The research is devoted to modelling of humidity transport in porous medium, with inserted point sources, and to search of optimal source power for a defined desired humidity function.
Filtration problem in porous medium is still topical today due to a large amount of related processes and parameters, which seriously improve its difficulty. Most of works are devoted to modelling the process particularly taking into account physical, chemical, or geological features of the porous medium. Optimization for Richards-Klute equation is particularly explored and usually requires additional assumptions for simplification. Solution of the optimal water distribution problem in porous medium would also lead to new results in alike fields of science – salt transfer during filtration, pollution spread and several other problems where porous medium is involved as a part.
The aim of dissertation work is:
1) Construction of mathematical model describing mass transfer in porous medium with point sources using modified Kirchhoff transformation.
2) Proof of correctness for the stated problem.
3) Construction of optimal control algorithm for mass transfer with point sources and proof of existence and uniqueness of optimal control.
4) Conducting computational experiments and studying the properties of the computational algorithm.
The object of the research is mathematical model of water transport with requested humidity values. The subject is generalized solutions of quasilinear differential equation with partial derivatives and singularity at the right-hand side of equation, their properties, and also solutions of the opposite problem – identification of point water source power humidifying the porous medium, represented by Richards-Klute equation.
Research methods: methods of the theory of generalized optimal control of linear distributed systems, methods of a priori estimates in negative norms, variational methods for solving inverse problems, apparatus of mathematical and functional analysis.
Firstly, the transition to dimensionless equation is done with help of Kirchhoff transformation and substituting variables. For one-dimensional and two-dimensional equations, the transformation with respect to the coefficients of water permeability along the corresponding coordinates and the correlation with the coefficients of other dimensions are taken into account. For the three-dimensional case the transformation is classical. As we consider a limited area with known initial and boundary conditions, scaling is introduced proportional to linear sizes of the area, period of time and possible source power. As a result, the initial nonlinear equation is transformed into linear dimensionless analogue for further calculations.
For the received equation the optimization criterion is introduced – the distribution of dimensionless analogue of humidity should be as near the desired function as possible at the last moment or during all process. Accordingly, the numerical optimization criterion is represented by the quality functional, which should be minimized. To check the result, one may take one source of humidification and find the optimal point source value by taking the functional equal to zero and solving this equation. After that the result is compared with the one received by the general method, represented in the dissertation. Theorems on correctness of the stated optimization problem, existence and uniqueness of its solution are proven.
