External force

In general, f(x) is part of the problem data and is usually provided by the specific physical problem you are trying to solve. Its form can vary:

In structural mechanics, f(x) could represent an external force applied over a domain.
In heat transfer, f(x) might represent an internal heat source distributed along a region.
In electromagnetic problems, it could correspond to a charge or current distribution.

Computing integrals

Quadrature

In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral. The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integration", especially as applied to one-dimensional integrals

The basic problem in numerical integration is to compute an approximate solution to a definite integral

$$ \int_a^bf(x)dx $$

Gaussian quadrature

Gaussian quadrature is one of the most used methods. It constructed to yield an exact result for polynomials of degree 2n − 1 or less by a suitable choice of the nodes x_i and weights w_i for i = 1, ..., n. It states that:

$$ \int_{-1}^1f(x)dx \approx \sum_{i=1}^n w_if(x_i) $$

Which is exact for polynomials of degree 2n − 1 or less.

An integral over [a, b] must be changed into an integral over [−1, 1] before applying the Gaussian quadrature rule. This change of interval can be done in the following way:

$$ \int_a^bf(x)dx = \int_{-1}^1 f(\frac{b - a}{2} \xi + \frac{a + b}{2} ) \frac{dx}{d\xi}d\xi $$