<i>gL</i>1 Scheme for Solving a Class of Generalized Time-Fractional Diffusion Equations

• Published in: Mathematics
• Main Authors: Xuhao Li, Patricia J. Y. Wong
• Format: Article
• Language: English
• Published: MDPI AG 2022-04-01
• Subjects: generalized fractional derivative; time-diffusion problem; generalized linear interpolation; numerical scheme
• Online Access: https://www.mdpi.com/2227-7390/10/8/1219
• ISSN: 2227-7390

In this paper, a numerical scheme based on a general temporal mesh is constructed for a generalized time-fractional diffusion problem of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>. The main idea involves the generalized linear interpolation and so we term the numerical scheme the <b><i>gL</i>1 scheme</b>. The stability and convergence of the numerical scheme are analyzed using the energy method. It is proven that the temporal convergence order is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>2</mn><mo>−</mo><mi>α</mi><mo>)</mo></mrow></semantics></math></inline-formula> for a general temporal mesh. Simulation is carried out to verify the efficiency of the proposed numerical scheme.