On the summability of the solutions of the inhomogeneous heat equation with a power-law nonlinearity and variable coefficients

Abstract

In this article, we investigate the summability of the formal power series solutions in time of the inhomogeneous heat equation with a power-law nonlinearity of degree two, and with variable coefficients. In particular, we give necessary and sufficient conditions for the 1-summability of the solutions in a given direction. These conditions generalize the ones given for the linear heat equation by W. Balser and M. Loday-Richaud in a 2009 article [5].

Gevrey regularity and summability of the formal power series solutions of the inhomogeneous generalized Boussinesq equations

“…For example, in the case of real variables, that is when the variables pt, xq belong to a subset of R 2 , we can quote, among the many existing techniques, the tanh-sech method, the F-expansion method, the exp-function method, the variational iteration method, etc. More recently, in the case of complex variables, that is when the variables pt, xq belong to a subset of C 2 , the summation theory has also been used succesfullly [19,25,26]. This theory, initially developed within the framework of the meromorphic ordinary differential equation with an irregular singular point (see for instance [10,16]), allows the construction of explicit solutions from formal solutions.…”

“…In this article, we propose to prove that these two results remain true in the case of the generalized Boussinesq equation (1.2). To do this, we shall use an approach similar to those already developed by the author in [17][18][19][23][24][25][26] for some nonlinear partial differential equations (see also [3,[20][21][22] for an approach in the linear case). Let us point out here that, as we shall see below, the terms u m pB x uq 2 of Eq.…”