Complex Equations Of Second Order Research Articles

Гідродинамічне обґрунтування структури режимної спостережної мережі у гірничодобувних регіонах

Study of changes in the grounwater regime in space and time on the technologically impaired territory is on the supervisory network of boreholes. This is the first mandatory and the most important part of the hydrogeological monitoring each of the three levels - global, regional and local. Previous studies made by other authors about justify the supervisory structure of the network has an recommender nature, or based on the methods of mathematical statistics. The proposed development has hydrodynamic rationale and based on the theory of mass transfer and filtration. Our additions to rationalisation of the supervisory structure of the regim minimally sufficient number of boreholes from the perspective of the best use of observations for reliable estimate geofiltrational and migration options are : single observational borehole characterize natural or disturbed mode, hydrogeological process at one point of the research area and it is not informative. With two observations in space it is possible to make a straight line, it is enough if hydrogeological process described by Laplace equation. But geofiltrational and migration processes are described by more complex equations of second order in partial derivatives of elliptic and parabolic types. They characterize the nonlinear processes, so the minimum number of observations must be at least three in space and time within each studied object or landscape elements (the flow and across groundwater). There are examples of solving inverse problems of migration and filtration. The hydrodynamic justification of the structure of the regime observations network enhances its scientific value and practical significance. This approach enables maximum accuracy to determine the filtration and migration parameters and solve the forecast problem for the scientific substantiation of a complex of environmental measures. The development of the theory of hydrogeological monitoring is ahead of its implementation, aimed at the result – sustainable use and protection of groundwater and surface water from pollution and depletion.

The oblique derivative problem for nonlinear elliptic complex equations of second order in multiply connected unbounded domains

In this article, we discuss that an oblique derivative boundary value problem for nonlinear uniformly elliptic complex equation of second order (0.1) $w_{z\bar z} = F(z,w,w_z ,\bar w_z ,w_{zz} ,\bar w_{zz} ) + G(z,w,w_z ,\bar w_z )inD$ , with the boundary conditions in a multiply connected unbounded domain D. The above boundary value problem will be called Problem P. Under certain conditions, by using the priori estimates of solutions and Leray-Schauder fixed point theorem, we can obtain some results of the solvability for the above boundary value problem (0.1) and (0.2).

A free boundary problem of parabolic complex equation†

We consider a free boundary problem of nonlinear parabolic complex equation of second order in a simply connected region in complex plane. This problem is originally motivated by a free boundary problem model of breast cancer. In this article we convert the free boundary problem to a sequence of mixed boundary value problems of nonlinear parabolic complex equations of second order with measurable coefficients, and solve them by applying results of Wen in his earlier works. †Dedicated to Professor Guochun Wen on the occasion of his 75th birthday.

Initial-irregular oblique derivative problems for nonlinear parabolic complex equations of second order with measurable coefficients

This paper deals with initial-irregular oblique derivative boundary value problems for nonlinear and nondivergence parabolic complex equations of second order in a multiply connected domain, where coefficients of equations are measurable. We first verify the uniqueness of solutions for the above problems, and then give a priori estimates of solutions for the problems. Finally, by using the above estimates and the Leray-Schauder theorem, the existence of solutions of the initial-boundary value problems is proved. The results in this paper are generalizations of corresponding theorems in [1],[4] and [5].

Estimates of solutions of initial-irregular oblique derivative problems for linear parabolic equations of second order with measurable coefficients

The initial-irregular oblique derivative boundary value problems for linear and nondivergence parabolic complex equations of second order in multiply connected domains are dealt with, where the coefficients of equations are measurable. Firstly the uniqueness of solutions for the above problems is introduced, and then somea priori estimates of solutions for the problems are given. By using the above estimates and the Leray-Schauder theorem, the existence of solutions of the initial-boundary value problems can be proved. The results are generalizations of corresponding theorems in literature.

Initial-oblique derivative problems for nonlinear parabolic complex equations of second order with measurable coefficients

This paper deals with initial-regular oblique derivative boundary value problems for nonlinear parabolic complex equations of second order in a multiply connected domain, where coefficients of equations are measurable. We first verify the uniqueness of solution for above problems, and then give a priori estimates of solutions for the problems. Finally, by using the above estimates and the method of parameter extension, the existence of solutions of initial-boundary value problems is proved. The results in this paper are the development of corresponding theorems in [1, 4, 5], here the condition (1.4) is weaker than the corresponding condition in [1, 5], i.e. the constant 4/3 in [1, 5] is replaced by 3/2 in (1.4).

Initial-Mixed Boundary Value Problems for Parabolic Complex Equations of Second Order with Measurable Coefficients

In [2], the authors introduced some results on initial-boundary value problems for divergence parabolic equations of second order with measurable coefficients. In [1], the authors considered the first boundary value problem for non-divergence parabolic equations of second order with discontinuous coefficients. In this paper, we consider initial-mixed boundary value problems for non-divergence parabolic complex equations of second order in a multiply connected domain. Firstly, we give a priori estimates of solutions of the above initial-boundary value problems by the method of symmetric extension, and then by using these estimates, the methods of auxiliary functions and parameter extension, we prove the solvability for the foregoing problems. Here the condition (1.3) is weaker than the corresponding one in [1] and [3], i.e. the constant 4/3 in [1] and [3] is replaced by 3/2 in (1.3).