# Convergence Outside the Initial Layer for a Numerical Method for the Time-Fractional Heat Equation

## Abstract

In this paper a fractional heat equation is considered; it has a Caputo time-fractional derivative of order \(\delta \) where \(0<\delta <1\). It is solved numerically on a uniform mesh using the classical L1 and standard three-point finite difference approximations for the time and spatial derivatives, respectively. In general the true solution exhibits a layer at the initial time \(t=0\); this reduces the global order of convergence of the finite difference method to \(O(h^2+\tau ^\delta )\), where *h* and \(\tau \) are the mesh widths in space and time, respectively. A new estimate for the L1 approximation shows that its truncation error is smaller away from \(t=0\). This motivates us to investigate if the finite difference method is more accurate away from \(t=0\). Numerical experiments with various non-smooth and incompatible initial conditions show that, away from \(t=0\), one obtains \(O(h^2+\tau )\) convergence.

## Keywords

Time-fractional heat equation Caputo fractional derivative Initial-boundary value problem L1 scheme Layer region Smooth and non-smooth data Compatibility conditions## Notes

### Acknowledgments

The research of José Luis Gracia was partly supported by the Institute of Mathematics and Applications (IUMA), the project MTM2016-75139-R and the Diputación General de Aragón. The research of Martin Stynes was supported in part by the National Natural Science Foundation of China under grants 91430216 and NSAF-U1530401.

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