Finite element methods

Chapter 1 The basis

1.1 Numerical differentiation• For a given discrete function

• Taylor expansion gives the following • Taking interval

With h = x i+1-xi

1.1 Numerical differentiation• Taking upto only second derivative term

• using above we can get

1.1 Numerical differentiation• Hence the differentiation rule called

central derivative (first)

Similarly, second derivative

1.1 Numerical differentiationFunction is u(x) vertical and the points are x on the horizontal

1.2 Ordinary differential equations• Initial value problem

Given a first order differential equation

Whose initial value is given i.e we know at a point x0 what is the value of the function y(x0) and we call the value y0

1.2 Ordinary differential equations• An example, given

,h = delta x = 0.1 , y(0) = - 0.2

The solution diagram

1.3 ODE beam of first , second, third and fourth order

1.4 Partial differential equation and plates