Method of mean weighted residuals
In applied mathematics, methods of mean weighted residuals (MWR) are methods for solving differential equations. The solutions of these differential equations are assumed to be well approximated by a finite sum of test functions formula_1. In such cases, the selected method of weighted residuals is used to find the coefficient value of each corresponding test function. The resulting coefficients are made to minimize the error between the linear combination of test functions, and actual solution, in a chosen norm. Notation of this page. It is often very important to firstly sort out notation used before  presenting how this method is executed in order to avoid confusion. Mathematical statement of method. The method of mean weighted residuals solves formula_6 by imposing that the degrees of freedom formula_5 are such that: is satisfied. Where the inner product formula_11 is the standard function inner product with respect to some weighting function formula_12 which is determined usually by the basis function set or arbitrarily according to whichever weighting function is most convenient. For instance, when the basis set is just the Chebyshev polynomials of the first kind, the weighting function is typically formula_13 because inner products can then be  more easily computed using a Chebyshev transform. Additionally, all these methods have in common that they enforce boundary conditions by either enforcing that the basis functions (in the case of a linear combination) individual enforce the boundary conditions on the original BVP (This only works if the boundary conditions are homogeneous however it is possible to apply it to problems with inhomogeneous boundary conditions by letting formula_14 and substituting this expression into the original differential equation and imposing homogeneous boundary conditions to the new solution being sought to find u(x) that is v(x) where L(x) is a function that satisfies  the boundary conditions imposed on u that is known.), or by explicitly imposing the boundary by removing n rows to the matrix representing the discretised problem where n refers to the order of the differential equation and substituting them with ones that represent the boundary conditions. Choice of test functions. The choice of test function, as mentioned earlier, depends on the specific method used (under the general heading of mean weighted residual methods). Here is a list of commonly used specific MWR methods and their corresponding test functions roughly according to their popularity: