96 Applied calculus of variations for engineers
The coeﬃcients are
As above, the unknown coeﬃcients are solved from the conditions
resulting in the linear system of equations
It may be shown that the system is nonsingular and always yields a nontrivial
solution assuming that the basis functions form a linearly independent set.
The computation of the terms of the equations, however, is rather tedious
and resulted in the emergence of the next method.
7.3 Galerkin’s method
The diﬀerence between the Ritz method and that of Galerkin’s is in the fact
that the latter addresses the diﬀerential equation form of a variational prob-
lem. Galerkin’s method minimizes the residual of the diﬀerential equation
integrated over the domain with a weight function; hence, it is also called the
method of weighted residuals.
This diﬀerence lends more generality and computational convenience to
Galerkin’s method. Let us consider a linear diﬀerential equation in two vari-
L(u(x, y)) = 0
and apply Dirichlet boundary conditions. Galerkin’s method is also based on
the Ritz approximation of the solution as