Blow-up solutions of the generalized Boussinesq equation obtained by variational iteration method

Abstract

This paper deals with obtaining explicit solutions of a generalized non-linear Boussinesq equation using He's variational iteration method. Both finite and blow-up solutions can be obtained.

Soliton-type and other travelling wave solutions for an improved class of nonlinear sixth-order Boussinesq equations

“…(15) and (16), when they are written in terms of the new variables X and T [see (22)]. This is made to replace recursively u and its derivatives by derivatives of η in the equation obtained from (19) after applying the change of variables (22). Namely, we have the following relations:…”

“…(25) and (26) are used recursively in the terms of order O(μ 4 , ε 2 ) and O(μ 2 , ε), respectively, in the equation obtained from (19) after applying the change of variables (22). Thus, we deduce the following equation:…”

“…The arbitrary function of integration g(T ) in (28) appears when we use the chain derivative rule to express the term ∂ 2 η ∂t 2 of (19) in the new variables X and T [see (22)]. Finally, we assume that the arbitrary functions of integration f (T ), g(T ) and g (T ) are slowly time-varying functions in what follows.…”

“…We start by introducing the dimensionless and linearized Airy equations (see, e.g. [31]) written in terms of X and T [see (22)],…”

“…[5,9,11,12,[18][19][20][21]). Several mathematical techniques have been used in the literature to integrate analytically nonlinear evolution equations such as the mapping method [12], variational iteration method [22], Hirota's bilinear method [23], Adomian decomposition method [24][25][26], Fan's F-expansion method [27], G /G-method [28], tanh-method [28], exp-method [29], inverse scattering method [30], Kudryashov's method [28] and the Weierstrass elliptic function method [12] as well as the Lie symmetry approach [12].…”