Methods of Integral Equations One-velocity Model

A second general approach to the problem of determining the angular distribution of the neutron flux may be developed by utilizing the methods of integral equations. These methods are generally quite powerful, and formal solutions are readily derived even for nonhomogeneous systems with complex source distributions. Unfortunately, the mathematical techniques required for these applications are of a somewhat sophisticated 

In the presentation which follows, we derive the formal solution to the one-velocity transport equation (7.15) for several media of practical importance. The general treatment follows that of Case, de Hoffmann, and Placzek. The reader is referred to the work of these authors for a much more complete development of the general methods and results. 

It is convenient for the purpose of integration to write the leakage term Q - as a partial derivative along the direction Q. Thus, if is a variable which increases along the direction Q, then 

The solution (7.180) states that the total directed flux 0(r,O) is the sum total of contributions from all sources which lie on the segment of a line through r parallel to Ck for which 0 ft 00. Clearly, sources off this line cannot contribute to the flux at r (in direction U), nor can sources on the line at stations ft 0. 

The integral equation (7.180) is the solution to the differential equation (7.175), and this general result will be very useful in the work which follows. As will be shown shortly, the neutron-balance equations for other, more complex systems may be reduced to the form (7.175) consequently, their solutions can be written down immediately from (7.180). Before proceeding with these calculations, however, we will develop an alternate expression for the solution (7.180) which is obtained by transforming the line integral to a volume integral. This form will be especially convenient for describing the more complex systems. 

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