Exact solutions of nonlinear heat equation $u_t=(F(u)u_x)_x+H(u)$

Abstract

A method for construction of exact solutions to nonlinear heat equation $u_t=(F(u)u_x)_x+H(u)$ which is based on ansatz $p(x)=w_1(t)\varphi(u)$ is proposed. Here the function $p(x)$ is a solution to one of the equations $(p')^2 = Ap^2 + B$, $(p')^2 = Ap^4 + Bp^2 +C$, and the functions $w_1(t)$ and $\varphi(u)$ can be found from the condition that this ansatz reduces the equation to an ordinary differential equation with unknown function $w_1(t)$.