The variational method is a commonly used mathematical approach to solving partial differential equations. However, in some cases, the integral that needs to be solved may be unsolvable. In this case, it is possible to use several strategies to solve the problem: 1. Use of numerical methods such as the finite element method, the finite difference method or the finite volume method. These methods make it possible to solve complex problems using numerical approximations. 2. Approximation of the functions of tes, This approximation can be performed using polynomial functions, Fourier functions or other types of functions. 3. Use of transformations which may include Laplace or Fourier transformations which allow the integral to be transformed into a simpler form. 4. Integration by parts: This technique consists of integrating with respect to one part of the integrand function and differentiating the other part. 5. Dimension reduction: If the problem can be reduced to a lower dimension, it may be possible to solve the integral using analytical methods. For example, if the problem is two-dimensional, it may be possible to reduce the problem to one dimension by using symmetry. The method of weighted residuals is a numerical method based on the variational method which consists in finding an approximate solution by minimizing a functional of weighted error. This method can be used to solve different types of differential equations, including partial differential equation

Bouzid Sihem Informative answer. Thank you.

you're welcome,

I will add that Galerkin method is one of weighted residuals method, where interpolation fonctions are choosen same weighted fonctions, you can consult this paper : Article Anisotropic Behavior of Different Three-Dimensional Structur...

and this book " The finit element method for Engineers ", Kenneth H.Huebner, 1982

Don't hesitate for more information

good continuation

It is typical to run across intractable integrals that are difficult to assess while using the variational approach to solve partial differential equations (PDEs). The following are some methods for handling such integrals:

Application of certain algebraic or trigonometric identities can occasionally be used to simplify the integrand of an intractable integral. The integral's complexity may be lowered as a result, making it easier to handle.