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NVariationalBound—Wolfram Documentation

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NVariationalBound[f,u[x],{x,x_min,x_max},u_t,{a,a₀},{b,b₀},…]

numerically searches for values of the parameters a, b, ... of a trial function u_t, 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,x_min,x_max},…},u_t,{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,x_min,x_max},u_t,{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.

• 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,a_min,a_max} searches for an extremum, stopping the search if a ever gets outside the range a_min to a_max.

Eigenvalue problem for a third-order ordinary differential equation:

The solution fits the equation well in this case:

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