By Furui S., Sandhi M.M. (eds.)
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Extra resources for Advances in Speech Signal Processing
2, so that the particles will be considered to be in motion relative to the piston. Let the tube be 100 units long, so that AO = 100, and 10 units high, so that AB = 10. 35 and of unit mass m, enter the tube at AB. Each such column is determined as follows. ,9, is either filled by a particle or left vacant by a random process, like the toss of a coin. Once it has been determined that a particle P. is at such an initial location, then its initial velocity vi)0= where ε. Τ and v. n i,U is determined by (ΙΟίΗ-ε^, e ± > 2 ) , "±92 are selected at random, but are small in magnitude relative to 100, thus assuring that the gas has a relative high speed in a relatively uni form direction.
Be the distance between P. and P. at t. Then J k _^ χ>J > k χ F. , , the force exerted on P. 48) and ( 2 . 4 9 ) , by p-2 F. Ρ-ζ-2 , G φI (r? ^ij^i^k+l 0 N 1 = m. Y m. k+l q-1 ξ=0 q-1 ( ij,krij,k+lVrij,k+l i = 1 , 2 , . ,N. 57) REMARK Only for simple forces, like gravity, do the continuous and the arithmetic ap proaches yield exactly the same dynamical behavior. In general (, ), the 3 two approaches yield results which differ by terms of order (At) in both posi tion and velocity. Recently , new numerical formulas have been developed which increase this order of magnitude difference to any prescribed exponent, but, for these, the conservation of angular momentum of systems which have more than one particle is still to be proved.
The special structures to be used are described in detail in the sections which follow. 4 THE STEFAN PROBLEM Stefan problems are free boundary problems associated with the processes of melting of solids and crystallizing of liquids. The name is derived from one of the early studies of Arctic ice formation. Specifically, the problem is that of describing the changing shape of the boundary between the liquid portion and solid portion when either, or both, of the above processes are occurring. Nonconservative Models 49 The Stefan problem is of such difficulty that classical mathematical studies have been restricted largely to existence, uniqueness, and asymptotic behavior of a one dimensional model based on the linear heat equation (, , ).
Advances in Speech Signal Processing by Furui S., Sandhi M.M. (eds.)