Numerical Discretization of Riemann–Liouville Fractional Derivatives with Strictly Positive Eigenvalues

Abstract

This paper investigates a unique and stable numerical approximation of the Riemann–Liouville Fractional Derivative. We utilize diagonal norm finite difference-based time integration methods within the summation-by-parts framework. The second-order accurate discretizations developed in this study are proven to possess eigenvalues with strictly positive real parts for non-integer orders of the fractional derivative. These results lead to provably invertible, fully discrete approximations of Riemann–Liouville derivatives.

1. Introduction

Fractional calculus, as an extension of differentiation and integration to non-integer orders, has emerged as a robust mathematical framework for modeling memory and hereditary effects in complex systems across science and engineering [1,2,3]. Recent applications encompass eco-epidemiology and predator–prey dynamics, including continuous-time, discrete-time, and disease-coupled models, underscoring the approach’s realism and flexibility [4,5,6,7]. Given that most fractional differential equations do not admit closed-form solutions, modern research places a strong emphasis on numerical methods. Comprehensive surveys detail a range of schemes, including finite-difference, spectral, wavelet, and decomposition methods, along with dedicated studies on stability and convergence theory [2,3]. Among the effective solvers are Haar-wavelet operational-matrix methods (including recent variants blended with optimization) [8,9,10], Adomian-type decompositions (comprising hybrid transforms and orthogonal–polynomial enhancements) [11,12], and analytical uniqueness frameworks built on fixed-point theorems and topological degree for fractional (integro-)differential boundary-value problems [13,14].

In numerical analysis, proving the stability of a numerical method is of paramount importance. A key tool for this is eigenvalue analysis. For instance, studies have demonstrated how the sign of the principal eigenvalue of a linearized operator can be used to prove the uniqueness and stability of positive solutions for semilinear elliptic systems [15]. Similarly, the positivity of eigenvalues has been employed to analyze the stability of difference schemes and justify the convergence of iterative methods for differential operators with nonlocal conditions [16]. The concept of summation-by-parts (SBP) operators, first developed by Kreiss and Scherer [17], offers a robust approach for constructing high-order finite difference methods with provable stability. These operators emulate the continuous property of integration by parts, providing a stable framework for solving initial boundary value problems, particularly when coupled with simultaneous approximating term (SAT) methods for imposing boundary conditions [18].

Recent studies have highlighted the central role of eigenvalue positivity in the stability of Summation-by-Parts (SBP) operators. [19] showed that the eigenvalue property, though not universally valid, can be restored through small perturbations, while [20] established eigenvalue constraints necessary for stable second-derivative operators. Stability has also been ensured through spectral radius control in entropy-stable discretizations [21], positive semidefinite formulations for variable coefficients [22], and positive-weight cubature in multidimensional settings [23]. Collectively, these works confirm eigenvalue positivity as a fundamental mechanism underlying SBP stability. Building upon these principles, this paper extends the SBP-SAT framework to fractional derivatives and introduces a novel numerical method for solving Riemann–Liouville fractional derivatives with guaranteed stability and accuracy.

Riemann–Liouville Fractional Derivative

Definition 1. 

(Riemann–Liouvville fractional integral and derivative) Let$t\in \mathbb{R}$ and $\alpha \in \mathbb{C}$, $\mathbf{Re}\left(\alpha \right)>0$, $n=\left[\mathbf{Re}\right(\alpha )+1]$, $n\in \mathbb{N}$ and $\alpha \notin \mathbb{N}$. Let $f\left(t\right)\in A{C}^n[a,b]$, the Riemann–Liouville fractional integral of order α, acting on the function $f, with-\infty <a<b<+\infty$, for $t\in [a,b]$ is defined by the following:

$${}_a{}^{RL}I_t^{\alpha }f\left(t\right)={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\alpha \right)}}{\int }_a^t{(t-\tau )}^{\alpha -1}f\left(\tau \right)d\tau ,$$

(1)

Below we state a very important property [24] of Riemann–Liouville factional derivative:

Definition 2. 

let ${}^{RL}I^{\alpha }$ with $\alpha \in [0,1]$ be the family of the fractional Riemann–Liouville integrals defined by (1). A one-parameter family ${}^{RL}D^{\alpha }$, of the linear operators, is called the fractional derivatives if and only if it satisfies the Fundamental Theorem of Fractional Calculus.

