VariationalMethods`
VariationalMethods`

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.

Details and Options

  • To use NVariationalBound, you first need to load the Variational Methods Package using Needs["VariationalMethods`"].
  • 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.

Examples

Basic Examples  (1)

Eigenvalue problem for a third-order ordinary differential equation:

The solution fits the equation well in this case: