AN ABSTRACT APPROACH TO A PROBLEM IN NEUTRON TRANSPORT THEORY
A mixed-initial and boundary valued problem was formulated for the transport equation, in the case in which the material, D, through which the neutrons travel, varies with time. By restricting the deformation to take place along rays emanating from some fixed point, 0, of D, some of the results of Kato, in his work on the abstract evolution equation, can be applied to get the existence of a unique solution. Thus it is possible to exhibit the solution in the form of an evolution operator applied to the initial data. This would be of physical interest since, if overheating causes D to expand, then it is reasonable to assume that the expansion occurs in just this manner. Next, an energy inequality was derived for this problem, in the general case of arbitrary but smooth deformations of D. The equation is approximated by step functions in the variable, v. The integral term of the equation degenerates irto a sum, and an associated mixed-problem for a symmetric hyperbolic system of equations was obtained. The components of the solution for this system are the steps of a step function that approximates the desired solution. Using the energy inequality, the limit can be passed to an appropriate norm. Thus, the sense, in which the transport equation may be regarded as a limiting case of hyperbolic equations, shows up nlcely. In particular, there exists a notion of characteristics for the transport equation, being the limit of the characteristics of the approximating hyperbolic equations. Finally these results would be relevant in cases of explosions, or of extending the critical size in such a way that catastrophic results do not occur. (Dissertation Abstr., 24: No. 9, March
- Research Organization:
- Originating Research Org. not identified
- NSA Number:
- NSA-18-021154
- OSTI ID:
- 4005362
- Resource Relation:
- Other Information: Thesis. Orig. Receipt Date: 31-DEC-64
- Country of Publication:
- Country unknown/Code not available
- Language:
- English
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