Backward-Euler convolution quadrature for a Caputo equation

Input $0<\alpha <1$, a step size $h>0$, an integer $N\ge 1$, $\lambda \ge 0$, an initial value $u_0$, and forcing values $F^1,\dots ,F^N$. If any admissibility condition fails, return invalid-input. Set ${\omega }_0=1$ and compute ${\omega }_1,\dots ,{\omega }_N$ in increasing order by

${\omega }_j:=\frac{j-1-\alpha }{j}{\omega }_{j-1}\mathrm{.}$

Set $V^0=0$ and $U^0=u_0$. For $n=1,\dots ,N$ in increasing order, compute

$V^n:=\frac{F^n-\lambda u_0-h^{-\alpha }{\sum }_{j=1}^n{\omega }_j{V}^{n-j}}{h^{-\alpha }+\lambda },U^n:=u_0+V^n\mathrm{.}$

For any sequence $V^0,\dots ,V^N$ with $V^0=0$, use the notation

${\bar{\mathrm{\partial }}}_h^{\alpha }{V}^n:=h^{-\alpha }{\sum }_{j=0}^n{\omega }_j{V}^{n-j}\mathrm{.}$

Return $U^0,\dots ,U^N$. A finite-precision implementation that cannot certify a scalar solve returns solve-failed without reporting a value. No FFT or truncated-memory error claim is included.

The coefficient sequence is identified with the principal-branch backward-Euler symbol in Principal-branch generating function for backward-Euler convolution quadrature. Its sign and partial-sum bounds used for scalar stability are proved in Backward-Euler fractional weights and positive partial sums.