The SPH is one kind of particle method, somewhat similar to particle grid method, but the main difference is that SPH's method needn't use any grid and can be replaced by the analysis of differential formula of the interpolation formula when calculating the derivative in space, thus we can avoid grid intertwist and deforming appearing in high degree Lagrange interpolation grid method, which makes us most headache.

Singular perturbation problems of systems of two ordinary analytic differential equations

Suppose that an analytic differential equation for which a first integral is known is subject to an analytical perturbation.

The underlying method is essentially algebraic and based on a certain nonlinear extension of similarity (gauge) transformations in the ring of analytic differential operators.

On the principle of subordination in the theory of analytic differential equations

Remarks on the topology of singular points of analytic differential equations in the complex domain and Ladis' theorem

In the paper,using the famous Lie-series formal solution of Cauchy problem of real analytic ordi- nary differential systems we derive firstly finite formulas for general solutions to two-dimensional systems of autonomuous linear inhomogencous diffe■cntial equations.Al- though such problems were known long ago to be solvable.But,however,it is unclear before weather or not the general solution can be formulated directly in terms of initial da- ta and the coefficients of the system under consideration.We discover...

In the paper,using the famous Lie-series formal solution of Cauchy problem of real analytic ordi- nary differential systems we derive firstly finite formulas for general solutions to two-dimensional systems of autonomuous linear inhomogencous diffe■cntial equations.Al- though such problems were known long ago to be solvable.But,however,it is unclear before weather or not the general solution can be formulated directly in terms of initial da- ta and the coefficients of the system under consideration.We discover that,there are six different cases distinguished by some coefficient conditions.In each case the general solution is given by a finite formula and can be calculated by simple substitution.Such results are new to the literature.

The sensitivity partial derivatives of doublet strength density with respect to the parameters of wing planfomi(bi-direction perturbation) which are needed to build the general senaitivity equations can be calculated with analytic method based on the intemal Dirichlet prob-leni fonnufation and analytical differentiation cascaded on singular integral8,and then the solu-tion of doublct strength distribution is generated for any perturbed wing planform by linear extrapolation.Thus the pressure distrlbuti on,lift...

The sensitivity partial derivatives of doublet strength density with respect to the parameters of wing planfomi(bi-direction perturbation) which are needed to build the general senaitivity equations can be calculated with analytic method based on the intemal Dirichlet prob-leni fonnufation and analytical differentiation cascaded on singular integral8,and then the solu-tion of doublct strength distribution is generated for any perturbed wing planform by linear extrapolation.Thus the pressure distrlbuti on,lift and pitehing moment coefflcients are deter-mined very rapidly.The comparison between the calculated results of the present perturbation panel method and the corresponding lower-order panel method shows that the present algorithm is of fine accuracy and can substantially reduce computing cost required(CPU time req uired for perturbation extrapolation is two orders of magnitude less than that of the corresponding lower-order panel method).

In this paper, a high order CMAC(HCMAC) neural network is proposed, in which the high order activation functions are utilized as the receptive field functions. The method of address mapping used by CMAC is adopted in the new network. Because of enhancement of the input pattern, the physical address in HCMAC is reduced highly, and by using HCMAC the approximation of complex functions can be obtained which is more continuous than using CMAC and has analytic derivatives. As a result of these characters, the computing...

In this paper, a high order CMAC(HCMAC) neural network is proposed, in which the high order activation functions are utilized as the receptive field functions. The method of address mapping used by CMAC is adopted in the new network. Because of enhancement of the input pattern, the physical address in HCMAC is reduced highly, and by using HCMAC the approximation of complex functions can be obtained which is more continuous than using CMAC and has analytic derivatives. As a result of these characters, the computing amount and learning time are reduced more than RBF neural networks. This paper also introduces originally the Kalman filter algorithm to the CMAC like networks, so learning effectiveness is improved further. By simulating, it is proved that HCMAC is feasible in many fields.