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Minimax approximate solutions of linear boundary value problems


Authors: Darrell Schmidt and Kenneth L. Wiggins
Journal: Math. Comp. 33 (1979), 139-148
MSC: Primary 65L10
DOI: https://doi.org/10.1090/S0025-5718-1979-0514815-X
MathSciNet review: 514815
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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\Vert {D[{p_k}] - {a_2}} \right\Vert = {\inf _{p \in {\mathcal{P}_k}}}\left\Vert {D[p] - {a_2}} \right\Vert = {\delta _k}$, where $ \left\Vert \cdot \right\Vert$ 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\Vert {p_k^{(i)} - {y^{(i)}}} \right\Vert = 0$ for $ i = 0,1,2$. If X denotes a closed subset of $ [0,\tau ]$ and

$\displaystyle {\delta _{k,X}} = \mathop {\inf }\limits_{p \in {\mathcal{P}_k}} \mathop {\max }\limits_{x \in X} \vert D[p](x) - {a_2}(x)\vert,$

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.

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DOI: https://doi.org/10.1090/S0025-5718-1979-0514815-X
Keywords: Minimax approximate solution, uniform approximation, boundary value problem
Article copyright: © Copyright 1979 American Mathematical Society