In other words, for the fractional derivatives ${}^{RL}D^{\alpha }$, with $\alpha \ge 0$ and the Riemann–Liouville fractional integrals ${}^{RL}I^{\alpha }$, the relation ${(}^{RL}{D}^{\alpha }{.}^{RL}{I}^{\alpha }f)\left(t\right)=f\left(t\right),t\in [0,1]$ holds true on appropriate nontrivial spaces of functions.

The Riemann–Liouville derivative of order $\alpha$, acting on the function $f,with-\infty <a<b<+\infty$, for $t\in [a,b]$ is given by the following:

$${}_a{}^{RL}D_t^{\alpha }f\left(t\right)={\displaystyle \frac{D^n}{\mathrm{\Gamma }(n-\alpha )}}{\int }_a^t{\displaystyle \frac{f\left(\tau \right)d\tau }{{(t-\tau )}^{n-\alpha -1}}}.$$

(2)

since $f\left(t\right)\in \mathcal{C}\left(\right[a,b\left]\right)$

Drawing from (1) and (2) we state the Fundamental Theorem of Fractional Calculus:

Theorem 1. 

Let $t\in \mathbb{R}$ and, $\mathbf{Re}\left(\alpha \right)>0$, $n=\left[\mathbf{Re}\right(\alpha )+1]$, $n\in \mathbb{N}$ and $\alpha \notin \mathbb{N}$, then with the Riemann–Liouville fractional integral and derivative of order α, acting on the function $f\in L^p[a,b]$, $1\le p<+\infty$, $-\infty <a<b<+\infty$, for $t\in [a,b]$ we have $\left(D_{a+}^{\alpha }{I}_{a+}^{\alpha }\right)f\left(t\right)=f\left(t\right)$ and ($D_{b-}^{\alpha }{I}_{b-}^{\alpha })f\left(t\right)=f\left(t\right).$

The fundamental theorem of fractional calculus extends the inverse relationship between differentiation and integration to non-integer orders. This theorem establishes the Riemann–Liouville fractional derivative as the left-inverse of the fractional integral.

2. Summation by Parts Operator

We begin by introducing the fundamental concepts of summation-by-parts simultaneous approximation term (SBP–SAT) schemes. To illustrate the main ideas, we consider the one-dimensional linear advection equation with unit wave speed, discretized on a uniform mesh. This model problem provides a clear setting for analyzing the stability and accuracy properties of SBP–SAT methods, which have become widely used for the high-order discretization of hyperbolic partial differential equations [25,26,27]. Let us define the inner product of two real-valued functions, $u,v\in L^2[a,b]$, as $(u,v)={\int }_a^{b}uvdx$ and the corresponding norm $(u,u)={\left|\right|u\left|\right|}^2$.

Consider a linear advection equation given by the following:

$$\begin{array}{cccc}\hfill u_t+a{u}_x& =& 0,\hfill & \\ \hfill u(0,x)& =& f\left(x\right),\hfill & 0\le x\le 1,\hfill \\ \hfill u(t,0)& =& u_l\left(t\right),\hfill & t\ge 0,\hfill \end{array}$$

(3)

where $f\left(x\right)$ is the initial condition and $u_l\left(t\right)$ is the boundary condition on the left. Through the energy method [10,20], the energy functional may be expressed as the following:

$$\begin{array}{cc}\hfill E\left(t\right)={\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{\int }_0^1{u}^{2}dx& ={\displaystyle \frac{1}{2}}{\int }_0^1{\displaystyle \frac{\partial u^2}{\partial t^2}}dx\hfill \\ \hfill & ={\int }_0^{1}u{\displaystyle \frac{\partial u}{\partial t}}dx\hfill \\ \hfill & =-{\int }_0^{1}au{\displaystyle \frac{\partial u}{\partial x}}dx\hfill \\ \hfill & =-{\displaystyle \frac{a}{2}}{\int }_0^1{\displaystyle \frac{\partial u^2}{\partial x}}dx\hfill \\ \hfill & =-{\displaystyle \frac{a}{2}}\left({\left.u^2\right|}_{x=1}-{\left.u^2\right|}_{x=0}\right)\hfill \end{array}$$

