e-mail: vatsala@usl.edu
The important idea of the method of quasilinearization
developed by Bellman, and Bellman and Kalaba is to provide an explicit analytic
representation for the solution of nonlinear differential equations. These
representation yields pointwise lower estimates for the solution of the
nonlinear problem, whenever the function involved is convex. The most important
application of this method has been to obtain either a sequence of lower or
upper bounds which are solutions of linear differential equations that converge
quadratically to the unique solution of the given nonlinear problem.
It
is also known that the method of lower and upper solutions together with
monotone iterative technique offers monotone sequences which converge to the
extremal solutions of the nonlinear problem. However, when we employ the
technique of lower and upper solutions coupled with the method of
quasilinearization and utilize the idea of Newton-Fourier, it is possible to
construct concurrently lower and upper bounding sequences. These sequences are
solutions of the corresponding linear problem and they converge quadratically
to the unique solution of the nonlinear problem. This unification provides a
mechanism to enlarge the class of nonlinear problem to which the method is
applicable. These idea have been refined, generalized and extended to various
type of nonlinear problems, and this technique is now referred to as generalized
quasilinearization method.
In this paper, we extend the method of generalized quasilinearization to scalar reaction diffusion equations with periodic conditions. We develop two sequences which are solutions of linear reaction diffusion equations which converge uniformly and monotonically to the unique solution of the nonlinear reaction diffusion equations, satisfying periodic conditions. Further, we prove the rate of convergence of the sequences is quadratic.