Absolute stability of nonlinear dynamical systems described by second-order partial differential equations

1969 ◽  
Vol 12 (3) ◽  
pp. 253-263
Author(s):  
V. A. Brusin
Keyword(s):  
Partial Differential Equations ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Absolute Stability ◽  
Nonlinear Dynamical Systems ◽  
Second Order ◽  
Partial Differential ◽  
Nonlinear Dynamical

scholarly journals Resolving the Multitude of Microscale Interactions Accurately Models Stochastic Partial Differential Equations

2006 ◽  
Vol 9 ◽  
pp. 193-221 ◽  
Author(s):  
A. J. Roberts
Keyword(s):  
Finite Element ◽  
Partial Differential Equations ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Nonlinear Dynamical Systems ◽  
Numerical Models ◽  
Noise Sources ◽  
Partial Differential ◽  
Manifold Theory

AbstractConstructing numerical models of noisy partial differential equations is a very delicate task. Our long-term aim is to use modern dynamical systems theory to derive discretisations of dissipative stochastic partial differential equations. As a second step, we consider here a small domain, representing a finite element, and derive a one-degree-of-freedom model for the dynamics in the element; stochastic centre manifold theory supports the model. The approach automatically parametrises the microscale structures induced by spatially varying stochastic noise within the element. The crucial aspect of this work is that we explore how a multitude of microscale noise processes may interact in nonlinear dynamical systems. The analysis finds that noise processes with coarse structure across a finite element are the significant noises for the modelling. Further, the nonlinear dynamics abstracts effectively new noise sources over the macroscale time-scales resolved by the model.

Series expansions for monogenic functions in Clifford algebras and their application

2020 ◽  
Vol 17 (3) ◽  
pp. 365-371
Author(s):  
Anatoliy Pogorui ◽  
Tamila Kolomiiets
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Vector Space ◽  
Clifford Algebra ◽  
Clifford Algebras ◽  
Monogenic Function ◽  
Second Order ◽  
Series Expansions ◽  
Monogenic Functions ◽  
Partial Differential

This paper deals with studying some properties of a monogenic function defined on a vector space with values in the Clifford algebra generated by the space. We provide some expansions of a monogenic function and consider its application to study solutions of second-order partial differential equations.

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