Wolfram Research

VariationalBound

Contributed by: Wolfram Research

Source Notebook

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]:=
eqn = y''''[x] + Pi y''[x] + 5  y'[x] == \[Lambda] y[x];
In[2]:=
sol = ResourceFunction["VariationalBound"][{y[x] eqn[[1]], y[x]^2}, y[x], {x, 0, Infinity}, E^(c x), {c}]
Out[2]=

The solution fits the equation well in this case:

In[3]:=
y[x_] = E^(c x) /. sol[[2]]
Out[3]=
In[4]:=
eqn[[1]] - eqn[[2]] /. {\[Lambda] -> sol[[1]]} // Chop
Out[4]=
In[5]:=
Clear[y]

Resource History

See Also