VariationalMethods`
VariationalMethods`

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.

Details and Options

  • 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.

Examples

Basic Examples  (1)

Eigenvalue problem for a fourth-order ordinary differential equation:

The solution fits the equation well in this case: