Accuracy of the Laplace transform method for linear neutral delay differential equations

Kerr, Gilbert; González-Parra, Gilberto

doi:10.1016/j.matcom.2022.02.017

Cited by 12 publications

(35 citation statements)

References 37 publications

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“…The non-neutral DDEs and NDDEs established in [12] are presented and discussed in the following two sections, then we employ Maple software to identify solutions based on the MoS technique.…”

Section: A Maple Procedures For Implementing a Methods Of Stepsmentioning

confidence: 99%

General Maple Code for Solving Scalar Linear Neutral Delay Differential Equations

Bahgat

2023

Sohag Journal of Sciences

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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.

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Section: A Maple Procedures For Implementing a Methods Of Stepsmentioning

confidence: 99%

General Maple Code for Solving Scalar Linear Neutral Delay Differential Equations

Bahgat

2023

Sohag Journal of Sciences

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“…and N j is the matrix formed by replacing column j in D with the vector v h . If all of the entries in the history vector H(t) are bounded: h 1 < H(t) < h 2 , and integrable for t ∈ (−τ, 0), then it can be shown that any singularities resulting from the integral term in v h are removable [25]. In which case all of the relevant poles are determined by the roots of the (characteristic) equation…”

Section: Solving Linear Systems Of Ddes By the Ltmmentioning

confidence: 99%

“…A key step in the LTM solution process, when it is applied to linear DDEs, is to evaluate the poles of the transformed equation, in the s-space. There are infinitely many poles and different methods can be used to find them [16,25,31,34]. The poles are then used to compute the ILT, by means of the Cauchy residue theorem [41,42].…”

Section: Introductionmentioning

confidence: 99%

“…The poles are then used to compute the ILT, by means of the Cauchy residue theorem [41,42]. The resulting solutions are then obtained, in the form of an infinite non-harmonic Fourier series [25,[43][44][45]. We present a variety of examples illustrating the methodology in order to facilitate a discussion on how the spectral properties affect the approach and the solutions.…”

Section: Introductionmentioning

confidence: 99%

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Analytical solutions of systems of linear retarded and neutral delay differential equations by the Laplace transform: featuring limit cycles

Kerr,

Lopez,

Gonzalez-Parra

2023

Preprint

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In this paper we develop an approach for obtaining the solutions to systems of linear retarded and neutral delay differential equations. Our analytical approach is based on the Laplace transform, inverse Laplace transform and the Cauchy residue theorem. The obtained solutions have the form of infinite non-harmonic Fourier series. The main advantage of the proposed approach is the closed-form of the solutions, which are capable of accurately evaluating the solution at any time. Moreover, it allows one to study the asymptotic behavior of the solutions. A remarkable discovery, which to the best of our knowledge has never been presented in the literature, is that there are some particular linear systems of both retarded and neutral delay differential equations for which the solution approaches asymptotically to a limit cycle. The well-known method of steps in many cases is unable to obtain the asymptotic behavior of the solution and would most likely fail to detect such cycles. Examples illustrating the Laplace transform method for linear systems of DDEs are presented and discussed. These examples are designed to facilitate a discussion on how the spectral properties of the matrices determine the manner in which one proceeds and how they impact the behavior of the solution. Comparisons with the exact solution provided by the method of steps are presented. Finally, we should mention that the solutions generated by the Laplace transform are, in most instances, extremely accurate even when the truncated series is limited to only a handful of terms and in many cases becomes more accurate as the independent variable increases.

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“…Using Raabe's test [38], we approximate an expression in the last term of inequality (40) in the following manner:…”

Section: Stability Analysismentioning

confidence: 99%

Numerical solutions, Convergence and Global stability of delayed differential system with delay in the state variable of the Caputo derivative

Kumar,

Sharma,

Singh

2023

Preprint

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The aim of this paper is to investigate the numerical solutions of delayed fractional differential equations (FDEs). A constant time delay is taken in the Caputo derivative of the state variable and also in the state variable. To obtain solutions, we extend the block boundary value method (BBVM) and show the convergence of the resulting scheme. Further, it is found that the order of convergence is $\min\{m,q-\delta+1\}$, where $m$ and $q$ are the order and block-size of the BBVM, respectively, and $\delta$ lies in between $0$ and $1$. In our methodology, we approximate the fractional-ordered derivative by a combination of the $m^{\text{th}}$-order BBVM and a $q^{\text{th}}$-order Lagrange interpolating polynomial. Lastly, we analyze the global stability of the numerical scheme and with the aid of some numerical examples, its computational effectiveness is proved. In examples, we analyse some fractional order delay differential systems and present a comparison of solutions for different fractional order derivatives. Furthermore, it is demonstrated that the global errors diminish as the order of the fractional derivative decreases, and this fact is supported by the theoretical order of convergence. MSC Classification: 34K37 , 34D23 , 65L20

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Accuracy of the Laplace transform method for linear neutral delay differential equations

Cited by 12 publications

References 37 publications

General Maple Code for Solving Scalar Linear Neutral Delay Differential Equations

General Maple Code for Solving Scalar Linear Neutral Delay Differential Equations

Analytical solutions of systems of linear retarded and neutral delay differential equations by the Laplace transform: featuring limit cycles

Numerical solutions, Convergence and Global stability of delayed differential system with delay in the state variable of the Caputo derivative

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