On the branching convolution equation E=Z⊛E

A Convolution Equation on a Compact Interval

Middle Convolution and Heun’s Equation

The parameter q is called an accessory parameter. Although the local monodromy (local exponent) is independent of q, the global monodromy (e.g. the monodromy on the cycle enclosing two singularities) depends on q. Some properties of Heun’s equation are written in the books [21, 23], but an important feature related with the theory of finite-gap potential for the case γ, δ, ǫ, α−β ∈ Z+ 12 (see [...

Euler Equation Branching

Some macroeconomic models exhibit a type of global indeterminacy known as Euler equation branching (e.g., the one-sector growth model with a production externality). The dynamics in such models are governed by a differential inclusion ẋ ∈ F (x), where F is a set-valued function. In this paper, we show that in models with Euler equation branching there are multiple equilibria and that the dynami...

The Time Domain Lippmann-Schwinger Equation and Convolution Quadrature

We consider time domain acoustic scattering from a penetrable medium with a variable sound speed. This problem can be reduced to solving a time domain volume Lippmann-Schwinger integral equation. Using convolution quadrature in time and trigonometric collocation in space we can compute an approximate solution. We prove that the time domain Lippmann-Schwinger equation has a unique solution and p...

Fast convolution quadrature for the wave equation in three dimensions

This work addresses the numerical solution of time-domain boundary integral equations arising from acoustic and electromagnetic scattering in three dimensions. The semidiscretization of the time-domain boundary integral equations by Runge-Kutta convolution quadrature leads to a lower triangular Toeplitz system of size N . This system can be solved recursively in an almost linear time (O(N logN)...