Asymptotic integration algorithms for nonhomogeneous, nonlinear, first order, ordinary differential equations

Original publisher: [Cleveland, Ohio : Lewis Research Center, 1991] OCLC Number: (OCoLC)62873671 Excerpt: ... whichisamoreaccurateapproximatiotno integral ( 17 ). Introducingthechangeinvariable ( 33 ) into ( 38 ) gives AI [ At ] __ e-( tq [ ' ] ' at + ½ u2 [ t ] ' at2 ) ey e ½ u2 [ t ] ' ' 2 ( V1 [ t ] + V2 [ t ]. z ) dy, fv, lq.A, _yy Jy = O which, upon expansion of the second exponential in the integrand into a power series, interchanging the order of summation and integration, and differentiating the change in variable ( 33 ), enables integral ( 38 ) to be written as £, r AZ [ Aq _ e-W, l, l. ", + _u_H ". e ) _ _ x n----O: v, t, fl ",, '-" v, E, l / ",,-' ) x Ul [ t ] Jy = o ey y2, dy + UI [ t ] 2 Jy = o eY y2n + l dy. ( 39 ) Using the fact that [ 16 ] ( 40 ) e-* = o e " yn dy = (-1 ) " n ! (-1 ) m _ ¢ ' n _ e-¢ ( 20 ) allows the two integrals in ( 39 ) to be evaluated, resulting in e--_U2 [ t ]. A ¢ 2 AI [ At ] ( U2 [ t ] _ ": ( 2_n ) ! V, [ t ] E (-1 ) ' _ m [ e-V ' [ q ' "-oo ( _. ( Ul [ t ]. At ) m ) 2u, [ t ] 2 ] [,, ! u, [ t ]., = o n----0 _ ( 2n + 1 ) ! V2 [ t ] _n + z,, k ( Ul [ t ] ' At ) k This leads to the desired approximation X [ t + At ] _ X [ t ] e-( U'lqA_ + ½ v_ [ tlA_ ) + e-_Ua [ t ]. At2 + ° ° ( U2 [ t ] ' _ ": ( 2n ), Vl [ t ] ( 2 " ( Ui [ t ]. At ) " e_V, tq.zs, ) _--_ 2U, [ t ] 2j [ n ! U, [ t ] _ (-1 ) " m ! n = O rn = O-k ! e-U ' [ t ] ' At, ( 41 ) ( 2n-l'l ) ' V2 [ t ] [ 2n + l ) } n ! U, [ t ] 2 [ _ = o (-llk ( U ' [ tl'At ) ' which is a quadratic, explicit, recursive solution to the first order differential equation ( 2 ). This is an explicit representation because the four parameters Uz, U2, 171, and V2 are all evaluated at the current time, t, and therefore their values are known. 6.3 Euler-Maclaurin Approximation Let us express the nonhomogeneous integral ( 15 ) in terms of an asymptotic Euler-Maclaurin expansion [ 15 ] in the form k ( _1 ) n Bn ( f2n-, [ t + At ]-f2n_l [ t ] ). At 2n + h.o.t. 1 [ At ] = ½ ( Y [ t + At ] + ...