Mixed-type functional differential equations: A numerical approach

Abstract

a b s t r a c tThe equations considered in this paper are mixed-type functional equations (sometimes known as forward-backward differential equations) that take the formWe consider basic existence and uniqueness properties when I = [t 1 , t 2 ] and we seekfor prescribed functions w 1 , w 2 absolutely continuous, respectively, on [t 1 − 1, t 1 ], [t 2 , t 2 + 1]. With arbitrary boundary conditions specified in this way, the problem turns out to be ill-posed and so existence and uniqueness questions have an important role to play in developing numerical schemes.We consider, with t 1 , t 2 ∈ N, numerical approximations of a solution when it exists. The numerical methods that we consider are linear θ-methods and we investigate computationally their effectiveness through some illustrative examples.

An abstract framework in the numerical solution of boundary value problems for neutral functional differential equations

“…BVPs for non-neutral delay differential equations [8,12,28,29,37,38,49,52] BVPs for non-neutral differential equations with deviating arguments [1,3,[9][10][11]19,20,50,51,53] BVPs for non-neutral differential equations with a state-dependent deviating argument θ(t, y(t)) [4] BVPs for non-neutral singularly perturbed differential equations with deviating arguments [31][32][33][34][35] BVPs for the determination of periodic solutions of non-neutral delay differential equations [15,16,18,39,54] BVPs for the determination of periodic solutions of non-neutral delay differential equations with a state-dependent delay [40] BVPs for the determination of periodic solutions of non-neutral differential equations with deviating arguments [6,7] BVPs for non-neutral Fredholm integro-differential equations [21,22,45] BVPs for non-neutral Fredholm integro-differential equations with weakly singular kernels [46][47][48] BVPs for non-neutral Volterra integro-differential equations [23] BVPs for the determination of periodic solutions of neutral delay differential equations [5,17] BVPs for neutral differential equations with st...…”

“…In this section, we study how x * and (v * , β * ) can be approximated by the approximations (18) and (19), respectively, obtained by some fixed point x * K of Φ K . Our analysis is based on studying of how x * is approximated by fixed points of the operator…”

Finite element solution of a linear mixed-type functional differential equation

“…This was the purpose of the authors of [7,8] who have developed a new approach to the analysis of the autonomous case. They have analyzed MTFDEs as boundary value problems (BVP), that is, they have considered the problem of finding a differentiable solution on a certain real interval [0, k − 1], given its values at the intervals [−1, 0] and (k − 1, k].…”

“…Here we provide its detailed description and error analysis. A comparative analysis of the algorithms presented in [7,11,12,[18][19][20] is also provided. We introduce a criterion for the existence of a solution of the considered boundary value problem and present some examples that illustrate the application of this criterion.…”

A lumped mucosal wave model of the vocal folds revisited: Recent extensions and oscillation hysteresis

“…It is not easy, in general, to solve an advance-delay differential equation (e.g., Ford and Lumb, 2009). Standard methods for initial value problems do not apply, because when computing the solution x at a certain time t, its value at t þ s is still unknown.…”