Relation of Integro-Partial Differential Equations With Delay Effect Based on the Maxwell Equations to the Heaviside and Pocklington Equations

We derive the integro-partial differential equations with delay effect based on the Maxwell equations for a two-line system under the thin-wire approximation and the approximation of neglecting the interwire distance in the delay term. We introduce the normal- and common-mode and write the integro-partial differential equations in terms of these two modes. For the case of the same radius, we show that the equation of the normal-mode reduces to the Heaviside telegraph equation in the partial differential form and the equation of the common-mode becomes the Pocklington equation in the integro-partial differential form. We perform numerical calculations of the coupled integral and partial differential equations in the time domain and compare the results with those of various approximations. We show that the Heaviside telegraph equation is valid for the normal-mode, while the full treatment of the delay term in the integro-partial differential equations is necessary for the common-mode.