Approximation first order differential using finite difference method

Approximation first order differential using finite difference method

Finite Difference Method is a numerical method used to approximate the derivatives of partial differential equations, to a system of equations using discretization techniques, this system of equations then can be solved using matrix algebra, Why and how we use this technique ?.

This process of reduction or approximation  (from PDE’s to system of equation), makes finding the solution to the problem (heat transfer for example) suited to the modern computers.

Let’s assume that we have a function at x, the value at x+h of this function is given by the following equation using Taylor series :

Were h is a step size in x direction, or Δx

First order Partial deferential approximation

Using the previous equation we can derive the approximation of a first order derivative for a partial deferential equation with forward, backward or central differentiating in simple steps.

Types of the finite difference method.

Forward finite difference scheme :

Rearranging : 

Or we can use the indexed form :

Were R(h) is the truncation error :

Backward finite difference scheme :

Rearranging : 

Or we can use the indexed form :

Were R(h) is the truncation error :

Central finite difference scheme :

Using a step forward and a step backward equation together as follows :

equation (1)

equation (2)

Multiplying the second (the backward step equation)  by (-1) we get :

equation (3)

We add equations (1) and  (3) side by side to get the following equation :

Rearranging : 

Or we can use the indexed form :

Were R(h) is the truncation error :