Numerical Analysis of Systems of Ordinary and Stochastic Differential Equations

Abstract

Part 1 Numerical solution of the Cauchy problem for systems of ordinary differential equations: the Cauchy problem for an ODE system linear systems stiff systems - single-step methods main definitions - Rosenbrock-type methods the convergence theorem - Taylor expansion of exact and numerical solutions of the Cauchy problem, consistency of RTMs, A-stability of RTMs, practical applications of RTMs to the solution of stiff ODEs, some generalizations of RTMs, numerical experiments. Part 2 Statistical simulation of the Cauchy problem solution for systems of stochastic differential equation: elements of probability theory, stochastic processes and statistical simulation - Cauchy problem for a SDE system main definitions - construction of SDEs with given probability characteristics of solution, linear SDE systems with additive and multiplicative noise, mean square stability of SDE solutions stiff in mean-square SDE systems oscillatory stochastic systems - simple numerical methods generalizing the explicity Runge-Kutta methods, families of numerical methods for solving SDE systems theorem of convergence - mean-square consistency of methods, mean-square stability of methods, numerical methods for solving linear SDE systems, variable-step algorithms for solving SDEs, numerical solution of SDE system with Poisson component, applying SDE for numerical solution of linear elliptic and parabolic equation, statistical simulation of the SDE solutions in problems of analysis and synthesis of automated control. (Part contents)