Exact solution to a multidimensional wave equation with delay

Jornet, Marc

doi:10.1016/j.amc.2021.126421

Cited by 5 publications

(3 citation statements)

References 29 publications

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“…An expression for the exact solution of ()–() was presented in Rodríguez et al [13], as a preliminary problem resulting from applying the method of separation of variables to an unidimensional wave equation with delay. In that context, the condition

a&lt;0

was assumed, writing

a&#x0003D;-{\alpha}&#x0005E;2

, but, as shown in Jornet [14], where it was applied to solve a more general multidimensional wave equation, it is straightforward to adapt the expression given in Rodríguez et al [13] to the case

a&gt;0

. Notwithstanding, these expressions do not seem to be suitable for constructing an exact numerical scheme, and so providing the basis for an efficient computational method.…”

Section: Introductionmentioning

confidence: 99%

Exact numerical solutions and high order nonstandard difference schemes for a second order delay differential equation

Castro

Mayorga

Sirvent

et al. 2023

Math Methods in App Sciences

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In this work, we obtain an expression, given in terms of some special functions, for the exact numerical solution of the second order delay differential equation with general initial condition for . Based on this exact solution, we propose a family of nonstandard finite difference methods that combine high order accuracy and efficient computational properties. We also show that the numerical solutions obtained with the new schemes are consistent with asymptotic stability properties of the exact solutions.

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a&lt;0

was assumed, writing

a&#x0003D;-{\alpha}&#x0005E;2

, but, as shown in Jornet [14], where it was applied to solve a more general multidimensional wave equation, it is straightforward to adapt the expression given in Rodríguez et al [13] to the case

a&gt;0

. Notwithstanding, these expressions do not seem to be suitable for constructing an exact numerical scheme, and so providing the basis for an efficient computational method.…”

Section: Introductionmentioning

confidence: 99%

Exact numerical solutions and high order nonstandard difference schemes for a second order delay differential equation

Castro

Mayorga

Sirvent

et al. 2023

Math Methods in App Sciences

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“…The time delay can make the equation unstable and periodic solutions can appear (Smith 2011). There are a variety of studies devoted to developing methods for finding analytical or numerical solutions of DDEs (Bellour et al 2020;Chamekh et al 2019;Cimen and Uncu 2020;Ebaid et al 2019;Eftekhari 2015;Jaaffar et al 2020;Jamilla et al 2020a;Jhinga and Daftardar-Gejji 2019;Jornet 2021;Peykrayegan et al 2021;Senu et al 2022;Shampine and Thompson 2009). For instance, in García et al (2018), the authors developed nonstandard numerical schemes for linear delay-differential models.…”

Section: Introductionmentioning

confidence: 99%

Analytical solutions of linear delay-differential equations with Dirac delta function inputs using the Laplace transform

2023

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In this paper, we propose a methodology for computing the analytic solutions of linear retarded delay-differential equations and neutral delay-differential equations that include Dirac delta function inputs. In numerous applications, the delta function serves as a convenient and effective surrogate for modeling high voltages, sudden shocks, large forces, impulse vaccinations, etc., applied over a short period of time. The solutions are obtained using the Laplace transform method, in conjunction with the Cauchy residue theorem. The accuracy of these solutions are assessed by comparing them with the ones provided by the method of steps. Numerical examples illustrating the methodology are presented and discussed. These examples show that the Laplace transform solution is very reliable for linear retarded delay-differential equations, because the analytic solution, for a single delta function input, is continuous. However, for linear neutral delay-differential equations with a delta function input the analytic solution is discontinuous. Consequently, the well-known Gibbs phenomenon is observed in the vicinity of the discontinuities. However, for neutral delay differential equations, we show that in some cases, the magnitude of the jumps at the discontinuities decrease, as time increases. Therefore, the Gibbs phenomenon of the Laplace solution dissipates.

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“…The wave equation is a partial differential equation for a scalar function that describes the propagation phenomenon in different areas of engineering, physics, and scientific applications [19,20]. Wazwaz [21] studied linear and nonlinear problems in bounded and unbounded domains using the variational iteration method.…”

Section: Introductionmentioning

confidence: 99%

A Novel Approach for the Approximate Solution of Wave Problems in Multi-Dimensional Orders with Computational Applications

Nadeem

Akgül

Guran

et al. 2022

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The main goal of this paper is to introduce a new scheme, known as the Aboodh homotopy integral transform method (AHITM), for the approximate solution of wave problems in multi-dimensional orders. The Aboodh integral transform (AIT) removes the restriction of variables in the recurrence relation, whereas the homotopy perturbation method (HPM) derives the successive iterations using the initial conditions. The convergence analysis is provided to study a wave equation with multiple dimensions. Some computational applications are considered to show the efficiency of this scheme. Graphical representation between the approximate and the exact solution predicts the high rate of convergence of this approach.

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Exact solution to a multidimensional wave equation with delay

Cited by 5 publications

References 29 publications

Exact numerical solutions and high order nonstandard difference schemes for a second order delay differential equation

Exact numerical solutions and high order nonstandard difference schemes for a second order delay differential equation

Analytical solutions of linear delay-differential equations with Dirac delta function inputs using the Laplace transform

A Novel Approach for the Approximate Solution of Wave Problems in Multi-Dimensional Orders with Computational Applications

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