Generalized Quasilinearization for Nonlinear Problems

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The problems of modern society are complex, interdisciplinary and nonlin­ ear. ~onlinear problems are therefore abundant in several diverse disciplines. Since explicit analytic solutions of nonlinear problems in terms of familiar, well­ trained functions of analysis are rarely possible, one needs to exploit various approximate methods. There do exist a number of powerful procedures for ob­ taining approximate solutions of nonlinear problems such as, Newton-Raphson method, Galerkins method, expansion methods, dynamic programming, itera­ tive techniques, truncation methods, method of upper and lower bounds and Chapligin method, to name a few. Let us turn to the fruitful idea of Chapligin, see [27] (vol I), for obtaining approximate solutions of a nonlinear differential equation u' = f(t, u), u(O) = uo. Let fl' h be such that the solutions of 1t' = h (t, u), u(O) = uo, and u' = h(t, u), u(O) = uo are comparatively simple to solve, such as linear equations, and lower order equations. Suppose that we have h(t, u) s f(t, u) s h(t, u), for all (t, u).