From these experimental results, an analytical model based on classical Westergaard well known equations has been developed to calculate the minimum thickness of RRFC for rigid pavements subjected to high density traffic, in order to obtain a durability of these rigid pavements of 106 cycles of 13 tons (127 kN) axle load.
Applying the Galerkin׳s method, the equation is converted into well known Duffing equation.
The Haasnoot-Altona equation (3) [12] and the Diez-Donders equation (4) [14] 14], both based on the well known Karplus equation (2), describe the relation between a 3J HH coupling and the corresponding exocyclic torsion angle (θ exo ) to a high level of accuracy.
It is well known that equation (1.2) shares many properties with the classical heat equation, u t = △ u, such as the bounded stationary solutions and the maximum principle [6].
It is well known that Equation 5 specifies a complete orthogonal system, and the integral transformation with a kernel as given by Equation 5 is the usual Fourier (inverse Fourier) transformation.
However, it is well known that equation (3.38) is oscillatory if and only if (q_{0}>9/16).
The well known Wilhelmy equation is used to explain the experimental results.
The model is based on single phase (bitumen) flow and uses the well known two equation turbulence model.
A nonlinear quadratic optimization is performed using the well known Hamilton Jacobi Bellman equation.
In addition, as is well known, the equation − ( a + b ∫ 0 l u x 2 d x ) u x x = f ( x, u ), 0 < x < l, (1.6).
It is well known that equation (1.2) has also a bi-Hamiltonian structure [3, 4] and is completely integrable (see [5, 6] and the in-depth discussion in [7, 8]).
