Random Processes for Classical Equations of Mathematical Physics

ISBN-13: 9780792300366 / Angielski / Twarda / 1989 / 282 str.

S. M. Ermakov; V. V. Nekrutkin; A. S. Sipin

Opis

'Et moi .... si j'avait su comment en revenir. One service mathema tics has rendered the je n'y serais point aIle.' human race. It has put common sense back Jules Verne where it belongs. on the topmost shelf next to the dusty canister labelled 'discarded non- The series is divergent; therefore we may be sense'. able to do something with it Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

Spis treści

1. Markov Processes and Integral Equations.- 1.1. Breaking-off Markov chains and linear integral equations.- 1.2. Markov processes with continuous time and linear evolutionary equations.- 1.3. Convergent Markov chains and some boundary values problems.- 1.4. Markov chains and nonlinear integral equations.- 2. First Boundary Value Problem for the Equation of the Elliptic Type.- 2.1. Statement of the problem and notation.- 2.2. Green formula and the mean value theorem.- 2.3. Construction of a random process and an algorithm for the solution of the problem.- 2.4. Methods for simulation of a Markov chain.- 2.5. Estimation of the variance of a random variable ???.- 3. Equations with Polynomial Nonlinearity.- 3.1. Preliminary examples and notation.- 3.2. Representation of solutions of integral equations with polynomial nonlinearity.- 3.3. Definition of probability measures and the simplest estimators.- 3.4. Probabilistic solution of nonlinear equations on measures.- 4. Probabilistic Solution of Some Kinetic Equations.- 4.1. Deterministic motion of particles.- 4.2. Computational aspects of the simulation of a collision process.- 4.3. Random trajectories of particles. The construction of the basic process.- 4.4. Collision processes.- 4.5. Auxiliary results.- 4.6. Lemmas on certain integral equations.- 4.7. Uniqueness of the solution of the (X, T?, H) equation.- 4.8. Probabilistic solution of the interior boundary value problem for the regularized Boltzmann equation.- 4.9. Estimation of the computational labour requirements.- 5. Various Boundary Value Problems Related to the Laplace Operator.- 5.1. Parabolic means and a solution of the mixed problem for the heat equation.- 5.2. Exterior Dirichlet problem for the Laplace equation.- 5.3. Solution of the Neumann problem.- 5.4. Branching random walks on spheres and the Dirichlet problem for the equation ?u = u2.- 5.5. Special method for the solution of the Dirichlet problem for the Helmholtz equation.- 5.6. Probabilistic solution of the wave equation in the case of an infinitely differentiable solution.- 5.7. Another approach to the solution of hyperbolic equations.- 5.8. Probabilistic representation of the solution of boundary value problems for an inhomogeneous telegraph equation.- 5.9. Cauchy problem for the Schrödinger equation.- 6. Generalized Principal Value Integrals and Related Random Processes.- 6.1. Random processes related to linear equations.- 6.2. Nonlinear equations.- 6.3. On the representation of a solution of nonlinear equations as a generalized principal value integral.- 6.4. Principal part of the operator and the Monte Carlo method.- 7. Interacting Diffusion Processes and Nonlinear Parabolic Equations.- 7.1. Propagation of chaos and the law of large numbers.- 7.2. Interacting Markov processes and nonlinear equations. Heuristic considerations.- 7.3. Weakly interacting diffusions.- 7.4. Moderately interacting diffusions.- 7.5. On one method of numerical solution of systems of stochastic differential equations.- Bibliographical Notes.- References.- Additional References.