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The neutron transport equation is a balance statement that conserves neutrons. Each term represents a gain or a loss of a neutron, and the balance, in essence, claims that neutrons gained equals neutrons lost. It is formulated as follows: [1]
Theory, computation and applications, Lecture Notes in Computational Science and Engineering, 11. Springer-Verlag, Berlin, 2000. P. Lesaint, and P. A. Raviart. "On a finite element method for solving the neutron transport equation." Mathematical aspects of finite elements in partial differential equations 33 (1974): 89–123.
The intensity field can in principle be solved from the integrodifferential radiative transfer equation (RTE), but an exact solution is usually impossible and even in the case of geometrically simple systems can contain unusual special functions such as the Chandrasekhar's H-function and Chandrasekhar's X- and Y-functions. [3]
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity , but it can be generalized to apply to any extensive quantity .
This involves computing exact or approximate solutions of the transport equation, and there are various forms of the transport equation that have been studied. Common varieties include steady-state vs time-dependent, scalar vs vector (the latter including polarization), and monoenergetic vs multi-energy (multi-group).
Because the Boltzmann equation is practical in solving problems in rarefied or dilute gases, it has been used in many diverse areas of technology. It is used to calculate Space Shuttle re-entry in the upper atmosphere. [42] It is the basis for Neutron transport theory, and ion transport in Semiconductors. [43] [44]
The Boltzmann equation can be used to determine how physical quantities change, such as heat energy and momentum, when a fluid is in transport. One may also derive other properties characteristic to fluids such as viscosity , thermal conductivity , and electrical conductivity (by treating the charge carriers in a material as a gas). [ 2 ]
He authored over a hundred papers and several books, most noticeably "Numerical methods in the theory of neutron transport" jointly with Gury Marchuk and "Functional Analysis in Computational Mathematics," based on his lectures. He graduated over 15 Ph.D.'s.