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

A Laplace transform-multiple scale procedure for the asymptotic solution of weakly non-linear partial differential equations

In this paper we formulate a Laplace-Transform multi-scale expansion procedure to develop asymptotic solutions of weakly nonlinear partial differential equations. The method is applied to some general non-linear wave and diffusion equations. 1 Introduction A common boundary value problem associated with the wave equation is the signaling problem, in which the boundary conditions prescribed at the origin x = 0 propagate into the quiescent region x ? 0. For example, the linear telegraph equation u tt \Gamma u xx + bu t + cu = 0 which governs the one dimensional propagation of waves u(x; t) in electric transmission lines is a classic equation treated in standard texts on partial differential equations (for example, [1], [5], [6]). However, in many application areas such as water waves, elastic and acoustic waves, the governing hyperbolic systems are nonlinear (See [7]). In this paper we are concerned with the asymptotic solution, in regions away from the boundary, of signaling problems ..

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

Associations between obesogenic risk and depressive symptomatology in Australian adolescents: A cross-sectional study

Background: Depression and obesity are significant health concerns currently facing adolescents worldwide. This paper investigates the associations between obesity and related risk behaviours and depressive symptomatology in an Australian adolescent population. Methods: Data from the Australian Capital Territory It's Your Move project, an Australian community-based intervention project were used. In 2012, 800 students (440 females, 360 males) aged 11-14 years (M=13.11 years, SD=0.62 years), from 6 secondary schools were weighed and measured and completed a questionnaire which included physical activity, sedentary behaviour and dietary intake. Weight status was defined by WHO criteria. A cut-off score ≥10 on the Short Mood and Feelings Questionnaire indicated symptomatic depression. Logistic regression was used to test associations. Results: After controlling for potential confounders, results showed significantly higher odds of depressive symptomatology in males (OR=1.22, p<0.05) and females (OR=1.12, p<0.05) who exceeded guidelines for daily screen-time leisure sedentary activities. Higher odds of depressive symptoms were seen in females who consumed greater amounts of sweet drink (OR=1.18, p<0.05), compared to lower female consumers of sweet drinks, and males who were overweight/obese also had greater odds of depressive symptoms (OR=1.83, p<0.05) compared to male normal weight adolescents. Conclusions: This study demonstrates the associations between obesogenic risks and depression in adolescents. Further research should explore the direction of these associations and identify common determinants of obesity and depression. Mental health outcomes need to be included in the rationale and evaluation for diet and activity interventions

Associations between obesogenic risk and depressive symptomatology in Australian adolescents: a cross-sectional study.

BACKGROUND: Depression and obesity are significant health concerns currently facing adolescents worldwide. This paper investigates the associations between obesity and related risk behaviours and depressive symptomatology in an Australian adolescent population. METHODS: Data from the Australian Capital Territory It's Your Move project, an Australian community-based intervention project were used. In 2012, 800 students (440 females, 360 males) aged 11-14 years (M=13.11 years, SD=0.62 years), from 6 secondary schools were weighed and measured and completed a questionnaire which included physical activity, sedentary behaviour and dietary intake. Weight status was defined by WHO criteria. A cut-off score ≥10 on the Short Mood and Feelings Questionnaire indicated symptomatic depression. Logistic regression was used to test associations. RESULTS: After controlling for potential confounders, results showed significantly higher odds of depressive symptomatology in males (OR=1.22, p<0.05) and females (OR=1.12, p<0.05) who exceeded guidelines for daily screen-time leisure sedentary activities. Higher odds of depressive symptoms were seen in females who consumed greater amounts of sweet drink (OR=1.18, p<0.05), compared to lower female consumers of sweet drinks, and males who were overweight/obese also had greater odds of depressive symptoms (OR=1.83, p<0.05) compared to male normal weight adolescents. CONCLUSIONS: This study demonstrates the associations between obesogenic risks and depression in adolescents. Further research should explore the direction of these associations and identify common determinants of obesity and depression. Mental health outcomes need to be included in the rationale and evaluation for diet and activity interventions