An Operator Approach for Physical Modelling of Distributed Parameters Systems

Many physical and information processes in various fields possess an identical mathematical structure, that can be described in a common operator form. Also, the operator approach is frequently used as an analysis tool to solve a wide variety of theoretical problems. An operator-based approach for solving initial-boundary-value problems with homogeneous boundary conditions has recently been introduced in (Dymkou, et al., 2004). The key point of the proposed method is that the solution of the considered system models can be expressed in terms of the corresponding semigroup. This semigroup, in turn, can be represented by chains of eigenfunctions and associated eigenfunctions generated by the spatial differential operator and its adjoint operator. These operators are unbounded and non-self-adjoint, in general. Similar to the Laplace transformation L used for the time variable t, the obtained decomposition of the solution is then used to determine the new multi-functional transformation T (MFT) which allows to replace the continuous spatial variable x by a discrete spatial frequency variable. Therefore, successive application of both transformations yields a solution that can be calculated by purely algebraic operations in the multidimensional frequency domain. This leads to major new results for the discrete simulation of many physical processes. Figure 1 illustrates the main concept of the application of MFT method to initial-boundary-value problems.