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 
, where the integrand f is a function of u, its derivatives, and x.
VariationalBound[{f,g},u[x],{x,xmin,xmax},ut,{a},{b},…]
finds values of the parameters that extremize the ratio 
, where the integrands f and g are functions of u, its derivatives, and x.
VariationalBound
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 
, where the integrand f is a function of u, its derivatives, and x.
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.
VariationalBound[{f,g},u[x],{x,xmin,xmax},ut,{a},{b},…]
finds 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 VariationalBound, you first need to load the Variational Methods Package using Needs["VariationalMethods`"].
- 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 (1)
Needs["VariationalMethods`"]Eigenvalue problem for a fourth-order ordinary differential equation:
eqn = y''''[x] + Pi y''[x] + 5 y'[x] == λ y[x];sol = VariationalBound[{y[x] eqn[[1]], y[x] ^ 2}, y[x], {x, 0, Infinity}, E ^ (c x), {c}]The solution fits the equation well in this case:
y[x_] = E ^ (c x) /. sol[[2]]eqn[[1]] - eqn[[2]] /. {λ -> sol[[1]]}//ChopClear[y]