Function Repository Resource:

VariationalBound

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

Contributed by: Wolfram Research
 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 , 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 , 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:

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The solution fits the equation well in this case:

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Requirements

Wolfram Language 11.3 (March 2018) or above