25 research outputs found
An Application of Spline and Piecewise Interpolation to Heat Transfer (Cubic Case)
An Application of Cubic spline and piecewise interpolation formula was applied to compute heat transfer across the thermocline depth of three lakes in the study area of Auchi in Edo State of Nigeria. Eight temperature values each for depths 1m to 8m were collected from the lakes. Graphs of these temperatures against the depths were plotted. Cubic spline interpolation equation was modelled. MAPLE 15 software was used to simulate the modelled equation using the values of temperatures and depths in order to obtain the unknown coefficients of the variables in the 21 new equations. Three optimal equations were found to represent the thermocline depth for the three lakes. These equations were used to obtain the thermocline gradients.....
Dynamics of a hyperbolic system that applies at the onset of the oscillatory instability
A real hyperbolic system is considered that applies near the onset of the oscillatory instability in large spatial domains. The validity of that system requires that some intermediate scales (large compared with the basic wavelength of the unstable modes but small compared with the size of the system) remain inhibited; that condition is analysed in some detail. The dynamics associated with the hyperbolic system is fully analysed to conclude that it is very simple if the coefficient of the cross-nonlinearity is such that , while the system exhibits increasing complexity (including period-doubling sequences, quasiperiodic transitions, crises) as the bifurcation parameter grows if ; if then the system behaves subcritically. Our results are seen to compare well, both qualitatively and quantitatively, with the experimentally obtained ones for the oscillatory instability of straight rolls in pure Rayleigh - Bénard convection
On the validity of mean-field amplitude equations for counterpropagating wavetrains
We rigorously establish the validity of the equations describing the
evolution of one-dimensional long wavelength modulations of counterpropagating
wavetrains for a hyperbolic model equation, namely the sine-Gordon equation. We
consider both periodic amplitude functions and localized wavepackets. For the
localized case, the wavetrains are completely decoupled at leading order, while
in the periodic case the amplitude equations take the form of mean-field
(nonlocal) Schr\"odinger equations rather than locally coupled partial
differential equations. The origin of this weakened coupling is traced to a
hidden translation symmetry in the linear problem, which is related to the
existence of a characteristic frame traveling at the group velocity of each
wavetrain. It is proved that solutions to the amplitude equations dominate the
dynamics of the governing equations on asymptotically long time scales. While
the details of the discussion are restricted to the class of model equations
having a leading cubic nonlinearity, the results strongly indicate that
mean-field evolution equations are generic for bimodal disturbances in
dispersive systems with \O(1) group velocity.Comment: 16 pages, uuencoded, tar-compressed Postscript fil
Finite size effects near the onset of the oscillatory instability
A system of two complex Ginzburg - Landau equations is considered that applies at the onset of the oscillatory instability in spatial domains whose size is large (but finite) in one direction; the dependent variables are the slowly modulated complex amplitudes of two counterpropagating wavetrains. In order to obtain a well posed problem, four boundary conditions must be imposed at the boundaries. Two of them were already known, and the other two are first derived in this paper. In the generic case when the group velocity is of order unity, the resulting problem has terms that are not of the same order of magnitude. This fact allows us to consider two distinguished limits and to derive two associated (simpler) sub-models, that are briefly discussed. Our results predict quite a rich variety of complex dynamics that is due to both the modulational instability and finite size effects
Initial Boundary-Value Problems for a Pair of Conservation Laws
We describe a multiple-scale technique for solving the initial boundary-value problem over the positive x-axis for a one-dimensional pair of hyperbolic conservation laws. This technique involves decomposing the solution into waves and incorporating slow temporal and stretched spatial scales in different parts of the solution domain. We apply these ideas to a wavemaker problem for shallow water flow and show why the presence of source terms in the conservation laws makes the analytic solution more complicated