A Numerical Scheme for Solving Fractional Regularized Long Wave Burgers Equation

Source abstract

Abstract This paper introduces a high-accuracy numerical framework for solving a generalized form of the fractional regularized long-wave Burgers (FRLWB) equation, incorporating the Caputo fractional derivative. The proposed method is based on a space-time pseudo-spectral collocation (PSC) approach, which leverages global interpolation via Lagrange polynomials and shifted Legendre-Gauss-Lobatto (LGL) nodes to achieve exponential convergence for smooth solutions. Unlike traditional schemes such as finite difference or finite element methods, the PSC technique offers superior spectral accuracy and computational efficiency, particularly in handling fractional operators. A key novelty of this work lies in the simultaneous spectral treatment of both spatial and temporal domains, enabling the construction of operational matrices that directly approximate fractional derivatives without resorting to discretization or auxiliary transformations. By collocating the governing equation along with its initial and boundary conditions, the problem is transformed into a system of nonlinear algebraic equations. These systems are solved efficiently using iterative solvers, ensuring stability and scalability across a range of fractional orders. The method also introduces a direct formulation for the mixed derivative term $$Q_{\eta \eta \kappa }$$ , which is often neglected or approximated in existing literature. This enhances the model’s fidelity and allows for more accurate simulation of wave propagation phenomena in complex media. A rigorous convergence analysis is provided to establish the reliability of the approach, and several benchmark problems are examined to demonstrate its precision and robustness. Overall, the proposed PSC framework represents a significant advancement in the numerical treatment of fractional PDEs. Its flexibility, accuracy, and ease of implementation make it a promising tool for future applications in fluid dynamics, nonlinear wave modeling, and fractional-order systems.