Picard’s Iteration Method

Differential equations can be notoriously difficult to solve. One of the reasons for this is the difficulty faced in integrating functions. Undaunted by difficult integrations, mathematicians have developed several numerical techniques that give us approximate solutions of a differential equation at any point of our choice in the domain. One of them is Picard s iteration method. Without further ado, let s see a demonstration of its superiority to other analytical methods of solving d.e. s.

Suppose is a differential equation we need to solve at x=1 with the given condition y(0)=0. Rewriting this equation we get, . Integrating, we get

The crucial process that will be followed now is that we substitute y in the R.H.S. of the above equation by a constant and follow this procedure many times until a sufficiently good approximation of y is obtained. In the first step it is necessary to substitute y by y(0)=0, the value of y at the lower limit of the integral. If the lower limit is a we need to substitute y by y(a). This gives,

Since, y(0)=0 we can further simplify this to,

This is our first approximation of y and we will denote it by . Next we substitute y by in equation (1). This gives us,