VARIATIONAL METHODS PACKAGE SYMBOL

# 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.

## DetailsDetails

• To use VariationalBound, you first need to load the Variational Methods Package using Needs["VariationalMethods`"].
• 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 to . A parameter specification of {a,amin,amax} can be used to restrict the range to the interval amin to amax.

## ExamplesExamplesopen allclose all

### Basic Examples  (1)Basic Examples  (1)

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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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