A New Higher-Order Convergence Laplace–Fourier Method for Linear Neutral Delay Differential Equations

In this article, a new higher-order convergence Laplace–Fourier method is developed to obtain the solutions of linear neutral delay differential equations. The proposed method provides more accurate solutions than the ones 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 method combines the Laplace transform method with Fourier series theory. All of the beneficial features from the original Laplace–Fourier method are retained. The 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 method, this improvement is small. As we shall show, with the improved higher-order convergence Laplace–Fourier method, the improvement is significantly larger. We show that the convergence rate of the new Laplace–Fourier solution has a remarkable order of convergence. The validity of the new technique is corroborated by means of illustrative examples. Comparisons of the solutions of the new method with those generated by the pure Laplace method and the unmodified Laplace–Fourier approach are presented.