Analytical calculation of collision integrals

 

• from A. Salat, Plasma Physics, Vol. 17, p. 589-607, (1975).

• common form of the integrals

  
  
  

• transformation of variables, new form

  
  
  

• sum of 288 terms up to the 9-th degree in , one term of the 9-th degree in gives after substitutions 19 683 terms, in one part of one collision integral is 259 584 terms (one term is a product appearing in a sum)

• integration over derived from

  
  
  

• integration over ; rotation and transformation into spherical coordinates

  
  
  

• integration over , products , elimination of

  
  
  

• integration over
  
  

• integration over

  
  
  

• final result contains only elementary functions and the error function

• algebraic program in REDUCE was unable to calculate all the collision integrals and so had to be rewritten into a more efficient symbolic program

• speed up of the symbolic program compared to the algebraic program depends on the size of a collision integral

  Size small medium large

  Speed up 3x 7x ?