Cubic spline on quintic spline method for nonlinear boundary value problems (Q1805280)

The problem considered is the scalar second-order equation \(d^2 c/dx^2=f(c,x)\) (with boundary conditions). The problem is recast as a 2-dimensional first-order boundary value problem and then treated numerically componentwise via splines over a uniform grid with mesh size \(h=1/n\) and \(x_i=ih\) for \(i=0, \dots, n\). Integrating the boundary value problem yields \[ Q_i(c) := c_i-c_{i-1}-\int^{x_i}_{x_{i-1}} f(x,c(x))dx=0\text{ for }i=0,\dots, n. \] The evaluation of the function \(Q(c)\) is performed via the Euler-Maclaurin formula: \(Q_i (c)=c_i -\) \(c_{i-1}-(h/2) (f_i-f_{i-1})+(h^2/12) (f^1_i-f^1_{i-1})-(h^4/720) (f^3_i-f^3_{i- 1})+(h^6/30240) (f^5_i-f^5_{i-1})+O(h^8)\). To determine \(f^1\) and \(f^3\) from \(f\), the author uses a quintic spline \(S(x)\) to interpolate \(f\), and then \(f^5\) is determined by a cubic spline \(\widehat {S} (x)\) using \(f^3\). This finally yields \(Q_i (c)=c_i-c_{i-1}-\) \((h/2) (f_i-f_{i-1})+(h^2/12) (S^1_i-S^1_{i-1})-(h^4/ 720) (S^3_i-S^3_{i-1})+(h^6/ 30240) (\widehat {S}^5_i-\widehat {S}^5_{i-1})+O(h^8)\). The nonlinear system which is numerically solved is \(u_i(c)+v_i(c)=0\) where \(u_i(c) := c_i-c_{i-1}-(h/2) (f_i-f_{i-1})\) and \(v_i(c) := (h^2/ 12) (S^1_i-S^1_{i-1})-(h^4/ 720) (S^3_i-S^3_{i-1})+(h^6/ 30240) (\widehat {S}^5_i-\widehat {S}^5_{i-1}).\) The system is solved via the modified Newton scheme of solving \((\partial u/\partial c)^{-1} (c^{(k)}) w=-Q(c^{(k)})\) and setting \(c^{(k+1)} := c^{(k)}+w\). Two numerical examples (one of which is nonlinear) are given with boundary conditions \(c'(0)=0\) and \(c(1)=0\).