Some Aspects of Quasilinearization

Abstract : Many of the fundamental nonlinear differential equations of mathematical physics and engineering can be written in the form L(u) = max (M(u,x,q) + a(x,q)), where L and M are linear differential operators on u, a scalar function of the vector x, and q is a decision variable. Among these equations are the Riccati equation, which plays a role in studies of wave propagation, neutron transport, and filter theory; the Emden-Fowler equation, of importance in astrophysical and nuclear studies; the Hamilton-Jacobi equation of mechanics; the eikonal equation of optics; and others. In addition, it is a basic equation of dynamic programming. In this paper a formula giving a representation for the solution of the above type of equation is presented. It involves use of 'max' operators applied to solutions of associated linear equations. In turn, this representation formula leads to the construction of quadratically convergent and monotonic sequences of functions which are of utility in the computational solution of nonlinear boundary value problems. Results of some numerical experiments involving both ordinary and partial differential equations are presented. (Author)