Improved generalized quasilinearization method and rapid convergence for reaction diffusion equations

Abstract

In the method of quasilinearization or the method of generalized quasilinearization we assume that ∂ k f ( t , x , u ) ∂ u k exists, where f ( t , x , u ) is the forcing function and obtain kth order of convergence. In this paper, we assume a weaker condition namely, ∂ k - 1 f ( t , x , u ) ∂ u k - 1 exists and Lipschitzian in u and develop generalized quasilinearization method to reaction diffusion equations. The iterates will be different depending on ∂ k - 1 f ( t , x , u ) ∂ u k - 1 is nondecreasing or nonincreasing in u and k being even or odd. Finally, we prove that the sequences generated by the generalized quasilinearization method converge to the unique solution of the nonlinear reaction diffusion equation and the convergence is of order k.