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u Subscript upper E upper X Baseline phi Subscript i Superscript double-prime Baseline d x period"/>

      Since phi double-prime Subscript i Baseline equals 0 for i equals 1 and i equals 2, we have:

StartLayout 1st Row 1st Column r 1 Superscript left-parenthesis k right-parenthesis Baseline equals 2nd Column minus StartFraction 1 Over script l Subscript k Baseline EndFraction left-parenthesis kappa u Subscript upper E upper X Baseline right-parenthesis Subscript x equals x Sub Subscript k plus 1 plus StartFraction 1 Over script l Subscript k Baseline EndFraction left-parenthesis kappa u Subscript upper E upper X Baseline right-parenthesis Subscript x equals x Sub Subscript k plus StartFraction script l Subscript k Baseline Over 2 EndFraction integral Subscript negative 1 Superscript 1 Baseline left-parenthesis c u Subscript upper E upper X Baseline right-parenthesis Subscript x equals upper Q Sub Subscript k Subscript left-parenthesis xi right-parenthesis Baseline upper N 1 d xi 2nd Row 1st Column r 2 Superscript left-parenthesis k right-parenthesis Baseline equals 2nd Column StartFraction 1 Over script l Subscript k Baseline EndFraction left-parenthesis kappa u Subscript upper E upper X Baseline right-parenthesis Subscript x equals x Sub Subscript k plus 1 minus StartFraction 1 Over script l Subscript k Baseline EndFraction left-parenthesis kappa u Subscript upper E upper X Baseline right-parenthesis Subscript x equals x Sub Subscript k plus StartFraction script l Subscript k Baseline Over 2 EndFraction integral Subscript negative 1 Superscript 1 Baseline left-parenthesis c u Subscript upper E upper X Baseline right-parenthesis Subscript x equals upper Q Sub Subscript k Subscript left-parenthesis xi right-parenthesis Baseline upper N 2 d xi EndLayout

      and for i greater-than-or-equal-to 3 we have:

      where upper P Subscript i minus 2 Baseline left-parenthesis xi right-parenthesis is the Legendre polynomial of degree i minus 2 and eq. (D.10) was used.

      Since the exact solution is known, the exact value of the potential energy can be determined for any set of values of α, κ, c and script l. When κ and c are both constants then

      (1.107)StartLayout 1st Row 1st Column pi left-parenthesis u Subscript upper E upper X Baseline right-parenthesis equals 2nd Column minus one half left-bracket kappa left-parenthesis StartFraction alpha squared Over 2 alpha minus 1 EndFraction script l Superscript 2 alpha minus 1 Baseline minus left-parenthesis alpha plus 1 right-parenthesis script l Superscript 2 alpha Baseline plus StartFraction left-parenthesis alpha plus 1 right-parenthesis squared Over 2 alpha plus 1 EndFraction script l Superscript 2 alpha plus 1 Baseline right-parenthesis period 2nd Row 1st Column Blank 2nd Column period right-bracket plus plus times times c left-parenthesis right-parenthesis plus plus minus minus times times 1 plus plus times times 2 alpha 1 script l plus plus times times 2 alpha 1 times times 1 plus plus alpha 1 script l times times 2 left-parenthesis right-parenthesis plus plus alpha 1 times times 1 plus plus times times 2 alpha 3 script l plus plus times times 2 alpha 3 period EndLayout

,
and
.

α pi left-parenthesis u Subscript upper E upper X Baseline right-parenthesis α pi left-parenthesis u Subscript upper E upper X Baseline right-parenthesis
0.600 −2.3728354978 1.000 −1.0000000000
0.700 −1.7571858289 1.500 −0.5104166667
0.800 −1.4176885916 2.000 −0.3047619048
0.900 −1.1799028822 3.000 −0.1420634921

      When α is a fractional number then derivatives higher than α will not be finite in x equals 0. In the range 0.5 less-than alpha less-than 1 the first derivative in the point x equals 0 is infinity. This range of α has considerable practical importance because the exact solutions of two‐ and three‐dimensional problems often have analogous terms.

      When α is an integer then all derivatives of Скачать книгу