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

TL;DR

The paper addresses solving linear neutral delay differential equations (NDDEs) by improving the Laplace-Fourier framework. It introduces a novel modified Laplace-Fourier method that derives an improved asymptotic expansion for the residues and refines pole approximations, thereby enhancing the Fourier-tail correction and achieving an $O(N^{-3})$ convergence. The key contributions include explicit formulas for the improved residue terms, a refined pole expansion, and demonstrations across three NDDE examples showing substantial accuracy gains over both pure Laplace and the original Laplace-Fourier method. The approach yields an essentially analytical solution valid over the entire time domain, with practical benefits in computational efficiency and precision, and is implementable in Maple or similar symbolic systems.

Abstract

In this article, a new modified Laplace-Fourier method is developed in order to obtain the solutions of linear neutral delay differential equations. The proposed method provides a more accurate solution than the one provided by the pure Laplace method and the original Laplace-Fourier method. We develop and show the crucial modifications of the Laplace-Fourier method. As with the original Laplace-Fourier method, the new modified method combines the Laplace transform method with Fourier series theory. All of the beneficial features from the original Laplace-Fourier method are retained. The modified solution still includes a component that accounts for the terms in the tail of the infinite series, allowing one to obtain more accurate solutions. The Laplace-Fourier method requires us to approximate the formula for the residues with an asymptotic expansion. This is essential to enable us to use the Fourier series results that enable us to account for the tail. The improvement is achieved by deriving a new asymptotic expansion which minimizes the error between the actual residues and those which are obtained from this asymptotic expansion. With both the pure Laplace and improved Laplace-Fourier methods increasing the number of terms in the truncated series obviously increases the accuracy. However, with the pure Laplace, this improvement is small. As we shall show, with the improved Laplace-Fourier method the improvement is significantly larger. We show that the convergence rate of the new modified Laplace-Fourier solution has a remarkable order of convergence $O(N^{-3})$. The validity of the modified technique is corroborated by means of illustrative examples. Comparisons of the solutions of the new modified method with those generated by the pure Laplace method and the original/unmodified Laplace-Fourier approach are presented.

Figures (9)

• Figure 1: Analytical solution obtained by the MoS. The NDDE is $y^{\prime }(t)\!=\!-2.1y(t)\!+\!0.9y^{\prime }(t-1)\!+\!2.12y(t-1), H(t)\!=\!2-48t(1+t), t\in \lbrack -1,0]$.
• Figure 2: Errors of the solutions obtained by pure Laplace (left), original Laplace-Fourier (center) and the new modified Laplace-Fourier (right) methods. The NDDE is $y^{\prime }(t)\!=\!-2.1y(t)\!+\!0.9y^{\prime }(t-1)\!+\!2.12y(t-1), H(t)\!=\!2-48t(1+t), t\in \lbrack -1,0]$.
• Figure 3: Relative errors of the imaginary (solid lines) and real (dashed lines) parts using the correct residues $c_{k}$ versus the approximate values $c_{k}^{a}$ for the original Laplace-Fourier (left) and for the new modified Laplace-Fourier method (right). The NDDE is $y^{\prime }(t)\!=\!-2.1y(t)\!+\!0.9y^{\prime }(t-1)\!+\!2.12y(t-1), H(t)\!=\!2-48t(1+t), t\in \lbrack -1,0]$.
• Figure 4: Analytical solution obtained by the MoS. The NDDE is $y^{\prime }(t)\!\!=\!\!-2.1y(t)\!+\!\frac{7}{11}y^{\prime }(t-2)\!-\!2y(t-2),H(t)\!=\!1\!+\!\frac{3}{2}(t+2)(0.5+t), t\in [-2,0]$.
• Figure 5: Errors of the solutions obtained by pure Laplace (left), original Laplace-Fourier (center) and the new modified Laplace-Fourier (right) methods. The NDDE is $y^{\prime }(t)\!\!=\!\!-2.1y(t)\!+\!\frac{7}{11}y^{\prime }(t-2)\!-\!2y(t-2),H(t)\!=\!1\!+\!\frac{3}{2}(t+2)(0.5+t), t\in [-2,0]$.