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Absolute and Convective Instability for Evolution PDEs on the Half‐Line
Author(s) -
Fokas A. S.,
Papageorgiou D. T.
Publication year - 2005
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/j.0022-2526.2005.01541.x
Subject(s) - instability , mathematics , mathematical analysis , boundary value problem , convection , initial value problem , plane (geometry) , invariant (physics) , physics , transformation (genetics) , mathematical physics , geometry , mechanics , biochemistry , chemistry , gene
We analyze evolution PDEs exhibiting absolute (temporal) as well as convective (spatial) instability. Let ω( k ) be the associated symbol, i.e., let exp[ ikx −ω( k ) t ] be a solution of the PDE. We first study the problem on the infinite line with an arbitrary initial condition q 0 ( x ) , where q 0 ( x ) decays as | x | → ∞ . By making use of a certain transformation in the complex k ‐plane, which leaves ω( k ) invariant, we show that this problem can be analyzed in an elementary manner. We then study the problem on the half‐line, a problem physically more realistic but mathematically more difficult. By making use of the above transformation, as well as by employing a general method recently introduced for the solution of initial‐boundary value problems, we show that this problem can also be analyzed in a straightforward manner. The analysis is presented for a general PDE and is illustrated for two physically significant evolution PDEs with spatial derivatives up to second order and up to fourth order, respectively. The second‐order equation is a linearized Ginzburg–Landau equation arising in Rayleigh–Bénard convection and in the stability of plane Poiseuille flow, while the fourth‐order equation is a linearized Kuramoto–Sivashinsky equation, which includes dispersion and which models among other applications, interfacial phenomena in multifluid flows.