EXPLANATION FILE OF PROGRAM STORMER =================================== Solve Y"=f(x,y,y') with initial conditions by Stormer's method -------------------------------------------------------------- The differential equation of order 2 can be replaced by a system of two equations of order 1: Given y"=f(x,y,y') with y(a) and y'(a), by calling u = y', the problem becomes | u'=f(x,y,u) | y'=u | with y(a), u(a) given We start from a particular form of the Taylor's formula, where the remainder is under the form of an integral: x+h y(x+h) = y(x) + h y'(x) + Sum (x+h-t) y"(t) dt x In the same way x-h y(x-h) = y(x) - h y'(x) + Sum (x-h-t) y"(t) dt x By summing x+h x-h y(x+h) - 2y(x) + y(x-h) = Sum ... + Sum ... x x For the second integral, we change the variable: u = 2x - t So x-h x+h Sum (x-h-t) y"(t) dt = - Sum (x+h-u) y"(u) du x x x+h du=-dt ==> = Sum (x+h-t) y"(2x-t) dt x Finally x+h y(x+h) - 2y(x) + y(x-h) = Sum (x+h-t)[y"(t)+y"(2x-t)] dt x We now use the interpolation polynomial recalling that y"(x)=f(x,y,y'): x n+1 y - y + y = Sum (x - t)[P(t)+P(2x - 1)] dt n+1 n n-1 x n+1 n n x 2 n+1 0 1 2 = h Sum (x - t) [O0 Div (fn) + O1 Div (fn) +O2 Div (fn)] dt x n+1 n m 1 |(-s) (m)| with O = (-1) Sum (1-s) |( ) + ( )| ds m 0 |(m ) (s)| (see file Adambash.txt). This leads to Stormer's formulas: Explicit: y - 2 y + y = (h²/12)[13f - 2 f + f ] n+1 n n-1 n n-1 n-2 Implicit: y - 2 y + y = (h²/12)[f + 10 f + f ] n+1 n n-1 n+1 n n-1 From [BIBLI 04] --------------------------------------- End of file Stormer.txt