New exact solutions of parabolic equations with quadratic non-linearities

Generalized procedure of separation of variables and reduction of nonlinear wave equations

“…Ansatz (4) contains an unknown function f x t ( , ), m unknown functions a x i ( ) , and m unknown functions ω i t ( ) , which are determined from the condition that ansatz (4) reduces the equation considered to a system of m ordinary differential equations with unknown functions ω i t ( ) . We consider the problem of finding this system by using the following nonlinear wave equations as an example:…”

“…In [3,4], the construction of solutions in the form of this finite sum was used for the analysis of various classes of nonlinear equations.…”

Generalized separation of variables and exact solutions of nonlinear equations

“…Exact solutions with generalized separation of variables that contain more than two terms were given in [2,3].…”

On the characteristics and solutions of the one-dimensional non-stationary seepage equation

“…Choosing the time t = t(x,p) as the dependent variable, we write Eq. (1.2) in the form 2 t2tpl~l 2 2 tp-+ p(t~xt p -2txtxpt p + txtpp ) = 0 (1.4) Ifx is chosen as the dependent variable, then, for the function x = x(p, t) we obtain the equation 2 xtxp + xp/~l -pxpp = 0 (1.5) Equations (1.2), (1.4) and (1.5) have a singularity accompanying the higher derivatives and are characterized by a finite rate of propagation of perturbations [2].Exact solutions, having a constant arbitrariness, have been obtained for such equations by different methods in a number of papers (see [3][4][5][6][7], for example) but the question remains as to why the exact solutions of Eqs (1.2), (1.4) and (1.5) only have a constant arbitrariness.A study of the characteristics of Eq. (1.1) enables us to answer this question.…”