Integral equation

In mathematics, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; u(x_1,x_2,x_3,...,x_n) ; I^1 (u), I^2(u), I^3(u), ..., I^m(u)) = 0where I^i(u) is an integral operator acting on u. Hence, integral equations may be viewed as the analog to differential equations where instead of the equation involving derivatives, the equation contains integrals. A direct comparison can be seen with the mathematical form of the general integral equation above with the general form of a differential equation which may be expressed as follows:f(x_1,x_2,x_3,...,x_n ; u(x_1,x_2,x_3,...,x_n) ; D^1 (u), D^2(u), D^3(u), ..., D^m(u)) = 0where D^i(u) may be viewed as a differential operator of order i. Due

A 1D Model For A Closed Rigid Filament In Stokes Flow

Philippe Caussignac

We adapt an existing asymptotic method to set up a one-dimensional model for the fall of a closed filament in an in finite fluid in the Stokes regime. Starting from the single-layer integral representation of the fluid velocity around the filament, we get, for a very slender filament, a Fredholm integral equation on the filament centerline. From this equation, we can compute the drag and force acting on the filament and consequently the resistance matrix. The integral equation is discretized with a collocation method. The study of a scalar model problem yields existence and uniqueness results together with an error estimate for the discretization scheme. Then, we compare the resistance matrix of thin ideal knots obtained from the discretization of the present model to a boundary element method; numerical convergence results and a good agreement of both methods validate our model.

2009

The non-linear stochastic wave equation in high dimensions

Daniel Conus

The main topic of this thesis is the study of the non-linear stochastic wave equation in spatial dimension greater than 3 driven by spatially homogeneous Gaussian noise that is white in time. We are interested in questions of existence and uniqueness of solutions, as well as in properties of solutions, such as existence of high order moments and Hölder-continuity properties. The stochastic wave equation is formulated as an integral equation in which appear stochastic integrals with respect to martingale measures (in the sense of J.B. Walsh). Since, in dimensions greater than 3, the fundamental solution of the wave equation is neither a function nor a non-negative measure, but a general Schwartz distribution, we first develop an extension of the Dalang-Walsh stochastic integral that makes it possible to integrate a wide class of Schwartz distributions. This class contains the fundamental solution of the wave equation, under a hypothesis on the spectral measure of the noise that has already been used in the literature. With this extended stochastic integral, we establish existence of a square-integrable random-field solution to the non-linear stochastic wave equation in any dimension. Uniqueness of the solution is established within a specific class of processes. In the case of a fine multiplicative noise, we obtain a series representation of the solution and estimates on the p-th moments of the solution (p ≥ 1). From this, we deduce Hölder-continuity of the solution under standard assumptions. The Hölder exponent that we obtain is optimal. For the case of general multiplicative noise, we construct a framework for working with appropriate iterated stochastic integrals and then derive a truncated Itô-Taylor expansion for the solution of the stochastic wave equation. The convergence of this expansion remains an open problem, so we conclude with some remarks that suggest an Itô-Taylor series expansion for the solution.

EPFL2008

The non-linear stochastic wave equation in high dimensions

Daniel Conus

EPFL2008

Boundary integral equation methods for solids and fluids

A 1D Model For A Closed Rigid Filament In Stokes Flow

The non-linear stochastic wave equation in high dimensions

The non-linear stochastic wave equation in high dimensions

Boundary integral equation methods for solids and fluids

A 1D Model For A Closed Rigid Filament In Stokes Flow