By George Adomian
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Extra info for Nonlinear Stochastic Operator Equations
Divided by the product of factorials of the superscripts for a given term. ) = 20. The last term of A4 has the coefficient 5 ! /(2 ! )(2 ! ) = 30. This makes a nice final check on our generation of the An. y 6 -f 20^>>2)>5 + 20yly3y4 + 2 0 y ^ ^ o + 20yly0y4 + 3 0 ^ y ^ 3 + 3 0 ^ ^ ^ ! + 30y%ylys + 60^37^2^4 + A 60yiyoy2y3 s = ^ . 5. >; 2 y 7 4- ó O ^ o ^ ^ ^ s 4- ö O y f r o ^ s + 6 ( W ^ 2 y 6 + ^20y0yly2y3y4 Case 2: Aty = y4. ) A0 = yî ^2 = *yly2 + öyg);? >;2 ^4 = yi + 4 y ^ 4 + 6yly\ 4- 1 2 y ^ ^ 3 4- 12yfr0)>2 ^ 5 = ^ 0 ^ 5 + 4j>î)>2 + 12 :VoJ>i}>4 + 12^gy 2 y3 + 1 2 ^ 0 ^ 3 + i 2y22y0y1 A6 = 4 ^ 6 + fyiy* + *yho + 6 ) ^ 1 4- 6 y ^ 4- 1 2 ^ !
Wiley, New York, 1959. 1. Fy — x, where χ(ί, ω) is a stochastic process defined on a suitable probability space (Ω, F, μ). if is an nth order linear differential operator which is conveniently written in the form if = Xv = o αν(Α ω) dy/df with the requirement that an is not stochastic. , an_l are stochastic processes on (Ω,F,μ). 1) We can write 5£ — L + 0t, where L= W = dn d? , each αν(ί, ω) = <αν(ί, ω)> + αν(ί, ω), where <αν> exists and is continuous on T and av is the fluctuating part of av but is not restricted to be a stationary or other special process.
SOLUTION OF DIFFERENTIAL EQUATIONS A two-term approximation φ2 — y0 + >Ί is therefore given by *' = Ι ί + 3-ΐ3 + 2 Ϊ5 + 63 Consequently , 2 , ί6 2t * 2 / ~ t2 + 1 - - - — - t2 , 2 , i6 2t4 ft3 V = i2 + l thus verifying the solution. It is worthwhile to remark further concerning these verifications. , in substituting for y 2 we use only y0t not y 0 + );χ as in the y' term. In approximating the derivative term up to yi9 the ^ * = 0 An should include only A0 since yY depends on A0. We will see that the nonlinear terms expanded in the An polynomials approach zero for high n so that we get an accurate solution and, as we will see, generally only a few terms are sufficient for most purposes.