An efficient computational technique for solving linear delay differential equations with a piecewise constant delay function is presented. The new approach is based on a hybrid of block-pulse functions and Legendre polynomials. A key feature of the proposed framework is the excellent representation of smooth and especially piecewise smooth functions. The operational matrices of delay, derivative, and product corresponding to the mentioned hybrid functions are implemented to transform the original problem into a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the proposed numerical scheme.
Delay differential equations (DDEs) naturally arise in diverse areas of science and engineering such as transmission lines, communication networks, biological models, population dynamics, and transportation systems [
In what follows, we describe some similarities and differences between our method and the procedures developed in [
The rest of the paper is organized as follows. In Section
Hybrid functions
In this section, we first present the operational matrix of derivative based on the weak representation of the derivative operator. We then state the operational matrices of delay and product corresponding to the proposed hybrid functions.
A function
We approximate the derivative of
The relationship between the two vectors
The structure of the operational matrix of derivative
In the case
Let
The goal of this subsection is to determine the operational matrix of delay associated with the developed hybrid functions. For this purpose, let
To obtain the operational matrix of delay corresponding to the proposed hybrid functions, we apply an approach analogous to the one devised in [
Define
Therefore, the problem is reduced to find the operational matrix of delay for the following delay function:
In order to find the matrix
With the use of (
If
As a consequence, if we expand
To illustrate the derivation process of the operational matrix of delay associated with the mentioned hybrid functions, we present an example. Define
Consequently
Consider the linear time-varying piecewise constant delay system described by
We approximate the system dynamics by hybrid functions as follows. Let
Now, using (
Therefore, the original problem is transformed into a system of linear algebraic equations. The aforementioned system with the associated initial condition can be easily solved by the well-known Tau method [
In this section, three examples are investigated to evaluate the performance of the method.
Consider the time-varying piecewise constant delay system
Consider the following piecewise constant delay system [
Results of maximum error for Example
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As a more complicated problem, consider the following nonlinear piecewise constant delay system:
An efficient procedure has been successfully developed for solving delay differential equations with a piecewise constant delay function. The method is based upon a hybrid of block-pulse functions and Legendre polynomials. The nice properties of the hybrid functions together with the associated operational matrices were used to convert the original problem into a system of algebraic equations. The proposed framework allows one to simultaneously make use of the best advantages of the two mentioned bases. After determining the appropriate value of
The authors declare that they have no competing interests.