Solving Nonlinear Differential Equations

Mathematica is great in solving analytically linear differential equations. It is also a good companion for computing numerical solutions to non–linear equations. We attack the reduced–gravity, shallow–water equation (RSE) problem. We compare the analytical solution to our problem without friction to the numerical solution obtained either with Mathematica or via Matlab. We exploit Mathematica ability in solving systems of non-linear Ordinary Differential Equations, on the way to identify some analytical solution to RSE when friction is non-negligible

Assessment of radiative heating errors in Tropical Atmosphere Ocean array marine air temperature measurements

We assess the radiative heating error affecting marine air temperature (MAT) measurements in the Tropical Atmosphere Ocean array. The error in historical observations is found to be ubiquitous across the array, spatially variable and approximately stationary in time. The error induces spurious warming during daytime hours, but does not affect night-time temperatures. The range encompassing the real, unknown daily- and monthly-mean values is determined using daytime and night-time mean temperatures as upper and lower limits. The uncertainty in MAT is less than or equal to 0.5 °C and 0.2 °C for 95% of daily and monthly estimates, respectively. Uncertainties impact surface turbulent heat flux estimates, with potentially significant influences on the quantification of coupled ocean-atmosphere processes

Simplified and accurate stiffness of a prismatic anisotropic thin-walled box.

Background: Beam models have been proven effective in the preliminary analysis and design of aerospace structures. Accurate cross sectional stiffness constants are however needed, especially when dealing with bending, torsion and bend-twist coupling deformations. Several models have been proposed in the literature, even recently, but a lack of precision may be found when dealing with a high level of anisotropy and different lay-ups. Objective: A simplified analytical model is proposed to evaluate bending and torsional stiffness of a prismatic, anisotropic, thin-walled box. The proposed model is an extension of the model proposed by Lemanski and Weaver for the evaluation of the bend-twist coupling constant. Methods: Bending and torsional stiffness are derived analytically by using physical reasoning and by applying bending and torsional stiffness mathematic definition. Unitary deformations have been applied when evaluation forces and moments arising on the cross section. Results: Good accuracy has been obtained for structures with different geometries and lay-ups. The model has been validated with respect to finite element analysis. Numerical results are commented upon and compared with other models presented in literature. Conclusion: For cross sections with a high level of anisotropy, the accuracy of the proposed formulation is within 2% for bending stiffness and 6% for torsional stiffness. The percentage of error is further reduced for more realistic geometries and lay-ups. The proposed formulation gives accurate results for different dimensions and length rations of horizontal and vertical walls.N/