Numerical methods of calculus of variations
In the last chapter we focused on analytical solutions. Application problems
in engineering practice, however, may not be easily solved by such techniques,
if solvable at all. Hence, before we embark on applications, it seems prudent to
discuss solution techniques that are amenable for practical problems. These
methods produce approximate solutions and are, as such, called numerical
It was mentioned in the introduction that the solution of the Euler-Lagrange
diﬀerential equation resulting from a certain variational problem may not be
easy. This gave rise to the idea of directly solving the variational problem.
The classical method is the Euler method.
The most inﬂuential method is that of Ritz. The methods of Galerkin and
Kantorovich, both described in , could be considered extensions of Ritz’s.
They are the most well-known by engineers and used in the industry. Finally,
the boundary integral method is also useful for certain kind of engineering
7.1 Euler’s method
Euler proposed a numerical solution for the variational problem of
f(x, y, y
)dx = extremum
with the boundary conditions
by subdividing the interval of the independent variable as
; i =1, 2,...,n.