Hermite‐Shannon‐Cosine spectral method for fractional partial differential equations

Abstract Shannon‐Cosine wavelet function possesses almost all excellent characteristics such as interpolation, compact support, and smoothness. As an interpolation wavelet function, it could be applied in fractional partial differential equations effectively. However, when solving engineering problems in a finite interval, the treatment of the boundary is still not smooth enough. So, the Hermite‐Shannon‐Cosine interval wavelet is constructed using the Hermite interpolation function to achieve smoother transitions at the boundary of the interval, thereby reducing boundary effects. Based on this, a method for solving Fractional PDEs is proposed, the method's performance and its processing capability at the interval boundary are verified by taking the Fractional Fokker‐Planck equation and the Time‐Fractional Korteweg‐de Vries equation as examples. Compared with the multi‐scale Faber‐Schauder wavelet collocation method, Point‐Symmetric interval wavelet spectral method, Dynamic interval wavelet spectral method, and so forth, the experimental results show that the method performs better in terms of numerical accuracy and effectiveness.