# Welcome Page

Finite Element Method
is a powerful method to compute numerical solutions to both Ordinary
Different Equations(ODEs)
, and Partial Differential Equations (PDEs).
However it involves quite a few advanced concepts in mathematics, such
as the functional spaces, minimizing a functional etc. These concepts are
hard to grasp .  These pages are  designed to help you to understand
the Finite Element Method in an application to ODEs.

Mathematical Concepts
: In this
page we will discuss several advanced concepts that appear in the development
of the Finite Element Method. The first of these concepts are functional–
or functions of functions. The second concept is minimizing  functionals,
and the third deals with the basis of a vector space and inner products.

Finite
Different Method ODE
—-
Rayleigh-Ritz
Method
we will introduce the Rayleigh-Ritz Method in solving boundary value problems.
We will derive, in detail, the method when the base functions are piecewise
linear functions. In the next two page we will apply the method  to
solve the boundary problem   -y” + 2y = x, for
0< x < 1 and y(0) = y(1) = 0.

Example
1
: In this page we will apply the method  to solve the boundary
problem   -y” + 2y = x, for  0< x < 1 and
y(0) = y(1) = 0, with two base functions. We will give details of the computation
and the Mathematica code as well. The graph of the approximated function
is also given.

Example
2
: Here we will apply the method  to solve the boundary problem
-y” + 2y = x
, for  0< x < 1 and y(0) = y(1) = 0, with four
base functions. We will give details of the computation and the Mathematica
code as well. The graph of the approximated function is also given.
With more basis function the computation is more complex and the approximation
is more accurate.