# VariationalBound

Contributed by: Wolfram Research

Find the values of parameters of a trial function that extremize a functional

 ResourceFunction["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 ∫xminxmaxfⅆx, where the integrand f is a function of u, its derivatives, and x. ResourceFunction["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. ResourceFunction["VariationalBound"][{f,g},u[x],{x,xmin,xmax},ut,{a},{b},…] finds values of the parameters that extremize the ratio ∫xminxmaxfⅆx/∫xminxmaxgⅆx, where the integrands f and g are functions of u, its derivatives, and x.

## Details and Options

ResourceFunction["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

Eigenvalue problem for a fourth-order ordinary differential equation:

 In[1]:=
 In[2]:=
 Out[2]=

The solution fits the equation well in this case:

 In[3]:=
 Out[3]=
 In[4]:=
 Out[4]=
 In[5]:=