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Additional resources for Nonlinear Partial Differential Equations. A Symposium on Methods of Solution
1954, On the Timoshenko theory of transverse beam vibrations, Quart. Appi. Math. 12, 175-187. DONNELLY, R. , 1958, Experiments on the stability of viscous flow between rotating cylinders. I. Torque measurements, Proc. Roy. Soc. London, Ser. A 246, 312-325. DONNELLY, R. , and SCHWARZ, K. , 1965, Experiments on the stability of vis cous flow between rotating cylinders. VI. Finite amplitude experiments, Proc. Roy. Soc. London, Ser. A 283, 531-556. 42 R. C. " Springer-Verlag, Berlin. " Wiley, New York.
43), are ace V oV 0 lo 1 + <5x + - ^ 1 + ôx + D*Ul0D*U20 ^20^10 + - / ) [ L / 1 0 D * 2 L / 2 0 2 + £(2L/20D*L/10 - - 2U20D*2UH L/10D*C/20)] + y [ - 4 D * ( l / 2 o l / i o ) + Ul0D*U20 - 2l/ 2 0 D*t/ 1 0 ], and -aL/ 1 0 D*K 0 0 - -[2L/ 2 0 D*F 1 0 + c/10D*K20 + Vl0D*U20 + 2F 2 0 D*I/ 1 0 ]. " Oxford Univ. Press, London and New York. CODDINGTON, E. A. " McGraw-Hill, New York. , 1965, Transition in circular Couette flow, / . Fluid Mech. 21, 385-425. DAVEY, A, 1962, The growth of Taylor vortices in flow between rotating cylinders, J.
The solution of Eq. (11) with the nonlinear terms neglected which satisfies the boundary conditions (14) and the initial conditions (15) is given by ν0(χ,τ)= f ("οΊ/οΧ^Κ^. (21) m=0 Since all of the a0m are positive, v0(x, τ) -> 0 as τ -+ oo irrespective of the initial conditions. When we consider the nonlinear problem for T > Tc, we will find this is no longer true. Next consider the eigenvalue problem (18). First note that the space of two dimensional vectors qn(x) with first and second components four times and two times continuously differentiable on — \ < x < \, respec tively, and satisfying the boundary conditions (18) form a pre-Hilbert space with inner product (q„ | O = H ' (1 + δχ) (/J/ + gninf) dx.