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An Application of Semigroup Theory to a Fragmentation Equation

  • Pamela N. Blair
  • Wilson Lamb
  • Iain W. Stewart

Abstract

The process of fragmentation arises in many physical situations, including polymer degradation, droplet breakage and rock crushing and grinding. Under suitable assumptions, the evolution of the size distribution c(x, t), where x represents particle size and t is time, may be described by the linear integro-differential equation
$$ \frac{{\partial c(x,t)}} {{\partial t}} = - a(x)c(x,t) + \int_x^\infty {a(y)b(x,y)c(y,t)dy,x > 0,t > 0;} $$
(4.1)
see [1] for details. In (4.1), a(x) describes the rate at which a particle of size x fragments and b(x, y) represents the rate of production of particles of size x due to the break-up of particles of size y > x. For the total mass in the system to remain constant during fragmentation, b must satisfy the condition
$$ \int_0^y {xb(x,y)dx = y.} $$

Keywords

Semigroup Theory Fragmentation Equation Abstract Cauchy Problem Perturbation Theorem Exte Nsion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Pamela N. Blair
  • Wilson Lamb
  • Iain W. Stewart

There are no affiliations available

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