Application of Mellin and Hankel Transforms to Networks with Time-Varying Parameters

Integral transform techniques for solving linear integro-differential equations can provide insight and flexibility in solving physical problems, especially network problems. The type of differential equation which describes the physical system will dictate the transform that should be applied to simplify the solution and this paper deals with two transforms, namely, the Mellin transform and the Hankel transform. The Laplace transform can be used to solve linear constant coefficient differential equations or networks which are represented by this type of equation. A familiarity with this transform is assumed and is not covered in this paper. Mellin transforms may be applied to networks which yield the Euler-Cauchy differential equation. This transform will simplify the solution of such an equation. A transform table, similar to that type used in Laplace transform theory, is developed and applied to network problems. Hankel transforms may be applied to networks which yield the Bessel differential equation or variations of this equation. Unlike the Laplace and Mellin transforms, the Hankel transform is symmetric and the transformed variable is a real, rather than a complex variable. A transform table of both operations and functions is developed anti applied to network problems as before. Three methods can be used to establish the table of transform pairs. They can be described as: performing the integral operation, applying the table of operations on known transform pairs, and deriving the Hankel transform from the Laplace transform. With both transforms, the applications are made to problems in analysis, instrumentation, and synthesis.