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The Richards equation is a nonlinear degenerate advection diffusion equation that models flow in saturated/unsaturated porous media, it's crucially important for prediction of disasters when heavy ...

In this paper, the well posedness of a sixth-order phase-field equation with degenerate phase-dependent diffusion mobility in three-dimensional space is studied. We first define a notion of weak solutions and establish a regularized problem. Then, by considering the limits of the regularized problem, we obtain the existence of such solutions.

The Richards equation is a degenerate nonlinear partial differential equation which serves as a model for describing a flow of water through saturated/unsaturated porous medium under the action of gravity. This paper develops a numerical method, with a mathematical support, for the one-dimensional Richards equation. Implicit schemes based on a backward Euler format have been widely used, but have a difficulty in insuring the stability, because of the strong nonlinearity and degeneracy. A linearized semi-implicit finite difference scheme that is faster than the backward Euler implicit schemes is established, the stability of this scheme is proved by adding a small perturbation to the coefficient function, and an error estimate is made. It is found that there is a linear relationship between the discretization error in a certain norm and the perturbation strength.

A stable explicit difference scheme, which is based on forward Euler format, is proposed for the Richards equation. To avoid the degeneracy of the Richards equation, we add a perturbation to the functional coefficient of the parabolic term. In addition, we introduce an extra term in the difference scheme which is used to relax the time step restriction for improving the stability condition. With the augmented terms, we prove the stability using the induction method. Numerical experiments show the validity and the accuracy of the scheme, along with its efficiency.

SOX9 plays a crucial, extensive and conservative role in the process of somatic tissue development and adult regeneration through the positive self-regulation mediated by SOM across all vertebrates. In this study, we have cloned SOX9 from the kidney of hatchling Alligator sinensis. The full-length of SOX9 cDNA is 3878 bp with an open reading frame encoding 494 amino acids. Amino acid alignment analyses indicated that the SOX9 exhibit highly conserved functional domains. Using the droplet digital PCR, the mRNA abundances of SOX9 during nephrogenesis in A. sinensis showed prominent changes in the embryonic development, suggesting that SOX9 might combines a vital role in the regulation of complex renal development. Interestingly, we detected the nucleocytoplasmic shuttling of SOX9 protein using immunofluorescence, implying that nucleocytoplasmic shuttling is critical to the regulation of SOX9 in the renal embryonic development. Collectively, these data provide an important foundation for further studies on renal developmental biology and molecular biology of non-mammalian SOX9. Furthermore, it provides new insights into the phenomenon of SOX9 nucleocytoplasmic shuttling in Alligator sinensis, which is probably of great significance to the development of kidney metanephros embryo.

In this paper, by using finite difference method, we consider the approximate solutions for a fourth-order parabolic equation describing crystal surface growth.

Domestic garbage is mainly treated by sanitary landfill.The main characteristics of landfill leachate produced in the sanitary landfill process are high concentration of COD and BOD, high ammonia nitrogen content, high levels of refractory organics, heavy metal ions, poor biodegradability, unstable water quality, difficult handling and so on. If the treatment is not carried out, the environment will be severely damaged. According to the needs of the project, this paper proposes a set of short-term landfill leachate treatment process, which involves ozone oxidation, internal electrolytic oxidation technology, coagulation precipitation, deep oxidation of the chain reaction, microwave field catalytic oxidation, adsorption and other treatment processes. Eliminating the biochemical process, the entire process flow is relatively short, and the processing efficiency is high. And the use of PLC-based control system to achieve automatic control of landfill leachate treatment. Since the leachate water quality has a wide range of variation, the PID control method based on BP neural network adaptive control strategy is adopted to achieve on-line adjustment of PID control parameters. The final effluent water quality can meet the emission standards.

In this paper, we consider a nonlinear model describing crystal surface growth. For the equation, the finite element method is presented and a nice error estimate is derived in the L 2 norm by means of a finite element biharmonic projection approximation.

In this paper, we consider the Fourier spectral method for numerically solving the modified Swift-Hohenberg equation. The semi-discrete and fully discrete schemes are established. Moreover, the existence, uniqueness and the optimal error bound are also considered.