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VariationalBound

VariationalBound[f, u[x], {x, xmin, xmax}, ut, {a}, {b}, ...]
finds values of the parameters a, b, ... of a trial function ut that extremize the functional , where the integrand f is a function of u, its derivatives, and x.
VariationalBound[f, u[x, y, ...], {{x, xmin, xmax}, {y, ymin, ymax}, ...}, ut, {a}, {b}, ...]
finds values of the parameters of a trial function of two or more variables.
VariationalBound[{f, g}, u[x], {x, xmin, xmax}, ut, {a}, {b}, ...]
finds values of the parameters that extremize the ratio , where the integrands f and g are functions of u, its derivatives, and x.
  • VariationalBound returns the extremal value of the functional as well as the optimal parameter values.
  • By default, the parameters a, b, ... may range over the interval -infty to infty. A parameter specification of {a, amin, amax} can be used to restrict the range to the interval amin to amax.
Needs["VariationalMethods`"]
Eigenvalue problem for a fourth-order ordinary differential equation:
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The solution fits the equation well in this case:
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