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 (12).

Recently, there has been much attention devoted to investigate better and more efficient analytical techniques such as homotopy decomposition method [1],

auxiliary equation method [2], homotopy perturbation method [3, 4], Taylor collocation method [5], Sumudu transform method [6] 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 of

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 [11].