The Generalized Quasi-linearization Method for Reaction Diffusion Equations on an Unbounded Domain

Abstract

The method of generalized quasi-linearization has been well developed for ordinary differential equations. In this paper, we extend the method of generalized quasi-linearization to reaction diffusion equations on an unbounded domain. The iterates, which are solutions of linear equations starting from lower and upper solutions, converge uniformly and monotonically to the unique solution of the nonlinear reaction diffusion equation in an unbounded domain. Initially an existence theorem for the linear nonhomogeneous reaction diffusion equation in an unbounded domain has been proved under improved conditions. The quadratic convergence has been proved by using a comparison theorem of reaction diffusion equations with ordinary differential equations. This avoids the computational complexity of the quasi-linearization method, since the computation of Green's function at each stage of the iterates is avoided.