Multiwavelets and multiwavelet packets of Legendre functions in the direct method for solving variational problems

A numerical technique for solving the linear problems of the calculus of variations is presented in this paper. Multiwavelets and multiwavelet packets of Legendre functions are used as basis functions in the Ritz method of formulation. An operational matrix of integration of multiwavelets and multiwavelet packets is introduced and is used to reduce the calculus of variation problem to the solution of the system of algebraic equations. The algorithm is applied to the analysis of mechanic problems which are formulated as functionals. Two examples are considered in this paper. The first example concerns the stability problem of a Euler–Bernoulli beam and the second one presents the calculation of the extreme value of the functional which defines the potential energy of an elastic string. The presented method yields the approximate solutions which are convergent to accurate results.