Finite Element Method
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. 

Advanced 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: In this page 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.