By Walter E Thirring
Mathematical Physics, Nat. Sciences, Physics, arithmetic
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Extra resources for A course in mathematical physics. Classical dynamical systems
Since u varies from 0 to h in the rarefaction wave, decay of the height of the top hat is given by max u = he−λt . 18) where we have inserted the 1C xF = 0 at t = 0. 19) which, on integration and use of 1C xs = X at t = 0, gives xs = X + hα (1 − e−λαt ). 20) gives the shock trajectory. 21) yielding t0 = − 1 (α + 1)λX ln 1 − . 22) Thus, the rarefaction wave catches up with the shock only if 1− (α + 1)λX > 0, hα that is, 1 h > [(α + 1)λX] α . 22). Suppose a characteristic in the rarefaction has a value u = c at time t.
17) is u = F (x, t, f (η)). 13). In the case of two independent variables, the resulting equation will be an ODE in f (η). Each solution of the ODE yields a similarity solution for the PDE. 14). In the following calculations, x, t, and u are regarded as independent variables. Thus, ∂X ∂X = Xx + Xu ux , = Xt + Xu ut . 14) we have ∂x ∂x ¯ = = ∂ [¯ x − X(x, t, u) + O( 2 )] ∂x ¯ ∂x 1 − (Xx + Xu ux ) + O( 2 ) ∂x ¯ or ∂x = 1 − (Xx + Xu ux ) + O( 2 ). 24) Similarly, we find that ∂x = − (Xt + Xu ut ) + O( 2 ) ∂ t¯ ∂t = − (Tx + Tu ux ) + O( 2 ) ∂x ¯ ∂t = 1 − (Tt + Tu ut ) + O( 2 ).
2) where h, α and β are positive constants; λ > 0 is the dissipative constant. If α = 0, the solution is a decaying travelling wave moving to the right with speed 1. 3) dx du = 0 along the characteristic curves = uα . 4) and so dx = uα which are straight dt lines in the (x, t) plane. The initial condition u = 0 for x < 0 and u = h for x > 0 give rise to a rarefaction wave centred at x = 0. All values from 0 to h propagate along the characteristics of the rarefaction wave. Since α > 0, dx the characteristic with value h for u has the highest speed: = hα .
A course in mathematical physics. Classical dynamical systems by Walter E Thirring