Awad, Sadia Kumail (2026) Discretization and Adaptive Control of a Variable-Order Fractional Burgers Equation with Short-Memory Acceleration. CENTRAL ASIAN JOURNAL OF MATHEMATICAL THEORY AND COMPUTER SCIENCES, 7 (4). ISSN 2660-5309

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Abstract

We study a fractional generalization of Burgers’ equation where the time derivative is replaced by a variable-order Grünwald–Letnikov (GL) fractional derivative . This naturally introduces a memory effect that depends on both space and time. We develop a consistent numerical scheme combining a discretization of the variable-order GL operator with a semi-implicit upwind finite-difference method for the nonlinear convection term, solved efficiently via a tridiagonal Thomas algorithm. Beyond the baseline scheme, we explore three practical extensions: a short-memory truncation that reduces runtime by an order of magnitude with minimal accuracy loss (≲1% deviation); a state-dependent adaptive order  that sharpens fronts automatically; and a careful discussion of the inherent ambiguity in defining variable-order GL derivatives. Numerical experiments validate the method and demonstrate its flexibility across a range of test cases.  ‎

Item Type:	Article
Uncontrolled Keywords:	Variable-order fractional calculus, Grünwald–Letnikov derivative, Burgers equation, Short-memory principle, Adaptive fractional order
Subjects:	H Social Sciences
Depositing User:	admin eprints
Date Deposited:	13 Sep 2026 10:20
Last Modified:	13 Sep 2026 10:20
URI:	http://eprints.umsida.ac.id/id/eprint/17075

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