Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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QUADRATIC B-SPLINE GALERKIN SCHEME FOR THE SOLUTION OF A SPACE-FRACTIONAL BURGERS' EQUATION

  • Khadidja Bouabid (Research Laboratory: Differential Equations Department of Mathematics Faculty of Exact Sciences University Freres Mentouri - Constantine 1) ;
  • Nasserdine Kechkar (Department of Mathematics Faculty of Exact Sciences University Freres Mentouri - Constantine 1)
  • Received : 2023.01.06
  • Accepted : 2024.03.19
  • Published : 2024.07.01
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Abstract

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

Acknowledgement

The authors would like to express their sincere thanks to the respected anonymous referees for their valuable comments and suggestions, which have greatly improved the quality of this paper. 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-C00L03UN250120200003).

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