Variational Methods Package 内置符号

# NVariationalBound

 NVariationalBound[f, u[x], {x, xmin, xmax}, ut, {a, a0}, {b, b0}, ...]numerically searches for values of the parameters a, b, ... of a trial function ut, starting from a=a0, b=b0, ..., that extremize the functional , where the integrand f is a function of u, its derivatives, and x. NVariationalBound[f, u[x, y, ...], {{x, xmin, xmax}, ...}, ut, {a, a0}, {b, b0}, ...] searches for values of the parameters of a trial function of two or more variables. NVariationalBound[{f, g}, u[x], {x, xmin, xmax}, ut, {a, a0}, {b, b0}, ...] searches for values of the parameters that extremize the ratio , where the integrands f and g are functions of u, its derivatives, and x.
• NVariationalBound returns the extremal value of the functional as well as the optimal parameter values.
• NVariationalBound uses FindMinimum to search for values of the parameters that extremize the functional.
• A parameter specification of {a, a0, a1} searches for an extremum using a0 and a1 as the first two values of a, avoiding the use of derivatives.
• A parameter specification of {a, a0, amin, amax} searches for an extremum, stopping the search if a ever gets outside the range amin to amax.
 例   (1)
Needs["VariationalMethods`"]
Eigenvalue problem for a third-order ordinary differential equation:
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The solution fits the equation well in this case:
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