Contents -- Foreword -- Preface -- 1. Gronwall inequalities Fanhai Zeng, Jianxiong Cao and Changpin Li -- 1.1 Introduction -- 1.2 The continuous Gronwall inequalities -- 1.3 The discrete Gronwall inequalities -- 1.4 The weakly singular Gronwall inequalities -- 1.5 Conclusion -- Bibliography -- 2. Existence and uniqueness of the solutions to the fractional differential equations Yutian Ma, Fengrong Zhang and Changpin Li -- 2.1 Introduction -- 2.2 Preliminaries and notations -- 2.3 Existence and uniqueness of initial value problems for fractional differential equations -- 2.3.1 Initial value problems with Riemann-Liouville derivative -- 2.3.2 Initial value problems with Caputo derivative -- 2.3.3 The positive solution to fractional differential equation -- 2.4 Existence and uniqueness of the boundary value problems -- 2.4.1 Boundary value problems with Riemann-Liouville derivative -- 2.4.2 Boundary value problems with Caputo derivative -- 2.4.3 Fractional differential equations with impulsive boundary conditions -- 2.5 Existence and uniqueness of the fractional differential equations with time-delay -- 2.6 Conclusions -- Bibliography -- 3. Finite element methods for fractional differential equations Changpin Li and Fanhai Zeng -- 3.1 Introduction -- 3.2 Preliminaries and notations -- 3.3 Finite element methods for fractional differential equations -- 3.4 Conclusion -- Bibliography -- 4. Fractional step method for the nonlinear conservation laws with fractional dissipation Can Li and Weihua Deng -- 4.1 Introduction -- 4.2 Fractional step algorithm -- 4.2.1 Discretization of the fractional calculus -- 4.2.2 Discretization of the conservation law -- 4.3 Numerical results -- 4.4 Concluding remarks -- Bibliography -- 5. Error analysis of spectral method for the space and time fractional Fokker-Planck equation Tinggang Zhao and Haiyan Xuan.

5.1 Introduction -- 5.2 Preliminaries -- 5.3 Spectral method -- 5.4 Stability and convergence -- 5.4.1 Semi-discrete of space spectral method -- 5.4.2 The time discretization of Caputo derivative -- 5.5 Fully discretization and its error analysis -- 5.6 Conclusion remarks -- Bibliography -- 6. A discontinuous finite element method for a type of fractional Cauchy problem Yunying Zheng -- 6.1 Introduction -- 6.2 Fractional derivative space -- 6.3 The discontinuous Galerkin finite element approximation -- 6.4 Error estimation -- 6.5 Numerical examples -- 6.6 Conclusion -- Bibliography -- 7. Asymptotic analysis of a singularly perturbed parabolic problem in a general smooth domain Yu-Jiang Wu, Na Zhang and Lun-Ji Song -- 7.1 Introduction -- 7.2 The curvilinear coordinates -- 7.3 Asymptotic expansion -- 7.3.1 Global expansion -- 7.3.2 Boundary corrector -- 7.3.3 Estimates of the solutions of boundary layer equations -- 7.4 Error estimate -- 7.5 An example -- Bibliography -- 8. Incremental unknowns methods for the ADI and ADSI schemes Ai-Li Yang, Yu-Jiang Wu and Zhong-Hua Yang -- 8.1 Introduction -- 8.2 Two dimensional heat equation and the AD scheme -- 8.3 ADIUSI scheme and stability -- 8.3.1 ADIUSI scheme -- 8.3.2 Stability study of the ADIUSI scheme -- 8.3.2.1 The minimum of P( p) is obtained in (-1, 1) -- 8.3.2.2 The condition P(1) > 0 -- 8.3.2.3 The condition P(-1) > 0 -- 8.4 Numerical results -- Bibliography -- 9. Stability of a colocated FV scheme for the 3D Navier-Stokes equations Xu Li and Shu-qin Wang -- 9.1 Introduction -- 9.2 Full discretization: finite volume scheme in space and projection method in time -- 9.3 The main result: stability of the scheme -- 9.3.1 Notations -- 9.3.2 Discrete weak formulation -- 9.3.3 Stability result -- 9.4 Technical lemmas -- 9.4.1 The Poincare inequality and an inverse inequality -- 9.4.2 Standard lemma.

