THE UNIFIED INTEGRAL TRANSFORMS (UNIT) ALGORITHM WITH TOTAL AND PARTIAL TRANSFORMATION

Abstract

The theory and algorithm behind the open-source mixed symbolic-numerical computational code named UNIT (unified integral transforms) are described. The UNIT code provides a computational environment for finding solutions of linear and nonlinear partial differential systems via integral transforms. The algorithm is based on the well-established analytical-numerical methodology known as the generalized integral transform technique (GITT), together with the mixed symbolic-numerical computational environment provided by the Mathematica system (version 7.0 and up). This paper is aimed at presenting a partial transformation scheme option in the solution of transient convective-diffusive problems, which allows the user to choose a space variable not to be integral transformed. This approach is shown to be useful in situations when one chooses to perform the integral transformation on those coordinates with predominant diffusion effects only, whereas the direction with predominant convection effects is handled numerically, together with the time variable, in the resulting transformed system of one-dimensional partial differential equations. Test cases are selected based on the nonlinear three-dimensional Burgers' equation, with the establishment of reference results for specific numerical values of the governing parameters. Then the algorithm is illustrated in the solution of conjugated heat transfer in microchannels.