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By R. Keith Dennis

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1. We have as usual u = Σ ^ = 0 w„(0 a n d Nu = uy = £ ^ = 0 An9 where the An are generated for this nonlinearity. Thus, Lu = — uy, L~lLu= — L _1 w y , then u = w(0) — L~ 1uy, consequently u0 = u(0) = k and ul = —L~ iA0(uy),... ^o = u o Ax =yuy0~1ul A2 = yuy0~lu2 + \y(y - \)uy0~2u\ A3 = yuy0'lu3 + y(y - 1)MJ" 2 M 1 M 2 + ^y(y - l)(y - 2)M&"3M? Λ 4 = 7^ _ 1 w 4 + 7(7 - ΐ)ιιδ" 2 (έ"2 + w i" 3 ) + h(y - i)(y - 2 K " 3 « ? (7-1)(7-2)(7-3Κ-χ Hence, "o = k "l -kyt = u2 = yk2y~l "3 t2 2! \)k3y- -2 = -7(27 IV "Xv — fi t3 L 3!

44 4. 13) where /c(i, τ) = Ζ(ί, τ)α(τ). 14) do k(t, T)/C(T, y)/c(y, a)F(a) + If Li has constant coefficients, /(ί, τ) = Ζ(ί — τ). F o r simplicity and clarity, let us consider the example L = Lx + a with a a constant and L\γ = \dt and the Green's function / = 1. O u r objective is to determine the Green's function G(i, τ) for L. 15) Thus, G can be found from the preceding equations by replacing x by the δ function. 4. 45 APPROXIMATING DIFFICULT GREENS FUNCTIONS Remembering that k(t, τ) = l(t — τ)α = a, Gl!

Y(v)(0) Consider now the linear stochastic differential equation in the form S£y = x; where x is a stochastic process on T x Ω, where (Ω, F, μ) is a probability space and j£? a stochastic operator, in this case, a linear ordinary differential stochastic operator. > since a simple Green's function is desirable. We are still decomposing i f into deterministic and random parts (but into L + R + 0t) and solving for Ly as before. , η — 1, so that 0t is zero-mean. 19) if we allow 0t to have a mean value and identify it as Σ"Ζΐαν(ί,ω)άν/άίν.

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