An Analysis of the Theta‐Method for Pantograph‐Type Delay Differential Equations

An innovative singular nonlinear sixth-order (SNSO) pantograph differential model (PDM), known as the SNSO-PDM, is the subject of this novel study along with its numerical investigation. The concepts of pantograph and conventional Emden-Fowler have been presented in the design of the novel SNSO-PDM. The models based on Emden–Fowler have huge applications in mathematics and engineering and are always difficult to solve due to singularity. For each class of the innovative SNSO-PDM, the singularity, shape and pantograph factors are described. A reliable stochastic Levenberg-Marquardt backpropagation neural network (LMBPNN) procedure is designed for the SNSO-PDM. The correctness of the SNSOs-PDM is observed through the comparison performances of the achieved and reference outputs. The obtained results of the SNSO-PDM are considered by applying the process of training, certification, and testing to reduce the mean square error. To authenticate the efficacy of the innovative SNSO-PDM, the numerical performances of the solutions are depicted in the sense of regression, error histograms and correlation.

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Exact and Numerical Analysis of the Pantograph Delay Differential Equation via the Homotopy Perturbation Method

Albidah

Kanaan

Ebaid

et al. 2023

Mathematics

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The delay differential equations are of great importance in real-life phenomena. A special type of these equations is the Pantograph delay differential equation. Generally, solving a delay differential equation is a challenge, especially when the complexity of the delay terms increases. In this paper, the homotopy perturbation method is proposed to solve the Pantograph delay differential equation via two different canonical forms; thus, two types of closed-form solutions are determined. The first gives the standard power series solution while the second introduces the exponential function solution. It is declared that the current solution agrees with the corresponding ones in the literature in special cases. In addition, the properties of the solution are provided. Furthermore, the results are numerically validated through performing several comparisons with the available exact solutions. Moreover, the calculated residuals tend to zero, even in a huge domain, which reflects the high accuracy of the current analysis. The obtained results reveal the effectiveness and efficiency of the current analysis which can be further extended to other types of delay equations.

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An Analysis of the Theta‐Method for Pantograph‐Type Delay Differential Equations

Cited by 3 publications

References 30 publications

A novel numerical approach and stability analysis for a class of pantograph delay differential equation

A novel numerical approach and stability analysis for a class of pantograph delay differential equation

A reliable neural network procedure for the novel sixth-order nonlinear singular pantograph differential model

Exact and Numerical Analysis of the Pantograph Delay Differential Equation via the Homotopy Perturbation Method

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