Three new and applicable approaches based on quasi-linearization technique,
wavelet-homotopy analysis method, spectral methods, and converting two-point
boundary value problem to Fredholm–Urysohn integral equation are proposed for
solving a special case of strongly nonlinear two-point boundary value problems,
namely Troesch problem. A quasi-linearization technique is utilized to reduce the
nonlinear boundary value problem to a sequence of linear equations in the first
method. Second method is devoted to applying generalized Coiflet scaling functions
based on the homotopy analysis method for approximating the numerical solution of
Troesch equation. In the third method we use an interesting technique to convert the
boundary value problem to Urysohn–Fredholm integral equation of the second kind;
afterwards generalized Coiflet scaling functions and Simpson quadrature are
employed for solving the obtained integral equation. Introduced methods are new
and computationally attractive, and applications are demonstrated through
illustrative examples. Comparing the results of the presented methods with the
results of some other existing methods for solving this kind of equations implies the
high accuracy and efficiency of the suggested schemes.