On closed-form solutions of some nonlinear partial differential equations (Q1975935)

Summary: This paper is devoted to closed-form solutions of the partial differential equation: \(\theta_{xx} + \theta_{yy} + \delta\exp(\theta)= 0\), which arises in the steady state thermal explosion theory. We find simple exact solutions of the form \(\theta(x,y) = \Phi(F(x) +G(y))\), and \(\theta(x,y) = \Phi(f(x +y)+g(x -y))\). Also, we study the corresponding nonlinear wave equation.