A smooth approximation for the solution of a fourth-order boundary value problem based on nonpolynomial splines
The authors consider smooth approximations of the solution of the fourth- order linear ordinary differential equation \(y^{(4)} + f(x) y = g(x)\), \(a \leq x \leq b\), subject to the boundary conditions \(y(a) - A_ 1 = 0 = y(b) - A_ 2\), \(y'(a) - B_ 1 = 0 = y'(b) - B_ 2\) whereby \(f(x)\) and \(g(x)\) are continuous functions on \([a, b]\) and \(A_ i\), \(B_ i\), \(i = 1,2,\) are finite real arbitrary constants. The aim of this paper is to construct a new spline method that is based on a nonpolynomial spline space. Indeed, since \(\{1, x, \dots, x^ m, \sin kx, \cos kx\}\) generates an extended complete Chebyshev space on \([a, b]\), it is possible to construct spline approximations that have a polynomial and a trigonometric part. The spline functions proposed in this paper have the form \[ a \sin kx + b \cos kx + c x^ 3 + d x^ 2 + e x + f. \] It is shown that the parameter \(k\) can be used to raise the order of accuracy of the method. Two numerical examples are given to sustain the autonomous parametric spline method and to compare it with a quintic polynomial spline method of second order.
fourth-order linear ordinary differential equation
scientific article; zbMATH DE number 699108