The field of numerical methods for partial differential equations is experiencing significant growth, with a focus on developing efficient and accurate algorithms for solving complex problems. Recent research has concentrated on improving spectral methods, introducing new techniques for handling non-periodic boundaries, and exploring the application of fractional calculus. Notably, the development of fast multipole methods and hybrid shifted Gegenbauer integral-pseudospectral methods has shown great promise in solving problems with high accuracy and efficiency.

Noteworthy papers include: The paper on Fourier Spectral Methods for Block Copolymer Systems on Sphere, which demonstrates the effectiveness of the OK model in real-world applications. The paper on Non-periodic Fourier propagation algorithms for partial differential equations, which presents a method that can treat vector fields with a combination of Dirichlet and/or Neumann boundary conditions in one or more space dimensions. The paper on Hybrid Shifted Gegenbauer Integral-Pseudospectral Method for Solving Time-Fractional Benjamin-Bona-Mahony-Burgers Equation, which achieves spectral accuracy and outperforms existing numerical approaches.