Journal of the
Korean Mathematical Society JKMS

In this study, the numerical solution of a space-fractional Burgers' equation with initial and boundary conditions is considered. This equation is the simplest nonlinear model for diffusive waves in fluid dynamics. It occurs in a variety of physical phenomena, including viscous sound waves, waves in fluid-filled viscous elastic pipes, magneto-hydrodynamic waves in a medium with finite electrical conductivity, and one-dimensional turbulence. The proposed QBS/CNG technique consists of the Galerkin method with a function basis of quadratic B-splines for the spatial discretization of the space-fractional Burgers' equation. This is then followed by the Crank-Nicolson approach for time-stepping. A linearized scheme is fully constructed to reduce computational costs. Stability analysis, error estimates, and convergence rates are studied. Finally, some test problems are used to confirm the theoretical results and the proposed method's effectiveness, with the results displayed in tables, 2D, and 3D graphs.

Keywords: Space-fractional Burgers' equation, fractional calculus, finite element method, Galerkin method, quadratic B-splines, Crank-Nicolson scheme

MSC numbers: Primary 65N30, 65D07, 74S05, 97N40

Supported by: The second author is also thankful to the General Directorate of Scientific Research and Technological
Development (Ministry of Higher Education and Scientific Research, Algeria)
for supporting him through a research grant (PRFU-C00L03UN250120200
003).