Approximate analytical solutions to a class of non-linear equations with complex functions

1992 ◽  
Vol 157 (2) ◽  
pp. 289-302 ◽  
Author(s):  
L. Cveticanin
Keyword(s):  
Analytical Solutions ◽  
Linear Equations ◽  
Approximate Analytical Solutions ◽  
Complex Functions ◽  
Non Linear ◽  
Non Linear Equations

scholarly journals A simplified treatment of surfactant effects on cloud drop activation

2011 ◽  
Vol 4 (1) ◽  
pp. 107-116 ◽  
Author(s):  
T. Raatikainen ◽  
A. Laaksonen
Keyword(s):  
Surface Tension ◽  
Analytical Solutions ◽  
Linear Equations ◽  
Computing Time ◽  
Cloud Droplets ◽  
Cloud Models ◽  
Non Linear ◽  
Solution Surface ◽  
Surfactant Effects ◽  
Non Linear Equations

Abstract. Dissolved surface active species, or surfactants, have a tendency to partition to solution surface and thereby decrease solution surface tension. Activating cloud droplets have large surface-to-volume ratios, and the amount of surfactant molecules in them is limited. Therefore, unlike with macroscopic solutions, partitioning to the surface can effectively deplete the droplet interior of surfactant molecules. Surfactant partitioning equilibrium for activating cloud droplets has so far been solved numerically from a group of non-linear equations containing the Gibbs adsorption equation coupled with a surface tension model and an optional activity coefficient model. This can be a problem when surfactant effects are examined by using large-scale cloud models. Namely, computing time increases significantly due to the partitioning calculations done in the lowest levels of nested iterations. Our purpose is to reduce the group of non-linear equations to simple polynomial equations with well known analytical solutions. In order to do that, we describe surface tension lowering using the Szyskowski equation, and ignore all droplet solution non-idealities. It is assumed that there is only one surfactant exhibiting bulk-surface partitioning, but the number of non-surfactant solutes is unlimited. It is shown that the simplifications cause only minor errors to predicted bulk solution concentrations and cloud droplet activation. In addition, computing time is decreased at least by an order of magnitude when using the analytical solutions.

scholarly journals A master exceptional field theory

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Guillaume Bossard ◽  
Axel Kleinschmidt ◽  
Ergin Sezgin
Keyword(s):  
Field Theory ◽  
Gauge Invariance ◽  
Sigma Model ◽  
Linear Equations ◽  
Field Theories ◽  
Gauge Invariant ◽  
Non Linear ◽  
Non Linear Equations

Abstract We construct a pseudo-Lagrangian that is invariant under rigid E11 and transforms as a density under E11 generalised diffeomorphisms. The gauge-invariance requires the use of a section condition studied in previous work on E11 exceptional field theory and the inclusion of constrained fields that transform in an indecomposable E11-representation together with the E11 coset fields. We show that, in combination with gauge-invariant and E11-invariant duality equations, this pseudo-Lagrangian reduces to the bosonic sector of non-linear eleven-dimensional supergravity for one choice of solution to the section condi- tion. For another choice, we reobtain the E8 exceptional field theory and conjecture that our pseudo-Lagrangian and duality equations produce all exceptional field theories with maximal supersymmetry in any dimension. We also describe how the theory entails non-linear equations for higher dual fields, including the dual graviton in eleven dimensions. Furthermore, we speculate on the relation to the E10 sigma model.

An elliptic boundary value problem for strongly non-linear equations in unbounded domains

1977 ◽  
Vol 77 (3-4) ◽  
pp. 217-230 ◽  
Author(s):  
Vesa Mustonen
Keyword(s):  
Boundary Value Problem ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Linear Equations ◽  
Unbounded Domains ◽  
Elliptic Boundary Value Problem ◽  
Elliptic Boundary Value ◽  
Non Linear ◽  
Linear Elliptic Boundary ◽  
Non Linear Equations

SynopsisThe existence of a variational solution is shown for the strongly non-linear elliptic boundary value problem in unbounded domains. The proof is a generalisation to Orlicz-Sobolev space setting of the idea introduced in [15] for the equations involving polynomial non-linearities only.

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