9.4.3 Specific lemmas for the Navier-Stokes equations -- 9.5 Apriori Estimate -- 9.6 Proof of stability -- Acknowledgement -- Bibliography -- 10. Computing the multiple positive solutions to p-Henon equation on the unit square Zhaoxiang Li and Zhonghua Yang -- 10.1 Introduction -- 10.2 Computation of D4 symmetric positive solutions -- 10.3 Computation of the symmetry-breaking bifurcation point -- 10.4 Branch switching to symmetric solutions -- 10.5 Numerical results -- Bibliography -- 11. Multilevel WBIUs methods for reaction-diffusion equations Yang Wang, Yu-Jiang Wu and Ai-Li Yang -- 11.1 Introduction -- 11.2 Multilevel WBIUs method -- 11.3 Approximate schemes and their equivalent forms -- 11.3.1 Approximate schemes -- 11.3.2 The equivalent forms of approximate schemes -- 11.4 Stability analysis -- 11.4.1 Lemmas for new vector norms -- 11.4.2 Stability analysis -- 11.5 Numerical results -- Bibliography -- 12. Models and dynamics of deterministically growing networks Weigang Sun, Jingyuan Zhang and Guanrong Chen -- 12.1 Introduction -- 12.2 A generation algorithm -- 12.3 Structural properties -- 12.3.1 Degree distribution -- 12.3.2 Clustering coefficient -- 12.3.3 Average path length -- 12.3.4 Degree correlations -- 12.4 Random walks on Koch networks -- 12.4.1 Evolutionary rule for first passage time -- 12.4.2 Explicit expression for average return time -- 12.4.3 Average sending time from a hub node to another node -- 12.5 An exact solution for mean first passage time -- 12.5.1 First passage time at the first step -- 12.5.2 Evolution scaling for the first passage time -- 12.5.3 Analytic formula for mean first passage time -- 12.6 Conclusions -- Bibliography -- 13. On different approaches to synchronization of spatiotemporal chaos in complex networks Yuan Chai and Li-Qun Chen -- 13.1 Introduction -- 13.2 Design of the synchronization controller.

13.3 Numerical results -- 13.4 Active sliding mode controller design -- 13.5 Numerical results -- 13.6 Master stability functions -- 13.7 Numerical results -- 13.8 Conclusion -- Bibliography -- 14. Chaotic dynamical systems on fractals and their applications to image encryption Ruisong Ye, Yuru Zou and Jian Lu -- 14.1 Introduction -- 14.2 Chaotic dynamical systems on fractals -- 14.2.1 Iterated function systems -- 14.2.2 Chaotic dynamical systems on fractals -- 14.3 A special shift dynamical system associated with IFS -- 14.4 The image encryption scheme based on the shift dynamical system associated with IFS -- 14.4.1 Permutation process -- 14.4.2 Diffusion process -- 14.4.3 Security analysis -- 14.4.3.1 Key space analysis -- 14.4.3.2 Statistical analysis -- 14.4.3.3 Differential attack -- 14.4.3.4 Resistance to known-plaintext and chosen-plaintext attacks -- 14.5 Conclusions -- Bibliography -- 15. Planar crystallographic symmetric tiling patterns generated from invariant maps Ruisong Ye, Haiying Zhao and Yuanlin Ma -- 15.1 Introduction -- 15.2 Planar crystallographic groups -- 15.2.1 Groups p2, pm, pmm -- 15.2.2 Groups pg, pmg, pgg, cm, cmm -- 15.2.3 Groups p4, p4g, p4m -- 15.2.4 Groups p3, p3m1, p31m -- 15.2.5 Groups p6, p6m -- 15.3 Rendering method for planar crystallographic symmetric tiling patterns -- 15.3.1 Description of colormaps -- 15.3.2 Description of orbit trap methods -- 15.3.3 Description of the rendering scheme -- 15.4 Conclusions -- Bibliography -- 16. Complex dynamics in a simple two-dimensional discrete system Huiqing Huang and Ruisong Ye -- 16.1 Introduction -- 16.2 Fixed points and bifurcations -- 16.2.1 The existence of fixed points -- 16.2.2 The stability of fixed points and bifurcations -- 16.3 Existence of Marotto-Li-Chen chaos -- 16.4 Numerical simulation results -- Bibliography.

17. Approximate periodic solutions of damped harmonic oscillators with delayed feedback Qian Guo -- 17.1 Introduction -- 17.2 Hopf bifurcation analysis -- 17.3 Lyapunov-Schmidt reduction approach for periodic solutions -- 17.3.1 Preliminary: reformulation and projection operators -- 17.3.2 Quadratic Taylor polynomial approximation -- 17.3.3 Bifurcation equations -- 17.3.4 Accuracy of approximation -- 17.4 Multiple scales analysis for periodic solutions -- 17.5 Simulation of period-doubling cascade -- Bibliography -- 18. The numerical methods in option pricing problem Xiong Bo -- 18.1 Introduction -- 18.2 Black-Scholes option pricing theory assumptions -- 18.3 Binomial tree methods -- 18.4 Finite difference method -- Bibliography -- 19. Synchronization and its control between two coupled networks Yongqing Wu and Minghai Lu -- 19.1 Introduction -- 19.2 Anti-synchronization between two coupled networks with nonlinear signal's connection and the inter-network actions -- 19.2.1 Two coupled networks with nonlinear signals -- 19.2.2 Two coupled networks with reciprocity -- 19.2.3 Numerical examples -- 19.3 Pinning anti-synchronization between two general complex dynamical networks -- 19.3.1 Pinning anti-synchronization criterion -- 19.3.2 Numerical simulations -- 19.4 Generalized synchronization between two networks -- 19.4.1 Generalized synchronization criterion -- 19.4.2 Numerical examples -- 19.5 Conclusion -- Bibliography -- Index.