Abstract: A complex medium is generally a multiphase mixture. Unlike the classical solid, liquid and gas, its mechanical behaviors exhibit anomalous features such as the memory and the path-dependence characteristics, which can hardly be well described with the classical mechanics models of integral-order derivatives. From the viewpoint of mathematical and physical modeling, the local limit definition of the integral-order derivative is not suitable to depict such non-local mechanical behaviors. The fractional derivative is essentially an integro-differential operator with underlying clear statistical physical explanation and can accurately describe the global correlation of complex mechanical behaviors. Since 1990s, the fractional derivative modeling for anomalous mechanical behaviors of complex media has attracted extensive attention due to its merits of fewer parameters with clear physical explanations. From the phenomenological modeling perspective, a review was made on the applications and developments of the fractional and fractal derivative models for the diffusion and energy dissipation behaviors of complex media.
Abstract: The reduced 3-wave and 4-wave Hamiltonian equations for ocean surface waves were widely used for the simplified structure with symmetric polynomial kernels and for the conservation of energy, etc. However, according to the related assumption for approximation in derivation, the range of applicability was limited to weakly nonlinear waves of small amplitude. Here the following issue was further studied: for nonlinear waves of finite amplitude within a certain range, was it also possible to obtain reduced Hamiltonian equations with symmetric polynomial kernels in a sense of sufficient accuracy? Because of complicated strongly nonlinear coupling, few development in this significant respect had been made as yet. A new approach was proposed based on the Chebyshev polynomials to best approximate the primitive water wave equations in the exact sense of strongly nonlinear coupling and derive new reduced Hamiltonian equations with symmetric polynomial kernels. The new results exhibit an extension from a weakly nonlinear case in which the product of the wave number and the wave steepness is small to a nonlinear case in which this product goes up to about 1.035. Moreover, within this range, the approximation errors are lower than 5%, and in particular, the new results prove exact whenever the said product lies close to 0.9.
Abstract: An extended precise integration method (EPIM) for solving inhomogeneous two-point boundary value problems (TPBVPs) of linear time-invariant systems was proposed. Firstly, the interval quantities of the interval matrices and vectors were introduced to describe the discretization of the differential equations for the TPBVPs. Thus a general framework for solving the TPBVPs was established, where the interval quantities for different intervals were computed in parallel, and the assembled algebraic equations for global analysis were independent of the boundary conditions. Secondly the interval response matrices corresponding to the interval vectors were used to deal with the inhomogeneous terms. The addition theorems for the interval response matrices were derived with the inhomogeneous terms in the forms of polynomial function, sine/cosine function, exponential function and their combinations. Then the extended precise integration method was proposed in combination with the incremental storage technique, of which the accuracy approached the machine precision for the inhomogeneous terms in the above forms. The general forms of the inhomogeneous terms can be approximated with the mentioned forms. In comparison with the analytical methods, two numerical examples of stiff problems give results showing the high accuracy and stability of the proposed method.
Abstract: A lattice Boltzmann model for the heat conduction equation with a nonlinear source term and a nonlinear diffusion term was presented. 2 differential operators related to the source term distribution function were added to the evolution equation, on which the ChapmanEnskog expansion was carried out. Then, through some further improvement of the evolution equation the macroscopic differential equation was recovered in 2 schemes with highorder truncation errors. Detailed numerical simulations of the nonlinear heat conduction equation with different parameter selections were performed. The numerical results agree well with the exact solutions. This model can also be directly used to numerically solve other partial differential equations in similar forms.
Abstract: Based upon the method of boundary displacement functions, the non-homogeneous state equations and the control equations for the solutions of rectangular laminated plates under various boundary conditions were derived. Through dimension expansion, the non-homogeneous equations were converted to homogeneous equations, thus the possible problems of numerical ill-conditions were avoided, and the calculation process was simplified for the homogeneous equations. An introduction of the hypothesis of nonlinear boundary displacement distribution along the thickness direction reasonably decreased the number of sub-layers for the convergence of numerical results. The numerical results in the examples provide standard referential solutions for other numerical or semi-analytical methods. The present method is suitable for the problems under more complex boundary conditions.
Abstract: Since the power type variational principle was established by CHIEN Wei-zang, the differences and relations between the power type variational principles and the work-energy type quasi-variational principles in theory and practice have been a hot topic in the academic circle. According to the Jourdain principle and the d’Alembert principle, the power type variational principles and the work-energy type quasi-variational principles were established for the incompressible viscous flow in liquid-filled systems with the variational integral operation method, so as to deduce their stationary condition and quasi-stationary condition, respectively. The applications of the power type variational principles and the work-energy type quasi-variational principles in the finite element method were studied. It shows that the work-energy type quasi-variational principles coincide with the d’Alembert principle and the power type variational principles do with the Jourdain principle. The power type variational principles directly work in the state space so that they not only omit some transforms in the time space during the building of the related variational principles, but also make it convenient to build numerical models for dynamic problems.
Abstract: The global stability of impulsive complex-valued neural networks with time delay on time scales was investigated. Based on the time scale calculus theory, both the continuous-time and discrete-time neural networks were described under the same framework. For the considered complex-valued neural networks, the activation functions need not be bounded. According to the homeomorphism mapping principle in the complex domain, a sufficient condition for the existence and uniqueness of the equilibrium point of the addressed complex-valued neural networks was proposed in complex-valued linear matrix inequality (LMI). Through the construction of appropriate Lyapunov-Krasovskii functionals, and with the free weighting matrix method and matrix inequality technique, a delay-dependent criterion for checking the global stability of the complex-valued neural networks was established in the complex-valued LMIs. Finally, a simulation example shows the effectiveness of the obtained results.
Abstract: A class of high-dimensional weakly perturbed breaking solitary wave equations were studied. Firstly, the corresponding typical breaking solitary wave equations were considered. The exact solitary wave solution was obtained with the throwing method of undetermined coefficients. Then, the travelling wave asymptotic solution to the original weakly perturbed breaking solitary wave equation was found through functional analysis based on the perturbation theories. Finally, with an example, the proposed travelling wave asymptotic solution to the weakly perturbed breaking solitary wave equation shows the merits of simpleness, validity and good accuracy.
Abstract: The solution of the fractional differentialalgebraic systems (FDASs) was studied with the Adomian decomposition method. The influence of the algebraic constraints on the Adomian decomposition method was investigated, and the main difficulty of transforming the FDASs into fractional differential systems through solving the algebraic constraints directly was pointed out. To determine the components of the algebraic variable series solution, a new method was presented with the Adomian decomposition implemented successfully to obtain the solution of the FDAS. The solution of the FDAS under linear algebraic constraints was particularly discussed with the Adomian decomposition method. It’s proved that the linear relationship between the variables under algebraic constraints could be equivalently transformed into the linear relationship between the components of the corresponding series solution. 2 examples were given to illustrate the convenience and effectiveness of the proposed method.