Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.

Stochastic Equations in Infinite Dimensions

The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

Differential-Difference Equations. By Richard Bellman and Kenneth L. Cooke. Pp. xvi, 465. 1963. (Academic Press)

One-dimensional Bose-gas One-dimensional Heisenberg magnet Massive Thirring model Classical r-matrix Fundamentals of inverse scattering method Algebraic Bethe ansatz Quantum field theory integral models on a lattice Theory of scalar products Form factors Mean value of operator Q Assymptotics of correlation functions Temperature correlation functions Appendices References.

Quantum Inverse Scattering Method and Correlation Functions

The study of waves can be traced back to antiquity where philosophers, such as Pythagoras (c.560-480 BC), studied the relation of pitch and length of string in musical instruments. However, it was not until the work of Giovani Benedetti (1530-90), Isaac Beeckman (1588-1637) and Galileo (1564-1642) that the relationship between pitch and frequency was discovered. This started the science of acoustics, a term coined by Joseph Sauveur (1653-1716) who showed that strings can vibrate simultaneously at a fundamental frequency and at integral multiples that he called harmonics. Isaac Newton (1642-1727) was the first to calculate the speed of sound in his Principia. However, he assumed isothermal conditions so his value was too low compared with measured values. This discrepancy was resolved by Laplace (1749-1827) when he included adiabatic heating and cooling effects. The first analytical solution for a vibrating string was given by Brook Taylor (1685-1731). After this, advances were made by Daniel Bernoulli (1700-82), Leonard Euler (1707-83) and Jean d’Alembert (1717-83) who found the first solution to the linear wave equation, see section (3.2). Whilst others had shown that a wave can be represented as a sum of simple harmonic oscillations, it was Joseph Fourier (1768-1830) who conjectured that arbitrary functions can be represented by the superposition of an infinite sum of sines and cosines now known as the Fourier series. However, whilst his conjecture was controversial and not widely accepted at the time, Dirichlet subsequently provided a proof, in 1828, that all functions satisfying Dirichlet’s conditions (i.e. non-pathological piecewise continuous) could be represented by a convergent Fourier series. Finally, the subject of classical acoustics was laid down and presented as a coherent whole by John William Strutt (Lord Rayleigh, 1832-1901) in his treatise Theory of Sound. The science of modern acoustics has now moved into such diverse areas as sonar, auditoria, electronic amplifiers, etc.

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Linear and nonlinear waves

Analysis for an N-stepped Rayleigh bar with sections of complex geometry

Parametrically defined nonlinear differential equations, differential–algebraic equations, and implicit ODEs: Transformations, general solutions, and integration methods

New methods of the mass and heat transfer theory—I. The method of asymptotic correction and the method of model equations and analogies

Systems of hydrodynamic type equations: Exact solutions, transformations, and nonlinear stability

The problem of diffusion of a substance, dissolved in a flow, to absorbing drops (bubbles) moving one after another in a viscous incompressible fluid is investigated. An approximate analytic expression is obtained for the differential and integral flows of the substance to the surface of each drop with consideration of the changes of the concentration and velocity fields due to the presence of other drops. A chain of spherical drops of equal radius arranged on the axis of a uniform forward flow is examined. It is shown that if the distance between drops, referred to the radius of the drops, satisfies the inequality 1≪l≪P1/2 (P is the Peclet number), then the integral inflow of the substance to the surface of the second drop of the chain is 2.41 times less than the integral inflow to the first (the drops are enumerated along the flow); the total diffusion flow to the surface of an arbitrary drop with number k is determined by the expression Ik=I1[k1/2 − (k−1)1/2], where Ik is the total flow to the first drop of the chain. The case of diffusion interaction of a solid particle and drop is examined. It is shown that for particles moving one after another with the same velocity in a fluid at rest the presence of a drop before the solid particle leads to a marked decrease of the total diffusion flow of the solid particle [by O(P1/6) times], whereas the presence of a solid particle before a drop does not affect (in the main approximation with respect to the characteristic diffusion parameter) the total flow of the latter.I
k=I
i[k
1/2−(k−1)1/2]

Andrei D. Polyanin

Papers

Analysis for an N-stepped Rayleigh bar with sections of complex geometry

Parametrically defined nonlinear differential equations, differential–algebraic equations, and implicit ODEs: Transformations, general solutions, and integration methods

New methods of the mass and heat transfer theory—I. The method of asymptotic correction and the method of model equations and analogies

Systems of hydrodynamic type equations: Exact solutions, transformations, and nonlinear stability

Diffusion to a chain of drops (bubbles) at large péclet numbers