By R. Karp
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The plasma is considered to be one dimensional with an induced electric field as the only body force. 64) Since $ = $(x, t) it is clear that d$/dt = 0 since This relation will be important as we change variables from the old (x, t) system to the new ($, t) system. We think now that v = v($, t), x = x($, t) and E = E($, t) and remark that this transformation will be satisfactory if the Jacobian J [ ( x , t)/($,t)] = axla$ # 0. Recalling that 32 2. , b d t f t ) 5 = v ? 65) -A in (x, t ) system in ($, t ) system However, d+/dt = 0 so Eq.
Math. Phys. 3, 1028 (1962). 15. , Encycl. Math. Wiss. 4, 349 (1910). 16. Nowinski, J. , Nonlinear vibrations of elastic circular plates exhibiting rectilinear orthotropy, 2. Angew. Math. Phys. 14, 112 (1963). 17. Nowinski, J. , “Nonlinear Transverse Vibrations of a Spinning Disk,” ASME Paper No. 63-APMW-15 (1963). 18. ” McGraw-Hill, New York, 1953. 19. , Methods of Mathematical Physics,” Vol. 1. Wiley (Interscience), New York, 1953. 20. Ferron, J. , Approximations for mass transfer with chemical reaction, A.
2) dC and when the right-hand side of Eq. 2) is compared with Eq. 1) it is clear that the proper choice of # is such that 1 d* -- K ( C ) dC or in integral form where C, is an arbitrary reference. 1) then reduces to the linear Laplace’s equation in # v2* = 0. 5) 22 2. TRANSFORMATION AND GENERAL SOLUTIONS We must be concerned also with the boundary conditions and what effect this transformation has upon them. 6) R,[C] - f then the transfer to ah, of these boundary conditions is usually easily carried out.