, l2 (0, 1], look at the attributes:(A1 ) : ( p, 1) = 1; (A2 ) : ( p, l1 ) = 0; (A3 ) : (., l
, l2 (0, 1], consider the attributes:(A1 ) : ( p, 1) = 1; (A2 ) : ( p, l1 ) = 0; (A3 ) : (., l1 ) = (., l2 ), whenever l1 = l2 .By the class of conformable Diversity Library custom synthesis functions C l , we mean the aggregate of all continuous functions p : [0, ) (0, 1] R attaining the attributes (A1 )A3 ) and also the steady function ( p, m1 ) = 1. Definition 2 ([6]). Suppose C l with ( p, l ) [0, ) (0, 1]. The action of the GDCO on a p function y : [0, ) R is recognized by the limit: Dlp y( p) = lim y( p + ( p, l )) – y( p) . 0 (1)The related integral operator for the GDCO is often as follows. Definition three ([6]). Let p [0, ), c [0, p] and l R. Further, let y : [c, p] R be a function. The generalized integral conformable operator at y is expressed by: Ilp y( p) = when the integral converges and C l . p Remark 1. If ( p, l ) = 1, then Dlp y( p) becomes the traditional integer-order derivative and l has no influence. Moreover, if ( p, l ) = p1-l , then Dlp y( p) agrees together with the differential conformable operator proposed in [13]. The subsequent outcomes offer some substantial traits of GDCOs. Theorem 1 ([6]). Let l (0, 1], y1 , y2 be l-differentiable functions at p R+ and C l . Then: p (i) Dlp (ry1 + sy2 ) = rDlp y1 + sDlp y2 , for all r, s R; (ii) Dlp ( pr ) = rpr-1 ( p, l ), for all r R; (iii) Dlp (y1 y2 ) = y1 Dlp y2 + y2 Dlp y1 ; y2 Dlp y1 – y1 Dlp y2 y (iv) Dlp 1 = ; y2 y2 2 dy (v) If y1 is differentiable, then Dlp (y1 ( p)) = ( p, l ) ; dp (vi) If y1 , y2 are differentiable, then Dlp (y1 y2 )( p) = ( p, l ) dy1 dy2 . dy2 dpp cy(t) dt, (t, l )(two)The GDCOs for multivariable functions may be defined partially as follows.l Definition 4 ([9]). Let k C p , ( pk , lk ) [0, ) (0, 1], and k = 1, . . . , n. In addition, let lk ,k y : [0, )n R be a function. The partial derivative y( p) at p = ( p1 , . . . , pn ) is defined by: l pkkMathematics 2021, 9,4 oflk ,kl pkky( p) = limy( p + (0, . . . , k ( pk , lk ), . . . , 0)) – y( p) ,(three)when it exists. Remark 2. If k ( pk , lk ) = 1, k = 1, . . . , n, then in Rn . lk ,k pkkly( p) will be the traditional partial derivative3. Methodology for Solving Stochastic NEEs with GDCOs This section explains our methodology for extracting exact wave options with the stochastic NEEs with GDCOs. This methodology combines the utilization of GDCOs, the tools of white noise evaluation, as well as the generalized Kudryashov scheme. Ahead of displaying our methodology, we equip the reader with some significant instruments of white noise analysis. Consider the Kondratiev stochastic space (k)n 1 with the orthogonal basis Hg g M , – exactly where M = g = ( g1 , g2 , . . . , gi , gi+1 , . . .) : gi N and i=1 gi [31]. If X and Y are n , then we’ve got X = GSK2646264 custom synthesis elements in (k )-1 g g Hg and Y = g g Hg with g , g Rn . The Wick multiplication of X and Y is expressed by: X Y=ig igHg+g.g, g i =n(4)Furthermore, the Hermite transform of X = g g Hg (k )n 1 has the expansion: -H( X ) = X (z) = g z g Cn ,g(5)where z = (zi )i1 CN and z g = i=1 z gi for ( gi )i1 M. The connection between the Wick multiplication and Hermite transform could be extracted by way of Equations (four) and (5) because the kind: X Y ( z ) = X ( z ) Y ( z ), (6)where X (z) and Y (z) are finite for all z along with the operation “is the bilinear multiplication n in Cn , which is specified by (z1 , . . . , zn ) (z1 , . . . , z ) = i=1 zi zi . For M , N 0, we n N because the form U ( N ) = z g [31]. Let X.