The method of successive approximations
uses the equivalent integral equation for (1) and an iterative method
for constructing approximations to the solution. This is a
traditional way to prove (1) and appears in most all differential
equations textbooks. It is attributed to the French
Emile Picard (1856-1941).
Theorem 2 (Successive Approximations - Picard Iteration). The solution to the I.V.P in (1) is found by constructing recursively a sequence of functions
Then the solution to (1) is given by the limit:
Begin by reformulating (1) as an
equivalent integral equation. Integration of both sides of
Applying the Fundamental
Theorem of Calculus to the left side of (4)
we have , which
can be rearranged to obtain
Observe that occurs on both the left and right hand sides of equation (5). We can use this formula and input in the integrand on the right and then output the next iteration for on the left side. This is a type of fixed point iteration, the most familiar form of which is Newton's method for root finding.
Start the iteration with the initial
then define the next function as follows
Next is used to construct as follows
The process is repeated, and once has been obtained, the next function is given recursively by
We must take the limit
(6). Assume that the limit (3) exists,
If there is "no problem" when taking limits on the right side then we might expect the following
This is the "intuitive proof" of equation (5). Q.E.D.
For More Proof. More details for the existence and uniqueness of can be found in textbooks and the literature.
(c) John H. Mathews 2005