此为 Mathematica 7 文档,内容基于更早版本的 Wolfram 语言
查看最新文档(版本11.2)

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.
  • 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.
Needs["VariationalMethods`"]
Eigenvalue problem for a third-order ordinary differential equation:
In[2]:=
Click for copyable input
In[3]:=
Click for copyable input
Out[3]=
The solution fits the equation well in this case:
In[4]:=
Click for copyable input
Out[4]=
In[5]:=
Click for copyable input
Out[5]=
In[6]:=
Click for copyable input