Adams-Moulton Methods : Adams methods are based on the idea of approximating the integrand with a polynomial within the interval (tn, tn 1). Using a kth order polynomial results in a k 1th order method. There are two types of Adams methods, the explicit and the implicit types. The explicit type is called the Adams-Bashforth (AB) methods and the implicit type is called the Adams-Moulton (AM) methods.
Consider the k 1 data values, (xi 1, fi 1), (xi , fi), (xi–1, fi–1), ..., (xi–k 1, fi–k 1) which include the current data point. For this data, we fit the Newton’s backward difference interpolating polynomial of degree k as
.....................1.1
where,
The expression for the error is given by
.....................1.2
where ξ lies in some interval containing the points xi 1, xi , ..., xn–k 1 and x. We replace f(x, y) by Pk(x) in
......................1.3
The limits of integration in (1.3) become
for x = xi, s = 0, and for x = xi 1, s = 1.
Also, dx = hds. We get
now,
Hence, we have
....................1.4