Variational Methods Package Symbol

NVariationalBound

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.

MORE INFORMATION

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.

EXAMPLES

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Basic Examples (1)

Eigenvalue problem for a third-order ordinary differential equation:

The solution fits the equation well in this case:

Needs["VariationalMethods`"]

Eigenvalue problem for a third-order ordinary differential equation:

In[2]:=

In[3]:=

Out[3]=

The solution fits the equation well in this case: