An adaptive finite element method is proposed for a class of two-dimensional nonlinear convection-diffusion equations. The characteristic method is used to deal with the convection term of the equation, and the numerical oscillation and numerical dispersion caused by the convection dominant are effectively solved. An adaptive finite element algorithm based on a posterior error estimates of gradient reconstructiontype is designed to further adjust the mesh and improve the precision on the basis of standard finite element method. Finally, numerical experiments are carried out to verify the effectiveness of the proposed method.

This paper investigates a stochastic differential equation driven by G-Brownian motion that satisfies non-Lipschitz conditions. Initially, the Euler method is employed to construct a numerical solution for the equation. Subsequently, the convergence of the Euler numerical solution to the analytical solution is proven in the mean-square sense. Finally, an example is provided to validate the theoretical results.

The equivalent transformation form of the *-Sylvester matrix equation AX+X^*B=D is studied. Using the basic properties of Kronecker product, vectorization operator and permutation matrix,and separating the real and imaginary parts of the matrix, we obtain the equivalent transformation form of the *-Sylvester matrix equation in two different cases. It is proved that it can be equitably converted to generalized Sylvester matrix equation under certain conditions.

For eigenvalue problems of differential equations, an improved physical information neural network solution and a two-stage training method are proposed. The new method can solve multiple minimum eigenvalues, the eigenvalue problem closest to the initial value and the multiple eigenvalue problem. Numerical examples of Laplace operator eigenvalue problems in 1D and 2D square regions as well as L-shaped regions show that the new method is more accurate than the existing methods.

First, this paper proves the equivalence among several definitions of the Moore-Penrose inverse of an operator. Second, by utilizing the spectral decomposition of a self-adjoint operator, we provide the spectral decomposition of the Moore-Penrose inverse of an operator with a closed range. Finally, by employing the Moore-Penrose inverse of an operator, we offer a general representation of the best approximation set, thereby proving that the best approximation set is an affine manifold.

The paper investigates the stability of a recursive neural network model composed of three neurons with three delays. Initially, the system is linearized around its equilibrium point, and the characteristic equation, which contains two exponential terms, is derived. Subsequently, the critical conditions for stability switch are discussed by zero-point distribution properties of exponential polynomials and methods of eigenvalue analysis. Finally, the sufficient conditions are given to ensure the system stable, and the critical values of time-delay are deduced.

This paper investigated multivariate approximation problems APP_d(d∈N₊)of Banach spaces equipped with zero-mean Gaussian measures in the average case setting, where covariance kernels of the zero-mean Gaussian measures had non-negative weighted sequences {α_j} and {γ_j}. In particular, the paper introduced covariance kernels with two different weights. We approximated the multivariate problems APPd by the algorithms that used finitely many continuous linear functionals. This paper discussed(s,t)-weak tractability for s>0 and t≥1 of the L₂-approximation problems APP={APP_d}_d∈N+ from the Banach spaces with the above two weighted covariance kernels under the absolute error criterion and the normalized error criterion. As a result, by the real analyzing the sufficient and necessary condition for(s,1)-weak tractability of these two L₂-approximation problems APP could be obtained as follows: the weight sequence {γ_j} tendde to 0 as j tends to infinity.

In this article, we prove the existence of weak solutions for a class of non-autonomous second-order delay evolution equations on unbounded domain by the standard Faedo-Galerkin approximation method.

To investigate the impact of virus-carrying pigs and environmental viruses on the spread of African swine fever(ASF), an ASF transmission model is constructed. The basic reproduction number R₀ is calculated by the next generation matrix method. The existence of the equilibrium point is discussed, and the local stability and global stability of the disease-free equilibrium and the endemic equilibrium are analyzed. By applying Pontryagins maximum principle and optimal control theory, the optimal control solution is obtained. Finally, the numerical simulations verify the correctness of the conclusion and reveal the influence of the virus carrying pigs and the virus in the environment on the transmission. The optimal control simulation results show that the number of infections is reduced under controlled conditions.

This paper is concerned with the nonlinear wave equations with time-dependent coefficients. Firstly, the well-posedness of the solution is proved based on the theory of the uniform sector operator. Secondly, the bounded dissipation of the process is obtained by constructing appropriate functionals. Finally, the asymptotic compactness of the process is proved by using the method of the compression function. In the research of the problem, since the time-dependent coefficients α₁ are decomposed into positive and negative parts, the operator decomposition technique cannot be applied when proving the asymptotic compactness of the process. Therefore, the method of compressing the function is chosen.

In this paper, we study a class of inverse problems that use additional conditions to reconstruct the radiation coefficients of nonlinear parabolic equations with variable coefficients, where the variable coefficient of the equation depends on the gradient of the solution. Firstly, the uniqueness and stability of the corresponding solution to definite solutions problem are proved by energy estimation. Secondly, based on the optimal control theory, the original problem is transformed into an optimization problem by using Tikhonov regularization method. Finally, the existence, uniqueness and stability of the minimum element are proved by using the necessary conditions satisfied by the minimum.

This paper is devoted to finding the existence of the response solution(i.e., quasi-periodic solutions with the same frequency as the forcing)for a quasi-periodically forced damping oscillator equationx_tt+μx_t+x-βx²=εf(ωt)with arbitrary frequency. When μ≠0 and it is far away from zero, the system is hyperbolic(the real parts of eigenvalues are not zero), there is no small divisor problem at this time. Therefore, without imposing any arithmetic conditions on the frequency ω, nor requiring the average of the forcing to be 0, this paper formulates the existence of the response solution of the original equation into a fixed point problem in the Banach space, and proves the existence of theresponse solution for the equation by using the contraction mapping principle in the case of analytic and higher-order differentiability.

The migrativity of aggregation functions plays a pivotal role in various applications, including decision making and image processing. In this paper, we take general grouping functions and general overlap functions, which possess non-trivial neutral elements, the aim is to study the migrativity of these functions over t-norm and t-conorms, and to provide the structures of general grouping functions and general overlap functions that satisfy such migrative functional equations.

Let M be a perfect matching of a graph G, S⊆M, S'⊆E(G)\M. If S is not included in any perfect matching of G except for M, then S is called a forcing set of M. The cardinality of a forcing set of M which has the least number of edges is called the forcing number of M, the sum of forcing numbers of all perfect matchings of G is called the degree of freedom of G. If M is the unique perfect matching of the graph G\S' obtained by deleting the edges of S' from G, then S' is called an anti-forcing set of M. The cardinality of an anti-forcing set of M which has the least number of edges is called the anti-forcing number of M, the sum of anti-forcing numbers of all perfect matchings of G is called the anti-degree of freedom of G. By calculating the forcing and anti-forcing polynomials, the degrees of freedom and anti-degrees of freedom of all hexagonal systems with a perfect matching generated by seven benzene rings are obtained.