Using a generalization of Newton's method, a non-linear parabolic equation of the form ut - uxx = g(u), and a non-linear elliptic equation uxx + uyy = eu, are solved numerically. Comparison of these results with results obtained using the Picard iteration procedure show that in many cases the quasilinearization method offers substantial advantages in both time and accuracy.
Some numerical experiments using Newton’s method for nonlinear parabolic and elliptic boundary-value problems
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