A combined method for a two-point boundary value problem (Q2755021)

Consider the differential equation NEWLINE\[NEWLINE x^{(2n)}(t) + \phi(t,x(t)) = 0, \quad t\in(0,1),NEWLINE\]NEWLINE with the homogeneous boundary conditions \(x^{(j)}(0)=x^{(j)}(1)=0\), \(j\in\{0,1,\ldots, r-1\}\). Using a previous result of the authors for nonlinear equations in partially ordered locally convex spaces, a combined method is obtained in order to solve numerically the above two-point boundary value problem. The combined method involves a modification of Newton's method and divided differences. A straightforward error estimation is an advantage of the method. A numerical example is also given.