Applications of Standard Methods for Solving the Electric Train Mathematical Model With Proportional Delay

Abstract

In electric trains, the current is collected via a certain device, called the Pantograph. The governing mathematical model of such physical problem is well-known as the Pantograph delay differential equation (PDDE): y’(t)=ay(t)+by(ct), where c is a proportional delay parameter. In the literature, a special case of the PDDE was analyzed when c=−1. The objective of this paper is to determine the general solution of the PDDE for arbitrary c. In addition, it will be shown that the obtained general solution reduces to exact one at c=−1. Such exact solution is expressed in terms of several types of functions, e.g., hyberbolic, Mittag-Leffler, and trigonometric functions. Moreover, it is declared that the exact trigonometric solution is periodic with periodicity 2π/√b2−a2 which agrees with the corresponding results in the literature. Furthermore, the solution of PDDE is provided at almost all possible cases of the involved parameters a, b, and c. Finally, the solution Ambartsumian delay equation (ADE), which has an application in the theory of surface brightness in the Milky Way, will be recovered as a special case of our results.