Separate node ascending derivatives expansion (SNADE) on complex plane

Abstract

\begin{abstract} In this study we focus on the method called \lq\lq Separate % Node Ascending Derivatives Expansion \rq\rq ( SNADE ) % \cite {MD1,NABMD1,NABEG1,EGNAB1}which is % obtained as a result of the studies recently carried out in % Group for Science and Methods of Computing ( G4S \&MC ) under % the leadership of Metin Demiralp . SNADE is considered as a new % type power series involving denumerable infinitely many nodes, % like Taylor Series Expansion. However, each position is accom % panied by a different derivative value in this method, different % ly from Taylor Series Expansion. SNADE is based on the use of % derivative integration formula for a univariate function in dif % ferent nodal values. In this constructed structure, when integ % ral operators defined as % %--------------------------------(1)----------------------------------% \begin {eqnarray} \label{eq:1} \mathcal {I} _m ( x_1 , \cdots,x_m )g(x) \equiv \int_ { x_1 }^x d \xi_1 % \int_ { x_2 }^{ \xi_1 }d \xi_2 % \cdots\int_ { x_m }^{ \xi_ {m-1}} d \xi_m g( \xi_m ), % \nonumber \\ % m=1,2, \cdots, \quad \mathcal {I} _0g (x) \equiv g(x) \nonumber \end {eqnarray} %--------------------------------(1)----------------------------------% are used, it is possible to write the following formula for SNADE . % %--------------------------------(2)----------------------------------% \begin {eqnarray} \label{eq:2} f(x)= \sum_ {i=0}^ \infty f^{(i)}( x_ {i+1}) \mathcal {I} _i ( x_1 , % \cdots,x_i ) 1_f \nonumber % \end {eqnarray} %--------------------------------(2)----------------------------------% Heretofore, relations related to constructions which are called as % SNADE polynomials were obtained and the theoretical structure of % the method was composed on these findings \cite { DBMD1 , % DBMD2 }. By using these constructs, convergence of the method was % investigated and findings were supported by numerical % implementations. In this presentation we will focus on the transition from real- val % uedness to complex- valuedness and relations related with this meth % od will be reconstructed by taking domains in complex plane into % account. Convergence issues will also be considered on a complex % plane. \end{abstract}