.2 The T-H equations 1124.3 Evaluation of TDC search scheme 1124.3.1 Projected closed orbit 1124.3.2 Closed loop control 1134.4 Numerical results 114Appendix 123References 1255.Local Variational Iteration Method 1275.1 VIM and its relationship with PIM and ADM 1305.1.1 VIM 1305.1.2 Comparison of VIM with PIM and ADM 1325.2 Local variational iteration method1345.2.1 Limitations of global VIM 1345.2.2 Variational homotopy method1385.2.3 Methodology of LVIM 1405.3 Conclusion 145References 1456.Collocation in Conjunction with the Local Variational Iteration Method 1476.1 Modifications of LVIM 1496.1.1 Algorithm-1 1506.1.2 Algorithm-2 1526.1.3 Algorithm-3 1546.2 Implementation of LVIM1566.2.1 Discretization using collocation 1566.2.2 Collocation of algorithm-11576.2.3 Collocation of algorithm-21576.2.4 Collocation of algorithm-31596.3 Numerical examples 1606.3.1 The forced Duffing equation 1626.3.2 The Lorenz system 1656.3.3 The multiple coupled Duffing equations 1686.4 Conclusion 172References 1737.Application of the Local VariationalIteration Method in Orbital Mechanics 1757.1 Local variational iteration method and quasi-linearization method 1767.1.1 Local variational iteration method1767.1.2 Quasi-linearization method1787.2 Perturbed orbit propagation 1817.2.1 Comparison of local variational iteration method with the modified Chebyshev picard iteration method 1827.2.2 Comparison of FAPI with Runge-Kutta 12(10) 1857.3 Perturbed Lambert's problem 1877.3.1 Using FAPI 1887.3.2 Using the fish-scale-growing method 1897.3.3 Using quasilinearization and local variational iteration method 1937.4 Conclusion 196References 1968.Applications of the Local Variational Iteration Method in Structural Dynamics 1998.1 Elucidation of LVIM in structural dynamics 2008.1.1 Formulas of the local variational iteration method 2008.1.2 Large time interval collocation 2028.1.3 LVIM algorithms for structural dynamical system 2038.2 Mathematical model of a buckled beam 2088.3 Nonlinear vibrations of a buckled beam 2118.3.1 Bifurcations and chaos 2118.3.2 Comparison between HHT and LVIM algorithms 2188.4 Conclusion 223Appendix A 224Appendix B 224Appendix C 225References 225Index 227