Analysis of a First-Order Delay Model under a History Function with Discontinuity

Abstract

This paper analyzes the first-order delay equation y′(t)=αy(t)+βy(t−τ) subject to a history function in addition to an initial condition that assumes discontinuity at t=0. The method of steps is successfully applied to derive the exact solution in an explicit form. In addition, a unified formula is provided to describe the solution in any finite sub-interval of the problem’s domain. The characteristics and properties of the solution are theoretically investigated and then confirmed through several plots. The behavior of the solution and its derivative are examined and interpreted. The results show that the method of steps is an effective method of solution to treat the current delay model. The present successful analysis can be used to investigate other delay models with complex initial conditions. Furthermore, the present approach can be generalized to include the inhomogeneous version of the current model without using numerical methods.