基于Hopf-Cole变换的Burgers方程初边值问题的Sinc-Galerkin法

利用Sinc-Galerkin法数值求解Burgers方程的初边值问题。首先,用Hopf-Cole变换将二阶非线性的Burgers方程变换为二阶线性方程,同时把第一类边界条件变为第二类边界条件。时间上的导数采用θ加权格式离散,空间导数采用Sinc-Galerkin法离散,端点处分别引入权函数处理变换后的第二类边界条件。最后,通过数值算例验证了Sinc-Galerkin法的指数收敛性,与精确解相比,本文构造的数值格式精度高,能够有效捕捉激波等物理现象。     

Recyclable MoO3 nanobelts for photocatalytic degradation of Rhodamine B by near infrared irradiation

Molybdenum trioxide (MoO₃) represented an excellent photocatalytic performance with many applications, including degradation of organic contaminants and splitting of water. This paper presented a new route to synthesize MoO₃ nanobelts with high aspect ratios and crystallinity by a hydrothermal technique. This work showed that the as-synthesized nanobelts exhibited strong photocatalytic activity to degrade an organic dye of Rhodamine B (RhB) in aqueous solution under the exposure of the light source in the near infrared wavelength range, significantly improving the photocatalytic activity of the nanobelts. The results also showed that for a small concentration of RhB at 7.5 mg/L a complete photodegradation (for a given MoO₃ nanobelts quantity of 0.1 g) can be reached after exposing for 60 min. For all concentrations of the RhB solution, the photodegradation exhibited an exponential dependence on the exposure time followed by a sudden shutdown, but no complete photodegradation can be reached. Also, the residual quantity of RhB in solution after the photocatalytic reaction was determined by the initial RhB concentration. The photocatalytic degradation can be interpreted by the pseudo–first-order equation for the absorption of liquid/solid based on solid capacity; thus, photocatalytic degradation can be attributed to the interaction between the photoexcited electrons in the substrate and the antibonding orbital of the RhB in solution. The sudden shutdown was due to the inability of the photoexcited electrons in the substrate hopping to the antibonding orbital of RhB in the presence of the RhB intermediate products from the degraded RhB. In addition, this work showed that the photocatalytic reaction can be recovered after a thermal treatment of postreacted MoO₃ nanobelts, enhancing the utilization efficiency of the catalysis.     

Transition probabilities of Lévy‐type processes: Parametrix construction

We present an existence result for Lévy‐type processes which requires only weak regularity assumptions on the symbol with respect to the space variable x. Applications range from existence and uniqueness results for Lévy‐driven SDEs with Hölder continuous coefficients to existence results for stable‐like processes and Lévy‐type processes with symbols of variable order. Moreover, we obtain heat kernel estimates for a class of Lévy and Lévy‐type processes. The paper includes an extensive list of Lévy(‐type) processes satisfying the assumptions of our results.     

On the Lipschitz equivalence of self‐affine sets

Recently Lipschitz equivalence of self‐similar sets on has been studied extensively in the literature. However for self‐affine sets the problem is more awkward and there are very few results. In this paper, we introduce a w‐Lipschitz equivalence by repacing the Euclidean norm with a pseudo‐norm w. Under the open set condition, we prove that any two totally disconnected integral self‐affine sets with a common matrix are w‐Lipschitz equivalent if and only if their digit sets have equal cardinality. The main methods used are the technique of pseudo‐norm and Gromov hyperbolic graph theory on iterated function systems.     

Sharp oscillation and nonoscillation tests for delay dynamic equations

In this paper, we give new sufficient conditions for both oscillation and nonoscillation of the delay dynamic equation where and satisfy τ(t) ≤ σ(t) for all large t and . As an important corollary, we obtain the time scale invariant integral condition for nonoscillation: for all large t. Also, with some examples, we show that newly presented results are sharp.     

Improving the approximation of the probability density function of random nonautonomous logistic‐type differential equations

In this paper, we address the problem of approximating the probability density function of the following random logistic differential equation: P^′(t,ω)=A(t,ω)(1?P(t,ω))P(t,ω), t∈[t₀,T], P(t₀,ω)=P₀(ω), where ω is any outcome in the sample space Ω. In the recent contribution [Cortés, JC, et al. Commun Nonlinear Sci Numer Simulat 2019; 72: 121–138], the authors imposed conditions on the diffusion coefficient A(t) and on the initial condition P₀ to approximate the density function f₁(p,t) of P(t): A(t) is expressed as a Karhunen–Loève expansion with absolutely continuous random coefficients that have certain growth and are independent of the absolutely continuous random variable P₀, and the density of P₀, , is Lipschitz on (0,1). In this article, we tackle the problem in a different manner, by using probability tools that allow the hypotheses to be less restrictive. We only suppose that A(t) is expanded on L²([t₀,T]×Ω), so that we include other expansions such as random power series. We only require absolute continuity for P₀, so that A(t) may be discrete or singular, due to a modified version of the random variable transformation technique. For , only almost everywhere continuity and boundedness on (0,1) are needed. We construct an approximating sequence of density functions in terms of expectations that tends to f₁(p,t) pointwise. Numerical examples illustrate our theoretical results.     

A problem of determining a special spatial part of 3D memory kernel in an integro‐differential hyperbolic equation

The inverse problem of determining 2D spatial part of integral member kernel in integro‐differential wave equation is considered. It is supposed that the unknown function is a trigonometric polynomial with respect to the spatial variable y with coefficients continuous with respect to the variable x. Herein, the direct problem is represented by the initial‐boundary value problem for the half‐space x>0 with the zero initial Cauchy data and Neumann boundary condition as Dirac delta function concentrated on the boundary of the domain . Local existence and uniqueness theorem for the solution to the inverse problem is obtained.