Milʹstejn, Grigorij N.
[VerfasserIn]
;
Tretyakov, Michael V.
[Sonstige Person, Familie und Körperschaft];
Tretʹjakov, Michail V.
[Sonstige Person, Familie und Körperschaft]
Stochastic numerics for mathematical physics
: with ... 28 tables
Beschreibung:
Stochastic differential equations are not only interesting in themselves, but also for their applications in the natural sciences. They also are seen as a way to reduce complex multi- dimensional partial differential equations to the Cauchy problem of (essentially one-dimensional) stochastic differential equations. This approach leads to powerful computational mathematics that are presented in this treatise. The authors construct many new special schemes, some published here for the first time. In the second part of the book the authors construct stochastic numerical algorithms for solving complicated problems for partial differential equations occurring in practical applications, both linear and non-linear. All the methods are presented with proofs and hence founded on rigorous reasoning, thus giving the book textbook potential. It addresses researchers and graduate students in stochastics, numerics, physics and mathematical biology, as well as chemistry and engineering and even financial mathematics TOC:Mean-Square Approximation for Stochastic Differential Equations.- Weak Approximation for Stochastic Differential Equations.- Numerical Methods for SDEs with Small Noise.- Stochastic Hamiltonian Systems and Langevin Type Equations.- Simulation of Space and Space-Time Bounded Diffusions.- Random Walks for Linear Boundary Value Problems.- Probabilistic Approach to Numerical Solution of the Cauchy Problem for Nonlinear Parabolic Equations.- Numerical Solution of the Nonlinear Dirichlet and Neumann Problems Based on the Probabilistic Approach.- Application of Stochastic Numerics to Models with Stochastic Resonance and to Brownian Ratchets.- Appendix: Practical Guidance to Implementation of the Stochastic Numerical Methods.- References.- Index