scholarly journals Finite-time boundary stabilization of fractional reaction-diffusion systems

2022 ◽  
Author(s):  
Run-Jie Zhang ◽  
Liming Wang ◽  
Kai-Ning Wu
Keyword(s):  
Finite Time ◽  
Mixed Boundary ◽  
Sufficient Conditions ◽  
Reaction Diffusion ◽  
Mixed Boundary Conditions ◽  
Diffusion Term ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Finite Time Stabilization ◽  
Fractional Reaction

This paper investigates the boundary finite-time stabilization of fractional reaction-diffusion systems (FRDSs). First, a distributed controller is designed, and sufficient conditions are obtained to ensure the finite-time stability (FTS) of FRDSs under the designed controller. Then, a boundary controller is presented to achieve the FTS. By virtue of Lyapunov functional method and inequality techniques, sufficient conditions are presented to ensure the FTS of FRDSs via the designed boundary controller. The effect of diffusion term of FRDSs on the FTS is also investigated. Both Neumann and mixed boundary conditions are considered. Moreover, the robust finite-time stabilization of uncertain FRDSs is studied when there are uncertainties in the system’s coefficients. Under the designed boundary controller, sufficient conditions are presented to guarantee the robust FTS of uncertain FRDSs. Finally, numerical examples are presented to verify the effectiveness of our theoretical results.

scholarly journals Isolating Patterns in Open Reaction–Diffusion Systems

2021 ◽  
Vol 83 (7) ◽  
Author(s):  
Andrew L. Krause ◽  
Václav Klika ◽  
Philip K. Maini ◽  
Denis Headon ◽  
Eamonn A. Gaffney
Keyword(s):  
Boundary Conditions ◽  
Initial Conditions ◽  
Mixed Boundary ◽  
Cell Signalling ◽  
Reaction Diffusion ◽  
Mixed Boundary Conditions ◽  
Three Dimensions ◽  
Spatiotemporal Pattern ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems

AbstractRealistic examples of reaction–diffusion phenomena governing spatial and spatiotemporal pattern formation are rarely isolated systems, either chemically or thermodynamically. However, even formulations of ‘open’ reaction–diffusion systems often neglect the role of domain boundaries. Most idealizations of closed reaction–diffusion systems employ no-flux boundary conditions, and often patterns will form up to, or along, these boundaries. Motivated by boundaries of patterning fields related to the emergence of spatial form in embryonic development, we propose a set of mixed boundary conditions for a two-species reaction–diffusion system which forms inhomogeneous solutions away from the boundary of the domain for a variety of different reaction kinetics, with a prescribed uniform state near the boundary. We show that these boundary conditions can be derived from a larger heterogeneous field, indicating that these conditions can arise naturally if cell signalling or other properties of the medium vary in space. We explain the basic mechanisms behind this pattern localization and demonstrate that it can capture a large range of localized patterning in one, two, and three dimensions and that this framework can be applied to systems involving more than two species. Furthermore, the boundary conditions proposed lead to more symmetrical patterns on the interior of the domain and plausibly capture more realistic boundaries in developmental systems. Finally, we show that these isolated patterns are more robust to fluctuations in initial conditions and that they allow intriguing possibilities of pattern selection via geometry, distinct from known selection mechanisms.

Influence of mixed boundary conditions in some reaction–diffusion systems

1997 ◽  
Vol 127 (5) ◽  
pp. 1053-1066 ◽  
Author(s):  
Robert H. Martin ◽  
Michel Pierre
Keyword(s):  
Global Existence ◽  
Polynomial Growth ◽  
Existence Of Solutions ◽  
Mixed Boundary ◽  
Reaction Diffusion ◽  
Mixed Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Systems ◽  
Global Existence Of Solutions ◽  
Diffusion Systems

SynopsisWe analyse global existence of solutions to a system of two reaction–diffusion equations for whicha ‘balance’ law holds. The main aim is to make clear the influence of different combinations ofboundary conditions on global existence under the assumption that the nonlinearities satisfy polynomial growth estimates.

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