Consequently, the zero solution of the auxiliary equation
(1.7) loses its stability; in particular, the existence of a stable cycle is possible.
The 2-D DGTD iterative formula and auxiliary equation
in elliptical cylindrical UPML are derived.
In this paper, we develop an auxiliary equation
method for solving fractional partial differential equations, where the fractional derivative is defined in the sense of the conformable fractional derivative.
Then, by using the adjoint equation (2.10b), the auxiliary equation
(3.2), and the dual problem (3.11), we obtain
Link (2) is the calculation of the volume fractions [Y.sub.1] and [Y.sub.2] according to the mass conservation equations (5) and the auxiliary equation
Recently, there has been much attention devoted to investigate better and more efficient analytical techniques such as homotopy decomposition method , auxiliary equation
method , homotopy perturbation method [3, 4], Taylor collocation method , Sumudu transform method  and Adomian decomposition method [7, 8].
Xu, "Extended auxiliary equation
method and its applications to three generalized NLS equations," Abstract and Applied Analysis, vol.
Integrating (8) once or several times with respect to [xi], setting the integration constant to zero if possible, multiplying both sides of the equation by u', and then integrating once again, we obtain different types of the auxiliary equation
[([phi]').sup.2] = [a.sub.0] + [a.sub.1][phi] + [a.sub.2][[phi].sup.2] + [a.sub.3][[phi].sup.3] + [a.sub.4][[phi].sup.4], where [a.sub.i] (i = 0, 1, ..., 4) are constant coefficients that can be determined in this step.
(6) is differential equation, with auxiliary equation
Now let us consider the following auxiliary equation
Substituting 14) in 11) to 13) and eliminating [??], [??], and [??], we obtain the following auxiliary equation
make use of the auxiliary equation
and the expanded mapping methods to find the new exact periodic solutions for (2 + 1)-dimensional dispersive longwave equations .