h(x) x ∈ [a, b]

u(x) u− h

D_{2} = ∫_{a}^{b}(u−h)^{2}dxD_{1} = ∫_{a}^{b} |u−h|dx

P = ∫_{a}^{b} p(u, u′, u′′)dx

stabilizer P u u(x) u(x) min_{u(x)}(D+ P) =

min_{u(x)} ∫_{a}^{b} [(u − h)^{q} + γp(u, u′, u′′)] dx81whereisapenaltycoefficient.Hereweconsiderseveralproblemsofthebestapproximation. p =

u^{2} u min_{u(x)} J_{1}(u), J_{1}(u) =

∫_{a}^{b} (u^{2} + (h − u)^{2}) dx122.2.1itminimizesthesumofsquare_{2}^{2} D_{2} u′ u (u^{2} + (h − u)^{2}) = 0

u_{(}x) =

1+h(x) u h → 0 h(x) → ∞ u_{=}0

lim_{→0} u_{(}x) = h(x), lim_{→∞} u_{(}x) = 0

h(x) u(x) min_{u} J_{2}(u), J_{2}(u) = ∫_{a}^{b} (u′ ^{2} + (h − u)^{2}) dx122.2.1T hefirsttermoftheintegrantrepresentsthepenaltyforthegrowthratetobenon−

constant, andthesecondtermdescribestheclosenessofand. u = u → 0 h(x) → ∞ h(x)

lim_{→0} u_{(}x) = h(x), lim_{→∞} u_{(}x) == b − a ∫_{a}^{b} h()d

u′′ u min_{u} J(u), J(u) =

∫_{a}^{b} ((u′′)^{2} + (h − u)^{2}) dx122.2.11T heequationforthebestapproximate(Eulerequationof122.2.11)isu^{IV} +

u− h = 0, u′′(a) =

u′′(b) = 0, u′′′(a) = u′′′(b) =

0,Here, thenaturalboundaryconditionsareassumedsincethereisnoreasontoassignspecialvaluesoftheapproximationcurveattheendsoftheinterval. → 0 h(x) → ∞ h u → ∞ (u) = h

=u′′− u [a, b]

u′(a) = u′(b) =

0 u eigen-

functions u = u, u′(0) = u′(1) = 0.

u(x) ≢

0 0,_{1} ,_{2} , …

u_{n}(x) u_{n}=_{n}u_{n},n=1,2,...,.

u_{n} ⟨u_{k}, u_{m}⟩ = 0, k ≠ m

⟨f, g⟩

⟨f, g⟩ = ∫_{a}^{b} f(x)g(x) dx.

u(x) = ∑_{n=0}^{∞}_{n}u_{n}(x)

n (u) =

h

(u) = (∑_{n=0}^{∞}_{n}u_{n}) = ∑_{n=0}^{∞}_{n}_{n}u_{n} = h