Mathematical Modeling and Analysis of Complex Processes

The chapter addresses some aspects of modern computational mathematics concerning the mathematical modeling of processes in multicomponent media, wave processes in homogeneous and inhomogeneous infinite waveguides, and new methods for solving ill-posed problems of linear algebra. A methodology is proposed for the analysis of multicomponent bodies with technological or natural thin layers. These layers in many cases have a considerable effect on physical processes throughout the body. New mathematical models of processes in multicomponent media formulated as various classes of boundary-value and initial–boundary-value problems for partial differential equations with conjugation conditions are constructed and analyzed. For these problems (with discontinuous solutions), highly accurate computational algorithms are developed and efficient procedures of gradient methods for solving inverse problems are proposed and analyzed. They form the theoretical basis for modern information technologies of the analysis of complex processes. The results of investigations in the mathematical modeling of the dynamics of distributed process of the environment are described. The structure of the developed system for the automated solution of environmental problems is presented. A technique is proposed and tools are created for mathematical modeling of the distribution and formation of acoustic fields based on wave equations with complex nonself-adjoint operators. The stability analysis with respect to the initial data is carried out and the stability conditions are established for explicit and implicit three-layer difference schemes with such operators. New results in the theory of weighted pseudoinversion are obtained and used to develop efficient methods to solve ill-posed problems of linear algebra.