Construction of $C^1$-smooth piecewise quadratic functions for solving boundary value problems of 4th order equations

In this paper, two approaches to the construction of piecewise quadratic functions were presented with their subsequent application in solving boundary value problems for equations of the 4th order
$${G_y} - {\frac{d}{dx}}G_{y'} + {\frac{d^2}{dx^2}} G_{y''} = 0,$$

$$y(a) = A, y(b) = B, y^{'}(a) = A^{'}, y^{'}(b) = B^{'}.$$
For the first approach, it was shown that it cannot be used to find approximate solutions of the considered equations by the variational method. Next, a description was given of the second approach, which is based on smoothing the corner points of a piecewise linear function. In the process of such constructions, it turned out that the smoothing polynomial has the second degree. To demonstrate the correctness of this approach, computational formulas of the gradient descent method were obtained and two numerical examples of solving boundary value problems of 4th order equations were given.