A new collocation algorithm for solving even-order boundary value problems via a novel matrix method
The authors analyse the application of the collocation method for solving the boundary value problem of even order \[ u^{2n}(x)= f(x, \mathbf{U}(x)),\qquad x\in [a,b], \] \[ u^{r}(a)=a_{r}, u^{r}(b)=b_{r}, \qquad r=0, 1, \dots,n-1, \] where \(\mathbf{U}(x)=(u(x),u^{(1)}(x),\dots, u^{(2n-1)}(x))\). The construction of an approximate scheme of this problem uses the system of Legendre polynomials \( P_{n}(x)\) defined by recursive formulas. Using the properties of the derivatives and repeated integration of \(P_{n}(x)\) this scheme is presented as a system of linear equations in matrix form. This fact allows to apply the software Mathematica for some examples to illustrate the behavior of the error, compared with those obtained by other known methods (spline method and reproducing kernel method).
scientific article; zbMATH DE number 6803991
scientific article