Unified approach to the impulse response and green function in the circuit and field theory part II : multi–dimensional case

In the circuit theory the concept of the impulse response of a linear system due to its excitation by the Dirac delta function δ(t) together with the convolution principle is widely used and accepted. The rigorous theory of symbolic functions, sometimes called distributions, where also the delta function belongs, is rather abstract and requires subtle mathematical tools [1-4]. Nevertheless, the most people intuitively well understand the delta function as a derivative of the (Heaviside) unit step function 1(t) without too much mathematical rigor. In the previous part [5] the concept of the impulse response of linear systems was approached in a unified manner and generalized to the time-space phenomena in one dimension (transmission lines). Here the phenomena in more dimensions (static and dynamic electromagnetic fields) are treated. It is shown that many formulas in the field theory, which are often postulated in an inductive way as results of the experiments, and therefore appear as “deux ex machina” effects, can be mathematically deduced from a few starting equations.

Unified approach to the impulse response and green function in the circuit and field theory, part I: one–dimensional case

In the circuit theory the concept of the impulse response of a linear system due to its excitation by the Dirac delta function ƍ(t) together with the convolution principle is widely used and accepted. The rigorous theory of symbolic functions, sometimes called distributions, where also the delta function belongs, is rather abstract and requires subtle mathematical tools [1], [2], [3], [4]. Nevertheless, the most people intuitively well understand the delta function as a derivative of the (Heaviside) unit step function 1(t) without too much mathematical rigor. The concept of the impulse response of linear systems is here approached in a unified manner and generalized to the time-space phenomena in one dimension (transmission lines), as well as in a subsequent paper [5] to the phenomena in more dimensions (static and dynamic electromagnetic fields).

Another Approach to the Extended Stokes’ Problems for the Oldroyd-B Fluid

The extended Stokes problems, which study the flow suddenly driven by relatively moving half-planes, are reexamined for the Oldroyd-B fluid. This topic has been studied (Liu, 2011) by applying the series expansion to calculate the inverse Laplace transform. The derived solution was correct but tough to perform the calculation due to the series expansion of infinite terms. Herein another approach, the contour integration, is applied to calculate the inversion. Moreover, the Heaviside unit step function is included into the boundary condition to ensure the consistence between boundary and initial conditions. Mathematical methods used herein can be applied to other fluids for the extended Stokes’ problems.

Vibration of Circular Rings With Local Deviation

An analytical method is presented for predicting the effects of local deviations such as a point mass, a sharp decrease in stiffness and variation of thickness on the free in-plane bending vibration characteristics of nearly axisymmetric rings. The approach is based on the Laplace transformation method and the expression of local deviation as the variation of heaviside unit step function. The effects of local deviation on the natural frequencies and mode shapes of the rings are predicted for the proposed nondimensional mass and stiffness parameters. The validity of the approach was examined through modal testings and/or finite element computation.