Exact separable solutions of delay reaction–diffusion equations and other nonlinear partial functional-differential equations

2014 ◽  
Vol 19 (3) ◽  
pp. 409-416 ◽  
Author(s):  
Andrei D. Polyanin ◽  
Alexei I. Zhurov
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Partial Functional Differential Equations ◽  
Functional Differential

TRANSITION WAVES THAT LEAVE BEHIND REGULAR OR IRREGULAR SPATIOTEMPORAL OSCILLATIONS IN A SYSTEM OF THREE REACTION–DIFFUSION EQUATIONS

1993 ◽  
Vol 03 (05) ◽  
pp. 1269-1279 ◽  
Author(s):  
JONATHAN A. SHERRATT
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Wave Front ◽  
Plane Waves ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Spatiotemporal Oscillations ◽  
Transition Wave ◽  
Periodic Plane

Transition waves are widespread in the biological and chemical sciences, and have often been successfully modelled using reaction–diffusion systems. I consider a particular system of three reaction–diffusion equations, and I show that transition waves can destabilise as the kinetic ordinary differential equations pass through a Hopf bifurcation, giving rise to either regular or irregular spatiotemporal oscillations behind the advancing transition wave front. In the case of regular oscillations, I show that these are periodic plane waves that are induced by the way in which the transition wave front approaches its terminal steady state. Further, I show that irregular oscillations arise when these periodic plane waves are unstable as reaction–diffusion solutions. The resulting behavior is not related to any chaos in the kinetic ordinary differential equations.

scholarly journals Oscillation for Certain Nonlinear Neutral Partial Differential Equations

2010 ◽  
Vol 2010 ◽  
pp. 1-12
Author(s):  
Quanwen Lin ◽  
Rongkun Zhuang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Smooth Boundary ◽  
Functional Differential Equations ◽  
Second Order ◽  
Piecewise Smooth Boundary ◽  
Linear Ordinary Differential Equations ◽  
Partial Functional Differential Equations ◽  
Functional Differential

We present some new oscillation criteria for second-order neutral partial functional differential equations of the form(∂/∂t){p(t)(∂/∂t)[u(x,t)+∑i=1lλi(t)u(x,t-τi)]}=a(t)Δu(x,t)+∑k=1sak(t)Δu(x,t-ρk(t))-q(x,t)f(u(x,t))-∑j=1mqj(x,t)fj(u(x,t-σj)),(x,t)∈Ω×R+≡G, whereΩis a bounded domain in the EuclideanN-spaceRNwith a piecewise smooth boundary∂ΩandΔis the Laplacian inRN. Our results improve some known results and show that the oscillation of some second-order linear ordinary differential equations implies the oscillation of relevant nonlinear neutral partial functional differential equations.

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