A two-sided functional-discrete method for second-order differential equations with general boundary conditions (Q2387974)

Necessary and sufficient conditions are obtained for the functional-discrete method (FD method) which is used in order to provide two-sided approximations to the solution of the boundary value problem \[ \begin{aligned} & u''(x)-q(x)u(x)=-f(x),\quad x\in (0,1),\\ & M_1(u)=0,\quad M_2(u)=0,\end{aligned} \] where the boundary conditions \(M_1, M_2\) are linear functionals of \(u(x)\) and \(u'(x)\). Two examples of boundary conditions are used in order to illustrate the theoretical results. To verify the convergence of the FD method, the spectral radius of the integral operator corresponding to the above problem is used. An algorithm of the method designed for modern mathematical software (Maple 7/8 and Mathematica 4.1) is given.