Abstract Functional Differential Equations and Reaction-Diffusion Systems

1990 ◽  
Vol 321 (1) ◽  
pp. 1 ◽  
Author(s):  
R. H. Martin ◽  
H. L. Smith
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Functional Differential

scholarly journals Nonlinear Pantograph-Type Diffusion PDEs: Exact Solutions and the Principle of Analogy

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 511
Author(s):  
Andrei D. Polyanin ◽  
Vsevolod G. Sorokin
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Functional Differential Equations ◽  
Reaction Diffusion ◽  
Proportional Delay ◽  
Type Reaction ◽  
Self Similar ◽  
Functional Differential ◽  
Similar Solutions ◽  
Varying Delay

We study nonlinear pantograph-type reaction–diffusion PDEs, which, in addition to the unknown u=u(x,t), also contain the same functions with dilated or contracted arguments of the form w=u(px,t), w=u(x,qt), and w=u(px,qt), where p and q are the free scaling parameters (for equations with proportional delay we have 0<p<1, 0<q<1). A brief review of publications on pantograph-type ODEs and PDEs and their applications is given. Exact solutions of various types of such nonlinear partial functional differential equations are described for the first time. We present examples of nonlinear pantograph-type PDEs with proportional delay, which admit traveling-wave and self-similar solutions (note that PDEs with constant delay do not have self-similar solutions). Additive, multiplicative and functional separable solutions, as well as some other exact solutions are also obtained. Special attention is paid to nonlinear pantograph-type PDEs of a rather general form, which contain one or two arbitrary functions. In total, more than forty nonlinear pantograph-type reaction–diffusion PDEs with dilated or contracted arguments, admitting exact solutions, have been considered. Multi-pantograph nonlinear PDEs are also discussed. The principle of analogy is formulated, which makes it possible to efficiently construct exact solutions of nonlinear pantograph-type PDEs. A number of exact solutions of more complex nonlinear functional differential equations with varying delay, which arbitrarily depends on time or spatial coordinate, are also described. The presented equations and their exact solutions can be used to formulate test problems designed to evaluate the accuracy of numerical and approximate analytical methods for solving the corresponding nonlinear initial-boundary value problems for PDEs with varying delay. The principle of analogy allows finding solutions to other nonlinear pantograph-type PDEs (including nonlinear wave-type PDEs and higher-order equations).

DYNAMICS OF LATTICE DIFFERENTIAL EQUATIONS

1996 ◽  
Vol 06 (09) ◽  
pp. 1605-1621 ◽  
Author(s):  
SHUI-NEE CHOW ◽  
JOHN MALLET-PARET ◽  
ERIK S. VAN VLECK
Keyword(s):  
Differential Equations ◽  
Plane Wave ◽  
Traveling Wave ◽  
Reaction Diffusion ◽  
Induced Anisotropy ◽  
Lattice Differential Equations ◽  
Reaction Diffusion Systems ◽  
Wave Solutions ◽  
Spatial Chaos ◽  
Diffusion Systems

In this paper recent work on the dynamics of lattice differential equations is surveyed. In particular, results on propagation failure and lattice induced anisotropy for traveling wave or plane wave solutions in higher space dimensions spatially discrete bistable reaction–diffusion systems are considered. In addition, analysis of and spatial chaos in the equilibrium states of spatially discrete reaction–diffusion systems are discussed.

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