The construction of cellular automaton analogues for differential equations
Research members： Dr. Mikio Murata
Research fields： Mathematics
Departments： Institute of Engineering
Keywords： Cellular automaton, Differential equation, Reaction-diffusion equation
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Summary
A systematic procedure to the construction of cellular automaton analogues for differential equations was given.
This method is tailored to first-order differential equations and reaction-diffusion equations.
This discretizing scheme was applied to the Allen--Cahn equation. Traveling wave solutions and entire solutions of the resulting cellular automaton were constructed.
A cellular automaton analogue of Gray-Scott model which gives various spatial patterns with appropriate initial data and parameters were proposed. A (2+1)D Gray-Scott cellular automaton that gives a ring pattern and a self-replication pattern were also constructed.
Reference articles and patents
Mikio Murata, Multidimensional traveling waves in the Allen--Cahn cellular automaton, Journal of Physics A: Mathematical and Theoretical, Vol.48 No.25, 2015, pp. 255202
Keisuke Matsuya and Mikio Murata, Spatial pattern of discrete and ultradiscrete Gray-Scott model, Discrete and Continuous Dynamical Systems-Series B, Vol.20, No.1, 2015, pp. 173-187
Mikio Murata, Discretization and ultradiscretization of non-integrable systems, RIMS Kôkyûroku Bessatsu, Vol.B41, 2013, pp. 85-99
Mikio Murata, Tropical discretization: ultradiscrete Fisher-KPP equation and ultradiscrete Allen-Cahn equation、Journal of Difference Equations and Applications, Vol.19, No.6, 2013, pp. 1008-1021
Contact
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