These include Lagergren first-order equation and pseudo- second-order equation
, Elovich equation, Bangham's equation and intra-particle diffusion equation.
In order to obtain solitary wave amplitude equation, we continue solving second-order equation
For each type of surface, the function in (1) turns out to be specified by a nonlinear second-order equation
. It will be shown finally that Lax pairs can subsequently be formulated for each of the two second-order systems.
For the second-order equation
, the half-release times range was between 5.64 * [10.sup.2] min and 9.96 * [10.sup.2] min.
But no one has discussed the general second-order equation
In addition, these two symbolic solutions are not required to be known explicitly but are onlyassumed to satisfythe second-order equation
. Infact, we have ultimately expressed the first integrals found solely in terms of q, y and their derivatives.
These kinetic models included the pseudo first-order equation, the pseudo second-order equation
and the Elovich equation.
We first introduce an auxiliary variable [gamma] = -[nabla] x (a(t)[nabla]u) to reduce the fourth-order parabolic equation to a coupled system of second-order equations
, then solve a second-order equation
by FE method.
Figure 3 shows conversion rate predicted basing on the second-order equation
and variable order calculated by Eq.
The sorption process could be best described by the second-order equation
can be transformed into a second-order equation
However, in our previous research  we found that the boundary value problems with the second-order equation
cannot tackle the beginning stage of pipeline lifting and the ending stage of pipeline lowing accurately.