In this paper, Legendre collocation method for the heat equation with nonlocal boundary conditions is developed. The stability and convergence for its semidiscrete and fully discrete schemes are set up under some conditions. Numerical tests confirm these results.

A proper orthogonal decomposition (POD) technique is employed to establish a PODbased reduced-order finite difference extrapolation iterative format for two-dimensional (2D) hyperbolic equations, which includes very few degrees of freedom but holds sufficiently high accuracy. The error estimates of the POD-based reduced-order finite difference solutions and the algorithm implementation of the POD-based reduced-order finite difference extrapolation iterative format are provided. A numerical example is used to illustrate that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the POD-based reduced-order finite difference extrapolation iterative format is feasible and efficient for solving 2D hyperbolic equations.

This paper presents a general technique to construct high order schemes for the numerical solutions of the second kind nonlinear two-dimensional Volterra integral equations. This technique is based on the so-called block-by-block approach, which is a common method for the integral equations. In this approach, the classical block-by-block approach is improved in order to avoiding the coupling of the unknown solutions at each block step with an exception at $u(x_1,y),u(x_2,y),u(x,y_1)$ and $u(x,y_2)$, while preserving the good convergence property of the block-by-block schemes. By using this new approach, a high order schema is constructed for the second kind nonlinear two-dimensional Volterra integral equations. The convergence of the schema is rigorously established. It is proved that the numerical solution converges to the exact solution with order $4$.

Geometry multigrid method is designed for solving the quadratic Lagrangian finite element equation. Firstly, quadratic Lagrangian finite element space and a series of linear Lagrangian finite element spaces are selected as finest grid and coarse grids, respectively. Secondly, a new restriction operator and a geometry multigrid (GMG01) method are proposed, and the calculation of GMG01 method is discussed. Numerical experiments are shown to verify accuracy and stability of GMG01 method, compared with usual GMG and AMG01 methods.

Reparametrization of conics can make the parameter as uniform as possible and improve the smoothness at the junction points. The common ways are to use linear rational polynomials or quadratic rational polynomials. In the paper, a cubic rational polynomial is used to reparametrize the conic section, which triples the degree of quadratic rational curve. Experimental results obtained by the parametrization of the circular arcs show that the continuity at the junction point of two circular arcs can reach $C^3$ and the deviation between the parametrization presented in the paper and the arc length parametrization has been reduced about two orders of magnitude, compared with the quadratic rational polynomial parametrization.

In this paper, the stability and bifurcation behavior of the pest and natural enemy model with piecewise constant arguments and refuge are investigated. First the discrete solution determined the dynamical behavior of the model is achieved by calculation. Thus, the linearized stability theorem is applied to find some sufficient conditions for the local asymptotic stability of equilibria. Secondly by choosing the intrinsic rate of increase or the refuge ratio as the bifurcation parameter, it is shown that the discrete solution of the model undergoes Flip bifurcation and NeimarkSacker bifurcation by using the bifurcation theory. Furthermore, the explicit formulaes determining the stability of bifurcating periodic solution are derived by applying the normal form and center manifold theorems. Finally, numerical simulations are performed to illustrate the analytic results and exhibit the complex dynamical behaviors.

The boundary value problem of fractional functional differential equations with $p$-Laplacian operator are studied. Using the fixed point theorem on cones, sufficient conditions are given for the existence of single and multiple positive solutions. The results generalize some previous results. Several examples are given to illustrate the results.

In this paper, a class of quasilinear elliptic equations is considered in $\mathbf{R}^{N}$. By virtue of critical point theory for nonsmooth functionals, the multiple weak solutions of the equations are obtained.

Some properties of the measurement topology on posets and relations with other intrinsic topologies are given. In terms of the measurement topology, continuity of posets are characterized. A counterexample is constructed to prove that the measurement topology on a completely distributive lattice need not be locally compact, revealing that the measurement topology itself needn’t be continuous.

For the sake of optimizing the Identical-Discrepant-Contrary(IDC) trend division method in the IDC grey correlation analysis , and improving the accuracy of the IDC trend classification results, this paper based on analyzing the deficiencies of the two kinds of traditional classification methods, improved it, and put forward the average-division iterative method and regression coefficients ration method. Combining with the example of relativity between the content of organic matter and the content of As in soil, this paper carried numerical simulation on the two improved methods. The results show that: The results getting from two classification methods that were improved both have a high reliability. The reliability of the result getting from average-division iterative method is 70%, and from regression coefficients ration method is 55%, slightly lower than the former. Because the content of the organic matter in the soil has “abnormal” data, it has a great influence on the coefficient of regression, and reduces the accuracy of division result getting from regression coefficients ration method.

The concept of LI-ideals in lattice implication algebras $L$ was further studied by using the principle and method of lattice theory. Firstly, the lattice operations $\Cap$ and $\Cup$, implication operation $\Longrightarrow$ and pseudo-complement operation $\circledast$ are defined on the set $\mathscr{I}_{LI}(L)$ which contains all LI-ideals in $L$, and $(\mathscr{I}_{LI}(L), \subseteq, \Cap, \Cup, \Longrightarrow, \{O\}, L)$ is proved to form a complete Heyting algebra. Secondly, some necessary and sufficient conditions of $(\mathscr{I}_{LI}(L), \subseteq, \Cap, \Cup, \Longrightarrow, \circledast, \{O\}, L)$ becoming a Boolean algebra are given by using properties of operation $\circledast$. Finally, some equivalent characterizations of prime elements in lattice $(\mathscr{I}_{LI}(L), \subseteq, \Cap, \Cup, \{O\}, L)$ are obtained by means of prime LI-ideals in $L$.

This paper deals with two unsolved problems raised by Dilworth and Crawley in 1973. The first one is about the existence of irredundant completely meet decomposition for an element in compactly generated lattice which is not the upper semimodular lattice. The second one is about the structure of compactly generated lattice which is not the strongly atomic lattice. This paper first proves that every element which has a cover in the compactly generated lattice has an irredundant completely meet decomposition. Finally, the concepts of locally strong modular and locally strong distributive lattice are introduced. By investigating the structures of such lattices, this paper partly comes to the conclusion to the second problem.