Mathematical Methods for Physics and Engineering - Matematica.NET

27.6 DIFFERENTIAL EQUATIONSThe forward difference estimate of y i+1 , namely( ) dyy i+1 = y i + h = y i + hf(x i ,y i ), (27.72)dxiwould give exact results if y were a linear function of x in the range x i ≤ x ≤ x i +h.The idea behind the Adams method is to allow some relaxation of this andsuppose that y can be adequately approximated by a parabola over the intervalx i−1 ≤ x ≤ x i+1 . In the same interval, dy/dx can then be approximated by a linearfunction:f(x, y) = dydx ≈ a + b(x − x i) for x i − h ≤ x ≤ x i + h.The values of a and b are fixed by the calculated values of f at x i−1 and x i ,whichwe may denote by f i−1 and f i :Thuswhich yieldsa = f i ,∫ xi+h[y i+1 − y i ≈x ib = f i − f i−1.hf i + (f i − f i−1 )h](x − x i ) dx,y i+1 = y i + hf i + 1 2 h(f i − f i−1 ). (27.73)The last term of this expression is seen to be a correction to result (27.72). Thatit is, in some sense, the second-order correction,12 h2 y (2)i−1/2 ,to a first-order formula is apparent.Such a procedure requires, in addition to a value for y 0 , a value for either y 1 ory −1 ,sothatf 1 or f −1 can be used to initiate the iteration. This has to be obtainedby other methods, e.g. a Taylor series expansion.Improvements to simple difference formulae can also be obtained by usingcorrection methods. In these, a rough prediction of the value y i+1 is made first,and then this is used in a better formula, not originally usable since it, in turn,requires a value of y i+1 for its evaluation. The value of y i+1 is then recalculated,using this better formula.Such a scheme based on the forward difference formula might be as follows:(i) predict y i+1 using y i+1 = y i + hf i ;(ii) calculate f i+1 using this value;(iii) recalculate y i+1 using y i+1 = y i + h(f i + f i+1 )/2. Here (f i + f i+1 )/2 hasreplaced the f i used in (i), since it better represents the average value ofdy/dx in the interval x i ≤ x ≤ x i + h.1025