Comparison of Symbolic Computations for Solving Linear Delay Differential Equations Using the Laplace Transform Method

In this paper, we focus on investigating the performance of the mathematical software program Maple and the programming language MATLAB when using these respective platforms to compute the method of steps (MoS) and the Laplace transform (LT) solutions for neutral and retarded linear delay differential equations (DDEs). We computed the analytical solutions that are obtained by using the Laplace transform method and the method of steps. The accuracy of the Laplace method solutions was determined (or assessed) by comparing them with those obtained by the method of steps. The Laplace transform method requires, among other mathematical tools, the use of the Cauchy residue theorem and the computation of an infinite series. Symbolic computation facilitates the whole process, providing solutions that would be unmanageable by hand. The results obtained here emphasize the fact that symbolic computation is a powerful tool for computing analytical solutions for linear delay differential equations. From a computational viewpoint, we found that the computation time is dependent on the complexity of the history function, the number of terms used in the LT solution, the number of intervals used in the MoS solution, and the parameters of the DDE. Finally, we found that, for linear non-neutral DDEs, MATLAB symbolic computations were faster than Maple. However, for linear neutral DDEs, which are often more complex to solve, Maple was faster. Regarding the accuracy of the LT solutions, Maple was, in a few cases, slightly better than MATLAB, but both were highly reliable.