Fréchet-Volterra variational equations, boundary value problems, and function space integrals

Chapter 1 Foundations of Elliptic Boundary Value Problems 1 . 1 Euler equations of variational problems

A membrane is a surfacic, in its ground state plane, elastic body, whose potential energy is directly proportional to the change of the surface area. Thus, the ground state can be described by a bounded domain Ω ⊂ lR with boundary Γ = ∂Ω which we assume to be piecewise smooth. Under the influence of a force with density f = f(x), x ∈ Ω, which acts perpendicular to the (x1, x2)-plane, the membra...

Solutions of initial and boundary value problems via F-contraction mappings in metric-like space

We present sufficient conditions for the existence of solutions of second-order two-point boundary value and fractional order functional differential equation problems in a space where self distance is not necessarily zero. For this, first we introduce a Ciric type generalized F-contraction and F- Suzuki contraction in a metric-like space and give relevance to fixed point results. To illustrate...

The variational iteration method for a class of tenth-order boundary value differential equations

Rough Volterra equations 2: Convolutional generalized integrals

We define and solve Volterra equations driven by a non-differentiable signal, by means of a variant of the rough path theory allowing to handle generalized integrals weighted by an exponential coefficient. The results are applied to a standard rough path x = (x,x) ∈ C 2 (R)×C 2γ 2 (R), with γ > 1/3, which includes the case of fractional Brownian motion with Hurst index H > 1/3.

Distributed order equations as boundary value problems

There has been a recent increase in interest in the use of distributed order differential equations (particularly in the case where the derivatives are given in the Caputo sense) to model various phenomena. Recent papers have provided insights into the numerical approximation of the solution, and some results on existence and uniqueness have been proved. In each case, the representation of the ...