Quasilinearization

Two numerical solutions of the nonlinear example boundary value problem $y''=y^{2}$, $y(-1)=y(1)=1$. Solved by a spectral Chebyshev method and quasilinearization. The top curve $u_{1}$ used 21 interpolation nodes, and the bottom curve $u_{2}$ used 34. Both used 3 iterations.

In mathematics, quasilinearization is a technique which replaces a nonlinear differential equation or operator equation (or system of such equations) with a sequence of linear problems, which are presumed to be easier, and whose solutions approximate the solution of the original nonlinear problem with increasing accuracy. It is a generalization of Newton's method; the word "quasilinearization" is commonly used when the differential equation is a boundary value problem.