A Highly Accurate Modified Laplace-Fourier Method for Solving Linear Neutral Delay Differential Equations

Core Concepts

A new modified Laplace-Fourier method is developed that significantly improves the accuracy of solutions for linear neutral delay differential equations compared to the pure Laplace and original Laplace-Fourier methods.

Abstract

The article presents a new modified Laplace-Fourier method for efficiently solving linear neutral delay differential equations (NDDEs). The key highlights are:

The new modified method combines the Laplace transform and Fourier series theory to obtain solutions, but with crucial improvements over the original Laplace-Fourier method.

The main enhancement is deriving a new asymptotic expansion formula that minimizes the error between the actual residues and those obtained from the approximation. This leads to much more accurate solutions.

The convergence rate of the new modified Laplace-Fourier solution is shown to have a remarkable order of O(N^-3), where N is the number of terms in the truncated series. This is a significant improvement over the O(N^-1) convergence of the pure Laplace method.

Illustrative examples are provided that demonstrate the superior accuracy of the new modified method compared to the pure Laplace and original Laplace-Fourier approaches, without increasing the number of terms in the solution.

The new modified Laplace-Fourier method retains all the beneficial features of the original method, such as accounting for the tail of the infinite series to obtain more accurate solutions.

The solutions generated by the new modified method are very close to the analytical solutions, with impressively small errors that effectively become negligible as time increases.

Overall, the new modified Laplace-Fourier method is shown to be a highly efficient and accurate approach for solving linear NDDEs.

Stats

The convergence rate of the new modified Laplace-Fourier solution has a remarkable order of O(N^-3).
The pure Laplace method has a convergence rate of O(N^-1).

Quotes

"The new modified Laplace-Fourier method generates more accurate solutions than the ones generated by the pure Laplace method and the original Laplace-Fourier method."
"We have shown that the convergence rate of the new modified Laplace-Fourier solution has a remarkable order of convergence O(N^-3)."

Key Insights Distilled From

A new modified highly accurate Laplace-Fourier method for linear neutral delay differential equations

by Gilbert Kerr... at arxiv.org 04-25-2024

https://arxiv.org/pdf/2404.15291.pdf

Deeper Inquiries

How can the new modified Laplace-Fourier method be extended to solve nonlinear neutral delay differential equations

The new modified Laplace-Fourier method can be extended to solve nonlinear neutral delay differential equations by incorporating nonlinear terms into the equations and adjusting the method to handle the additional complexity. When dealing with nonlinear NDDEs, the key lies in adapting the asymptotic expansion for the residues to account for the nonlinearities in the equations. By appropriately modifying the formula for the residues and considering the impact of nonlinear terms on the solution, the method can be extended to handle a wider range of NDDEs. Additionally, the inclusion of higher-order polynomials and more terms in the series can help capture the nonlinear dynamics more accurately.

What are the potential limitations or drawbacks of the new modified method compared to other numerical approaches for solving NDDEs

While the new modified Laplace-Fourier method offers significant improvements in accuracy for solving linear neutral delay differential equations, there are potential limitations and drawbacks compared to other numerical approaches for solving NDDEs. One limitation could be the computational complexity of the method, especially when dealing with highly nonlinear NDDEs. The need to compute accurate asymptotic expansions for the residues and handle the nonlinear terms can increase the computational burden. Additionally, the method may require more manual intervention and fine-tuning compared to some numerical methods, which could be seen as a drawback in terms of automation and ease of use. Furthermore, the method's effectiveness may vary depending on the specific characteristics of the NDDE being solved, and it may not always outperform other numerical approaches in all scenarios.

What insights from the development of this new modified method could be applied to improve the accuracy of solutions for other types of delay differential equations beyond just the neutral case

Insights from the development of the new modified Laplace-Fourier method can be applied to improve the accuracy of solutions for other types of delay differential equations beyond just the neutral case. One key insight is the importance of refining the asymptotic expansion for the residues to enhance the accuracy of the solutions. This approach can be extended to nonlinear delay differential equations (DDEs) and fractional DDEs to improve the convergence rate and accuracy of the solutions. Additionally, the concept of combining Laplace transform with Fourier series theory can be applied to various types of DDEs to capture the dynamics more effectively. By optimizing the approximation of residues, adjusting the degree of polynomials, and considering the impact of nonlinearities, the accuracy of solutions for different types of DDEs can be significantly enhanced.