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Variable-order Scarpi operators with approximated kernels
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DOI：10.1007/s13540-026-00577-8.png)
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Abstract
En 中文
Variable-order time-fractional derivatives provide a flexible framework for modeling systems with time-dependent memory. Most existing variable-order formulations are obtained by formally replacing the constant order with a time-dependent function, a procedure that typically leads to the loss of fundamental structural properties such as the existence of a left inverse and the validity of a fundamental theorem of calculus. In contrast, the Scarpi derivative, defined through the Laplace transform and based on Sonine kernels, preserves these essential properties in the variable-order setting. Since explicit time-domain expressions of kernels are difficult to obtain, in this work we develop analytical time-domain approximations for order-modulated Scarpi-Sonine kernels by means of asymptotic expansions of their Laplace transforms. The accuracy of the proposed approximations is assessed through systematic comparisons with kernels obtained via numerical inversion of the Laplace transform. Furthermore, we apply inverse Laplace transform techniques to compute numerical solutions of some scalar and coupled linear initial-value problems governed by variable-order Scarpi derivatives to illustrate the influence of variable orders on the solution behavior.
Keywords:
Variable-order
Sonine kernel
Scarpi operators
Relaxation equation
Numerical inverse Laplace transform
Journal
IF:
2.9
Papers:
215
Citations:
3.4K