A variational formulation of vertical slice models

A variational formulation of vertical slice models

A variational framework is defined for vertical slice models with three-dimensional velocity depending only on x and z. The models that result from this framework are Hamiltonian, and have a Kelvin– Noether circulation theorem that results in a conserved potential vorticity in the slice geometry. These results are demonstrated for the incompressible Euler– Boussinesq equations with a constant t...

A variational formulation for PDEs

Definition 1. A classic solution (sometimes called strong solution) to the problem is a function u ∈ C2([a, b]) such that it vanishes at the endpoints and pointwise it satisfies the equation −u′′(x)+ u(x) = f(x). Let’s change approach. Suppose that we formally multiply the differential equation by an arbitrary function φ ∈ C1([a, b]) with φ(a) = φ(b) = 0 and we integrate over the interval [a, b...

Variational Formulation of Problems and Variational Methods

A Variational Formulation of Optimal Nonlinear Estimation

We propose a variational method to solve all three estimation problems for nonlinear stochastic dynamical systems: prediction, filtering, and smoothing. Our new approach is based upon a proper choice of cost function, termed the effective action. We show that this functional of time-histories is the unique statistically well-founded cost function to determine most probable histories within empi...

A novel variational formulation for thermoelastic problems

Article history: Received 26 March 2014 Received in revised form 25 September 2014 Accepted 27 September 2014 Available online 15 October 2014