Minimax approximate solutions of linear boundary value problems

Abstract:

Define the operator $D:C”[0,\tau ] \to C[0,\tau ]$ by $D[u] = u” - {a_0}u\prime - {a_1}u$ where ${a_0},{a_1} \in C[0,\tau ]$ and consider the two point boundary value problem $({\text {BVP}})\;D[y](x) = {a_2}(x)$, $x \in [0,\tau ]$, ${N_0}[y] = {\alpha _0}y(0) + {\alpha _1}y\prime (0) = {\alpha _2}$, ${N_\tau }[y] = {\beta _0}y(\tau ) + {\beta _1}y\prime (\tau ) = {\beta _2}$ where ${a_2} \in C[0,\tau ]$, $\alpha _0^2 + \alpha _1^2 \ne 0$ and $\beta _0^2 + \beta _1^2 \ne 0$. Let ${\Pi _k}$ denote the set of polynomials of degree at most k and define the approximating set ${\mathcal {P}_k} = \{ p \in {\Pi _k}:{N_0}[p] = {\alpha _2},{N_\tau }[p] = {\beta _2}\}$. Then for each $k \geqslant 3$ there exists ${p_k} \in {\mathcal {P}_k}$ satisfying $\left \| {D[{p_k}] - {a_2}} \right \| = {\inf _{p \in {\mathcal {P}_k}}}\left \| {D[p] - {a_2}} \right \| = {\delta _k}$, where $\left \| \cdot \right \|$ denotes the uniform norm on $C[0,\tau ]$. If the homogeneous BVP $D[y] = 0$, ${N_0}[y] = {N_\tau }[y] = 0$ has no nontrivial solutions, then the nonhomogeneous BVP has a unique solution y and ${\lim _{k \to \infty }}\left \| {p_k^{(i)} - {y^{(i)}}} \right \| = 0$ for $i = 0,1,2$. If X denotes a closed subset of $[0,\tau ]$ and \[ {\delta _{k,X}} = \inf \limits _{p \in {\mathcal {P}_k}} \max \limits _{x \in X} |D[p](x) - {a_2}(x)|,\] then for each $\varepsilon > 0$ there exists $\delta > 0$ such that $d(x) \leqslant \delta$ implies that $0 \leqslant {\delta _k} - {\delta _{k,X}} \leqslant \varepsilon$, where $d(X)$ denotes the density of X in $[0,\tau ]$. Several numerical examples are given.