Mixed Functional Differential Equations Research Articles

Oscillation of second order mixed functional differential equations with sublinear and superlinear neutral terms

In this paper, we shall establish some new oscillation theorems for the functional differential equations with sublinear and superlinear neutral terms of the form $$ (r(t)(z'(t))^\alpha)'=q(t)x^\alpha(\tau(t)), $$ where $z(t)=x(t)+p_1(t)x^\beta(\sigma(t))-p_2(t)x^\gamma(\sigma(t))$ with $0

A spectral collocation method for mixed functional differential equations

We propose and investigate a Chebyshev spectral collocation method for solving mixed functional differential equations. One can usually not solve these equations analytically, and hence one must employef numerical methods. Our method is for boundary value problems which include delay and advance terms in the solution or a derivative. Even though, in general, solutions are not very smooth, spectral collocation methods are well suited for these types of problems as they allow easy and accurate evaluation of the approximated solution at points which are not grid points. We present numerical results and examine smoothness related convergence behaviour.

Computing Transitional Cycles for a Deterministic Time-to-Build Growth Model

Capital investment requires a gestation lag for being productive. This time-to-build feature can lead to an autonomous system of mixed-type functional differential equations (FDEs), causing aggregate fluctuations for a deterministic economy. We present a continuation method to solve a system of mixed FDEs by solving a sequence of boundary value problems for systems of ordinary differential equations (ODEs). Unlike other iteration techniques, this method can avoid the need to predetermine both forward-looking and backward-looking difference terms. The strategy is to form a homotopy that can deform a simpler ODE system into a target FDE system, while treating the deformation process as a sequence of ODEs. We can thereby compute oscillatory cycles for a time-to-build economy while it is in transition to the steady state.

Traveling Waves Connecting Equilibrium and Periodic Orbit for a Delayed Population Model on a Two-Dimensional Spatial Lattice

This paper is concerned with the existence of fast traveling waves connecting an equilibrium and a periodic orbit in a delayed population model with stage structure on a two-dimensional spatial lattice, under the assumption that the corresponding ODEs have heteroclinic orbits connecting an equilibrium point and a periodic solution. In this work, we rewrite the mixed functional differential equation as an integral equation in a Banach space and analyze the corresponding linear operator. Our approach eventually reduces a singular perturbation problem to a regular perturbation problem. The existence of traveling wave solution therefore is obtained by using the Liapunov–Schmidt method and implicit function theorem.

Stability of Linear Mixed Functional-Differential Equations with Commensurable Deviations of the Space Argument

Stability of Linear Mixed Functional-Differential Equations with Commensurable Deviations of the Space Argument

L 2 -stability of linear spatially homogeneous mixed functional-differential equations

L 2 -stability of linear spatially homogeneous mixed functional-differential equations

Mixed functional-differential equations

Mixed functional-differential equations

Asymptotic and Periodic Boundary Value Problems of Mixed FDEs and Wave Solutions of Lattice Differential Equations

We discuss the existence and approximation of solutions of asymptotic or periodic boundary value problems of mixed functional differential equations. Our approach is via monotone iteration and non-standard ordering in the profile set for asymptotic boundary value problems and viaS1-degree and equivariant bifurcation theory for periodic boundary value problems. Applications will be given to wave fronts and to slowly oscillatory spatially periodic traveling waves of lattice delay differential equations arising from population genetics, population dynamics, and neural networks.