A new method based on the Laplace transform and Fourier series for solving linear neutral delay differential equations

Kerr, Gilbert; González-Parra, Gilberto; Sherman, Michele

doi:10.1016/j.amc.2021.126914

Cited by 8 publications

(10 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The traditional method for solving DDEs is with an elementary method called the method of steps [17,34,35,[48][49][50]. It is well-known for being a tedious process when being worked out by hand, but the complexity can be drastically simplified with symbolic toolboxes in computer programs [15]. Let us consider a NDDE with fixed delay τ:…”

Section: Methods Of Steps (Mos)mentioning

confidence: 99%

“…In this paper, we focus on comparing the symbolic mathematical computations of Maple and MATLAB to obtain analytical solutions of linear neutral and non-neutral DDEs by using the LT method. The accuracy of the LT solutions may be determined by comparing them with the MoS solutions [12][13][14][15][16][17]. However, the MoS has a very important limitation since it is a stepwise process that requires solving the relevant differential equation in each respective time interval, which can be computationally expensive.…”

Section: Introductionmentioning

confidence: 99%

“…Thus, we can compute the approximated solution by evaluating it at any point. Other advantages of the analytical solutions, such as the long-term behavior of the solution, can be seen in [14,15,[18][19][20][21][22]. The errors of the LT solutions usually decrease as time increases.…”

Section: Introductionmentioning

confidence: 99%

“…Regarding solving ordinary differential equations (ODEs) and DDEs, the latter is usually more difficult to solve since the derivatives of the state variables depend on the values of the state variable from the past. The classic approach used to obtain the solutions of DDEs is the MoS, but, in many cases, it is unfeasible to advance further in time and obtain the solution in the long term [14,15,[34][35][36]. In [14], the authors studied the accuracy of the LT for linear DDEs and, in particular, for NDDEs.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Sherman

Kerr

González-Parra

2022

MCA

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Methods Of Steps (Mos)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Sherman

Kerr

González-Parra

2022

MCA

Self Cite

View full text Add to dashboard Cite

show abstract

“…The imitative method for solving DDEs is a primary method called the method of steps [1][2][3][4][5][6][7][8][9][10]. When performed manually, it is generally known to be a laborious operation; however, using symbolic toolboxes in computer systems, the complexity can be greatly reduced [11]. Consider the following neutral delay differential equation with a fixed delay 𝛽:…”

Section: Introductionmentioning

confidence: 99%

General Maple Code for Solving Scalar Linear Neutral Delay Differential Equations

Bahgat

2023

Sohag Journal of Sciences

View full text Add to dashboard Cite

The objective is to demonstrate how to create straightforward Maple programs for numerical computations and programs for condensing or changing mathematical formulas, polynomials, or symbolic expressions. It is assumed that readers are accustomed to using interactive Maple. The programming language used in Maple is interpreted and interactive. Due to the overhead of the interpreter, Maple is not appropriate for running programs that require a lot of numbers. Although it can be used to create numerical codes and for high-precision numerical calculations. This paper uses Maple's general code to compute the method of steps (MoS) solution of linear neutral and delay differential equations (DDEs). The paper relies on entering simple inputs to get a quick result explained by theoretical solutions and their graphics. A Maple symbolic computation was more efficient than a programming language computation for linear non-neutral DDEs. Maple was faster at solving linear neutral DDEs, which are usually more challenging. Using the MoS methodology, we list and talk about various examples of non-neutral DDEs and NDDEs reported in the literature.

show abstract

An approaching method based on integral for linear neutral delay differential equations by using Hermite polynomials

Yüzbaşı,

Karaağaçlı

2024

Int J Numerical Modelling

View full text Add to dashboard Cite

This article presents a approximation method for linear neutral delay differential equations using Hermite polynomials. The method ensures the unknown function to be found by approximating the first derivative with the help of the finite Hermite series. This method reduces the problem to a linear algebraic system using the first derivative approach, matrix operations and collocation points. Also error of the solution is estimated by constructing an error problem. Numerical examples are solved to explain the method and error estimation technique. The results demonstrate the effectiveness and accuracy of the current study. Calculations have been made using MATLAB.

show abstract

A new method based on the Laplace transform and Fourier series for solving linear neutral delay differential equations

Cited by 8 publications

References 39 publications

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

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

General Maple Code for Solving Scalar Linear Neutral Delay Differential Equations

An approaching method based on integral for linear neutral delay differential equations by using Hermite polynomials

Contact Info

Product

Resources

About