Plenary lecture III: space extension based extended fluctuationlessness theorem
Metin Demіralp · International Conference on Microelectronics · 2008
Fluctuationlessness Theorem is a recently created very efficient tool for matrix representations. It dictates us that the matrix representation of an algebraic operator which multiplies its argument by a scalar univariate function is identical to the the image of the independent variable's matrix representation over the same space via same basis set, under that univariate function. This helps us to create very rapidly converging univariate numerical integration schemes which can be used in many diversive areas of science and engineering. The multivariate counterpart of this theorem has also been conjectured and proven quite recently. In these theorems, the matrix representations are defined on Hilbert spaces which are defined through certainappropriate inner products. On the other hand, the space extension methods aim at the building of simple structures which can be handled by well-known techniques to get solutions many applied mathematical problems. There the number of unknowns are increased to put the equation to be solved to an amenable form. Quite recently, this method is applied to the solution of ordinary differential equations and a universal form which leads us to use two-term recursions is obtained. This was for linear case. By using same approach in an indirect way via partial differential operators we could be able to deal with a quite large class nonlinear ODEs. This talk is about the extension of two fluctuationlessness theorems for higher accuracy with the aid of space extension.