(4)

which is non-positive for $a>0$ whenever $u_0=0$

To discretize (3) we begin by dividing the temporal domain $0\le t\le T$ into $n+1$ equidistant grid points, $\{t_j=j\Delta t,j=0,1,2,\dots ,n,\Delta tn=T\}$ where $\Delta t=1/n$. We define the vector $\overrightarrow{v}\left(x\right)={[v_0,v_1,\dots ,v_n]}^T$ as an approximation of the continuous solution $u(t_j,x)$ at each point $t_j$. A second-order accurate central difference method may be used at the interior points. We define an inner product and norm for the discrete real-valued vector fuctions $u,v\in {\mathbb{R}}^{\mathbb{n}}$ as follows:

$${(\overrightarrow{u},\overrightarrow{v})}_P={\overrightarrow{u}}^{T}P\overrightarrow{v},\left|\right|\overrightarrow{v}{\left|\right|}_P^2={\overrightarrow{v}}^{T}P\overrightarrow{v},{(\overrightarrow{u},\overrightarrow{v})}_P={\int }_0^T{\overrightarrow{u}}^T\overrightarrow{v}dt$$

(5)

where $P=P^T.$ is a positive definite matrix.

The Summation By Parts (SBP) operator behaves in a discretized space similar to how integration by parts behaves in a continuous space. This is a key property that allows us to achieve energy estimates in a discretized space similar to a continuous space. Initially, SBP operators were developed for centered finite difference methods on equally spaced grids [17].

Definition 3. 

We define D an n-dimensional square matrix as the (SBP) finite difference operator for the first derivative as $PD\overrightarrow{v}=Q\overrightarrow{v}$ i.e $D\overrightarrow{v}=P^{-1}Q\overrightarrow{v}\approx {\displaystyle \frac{\partial \overrightarrow{v}}{\partial t}}$ if the following are met:

1. 

$D{t}^k\approx k{t}^{k-1}$;

2. 

$D=P^{-1}Q$;

3. 

$D\overrightarrow{1}=\overrightarrow{0};$

4. 

$Q+Q^T=E_N-E_0$.

Q is an almost skew-symmetric matrix allowing for the following expression:

$$Q+Q^T=E_N-E_0$$

(6)

$$E_N-E_0=\left(\begin{array}{ccccc}-1& & & & \\ & 0& & \\ & & \ddots & \\ & & & 0& \\ & & & & & 1\end{array}\right).$$

The matrices D, P, and Q are given by the following:

$$D={\displaystyle \frac{1}{2\Delta x}}\left(\begin{array}{cccccc}-2& 2& 0& \dots & 0& 0\\ -1& 0& 1& 0& 0& 0\\ 0& -1& 0& 1& 0& 0\\ 0& 0& \ddots & \ddots & \ddots & 0\\ ⋮& 0& 0& -1& 0& 1\\ 0& \dots & 0& 0& -2& 2\end{array}\right)$$

(7)

and

$$P=\Delta x\left(\begin{array}{ccc}1/2& & \\ & & 1& & \\ & & & \ddots & & \\ & & & & & & 1& \\ & & & & & & & 1/2\end{array}\right).$$

(8)

Finally, we have that

$$\begin{array}{cc}\hfill Q& =PD\hfill \\ \hfill & =\left(\begin{array}{ccccc}-1/2& 1/2& & & \\ -1/2& 0& 1/2& & \\ & -1/2& 0& 1/2& & \\ & & \ddots & \ddots & \ddots \\ & & & -1/2& 0& 1/2\\ & & & & -1/2& 1/2\end{array}\right).\hfill \end{array}$$

(9)

We can now write the semi-descretized version of (3) as follows:

$$\begin{array}{cc}\hfill {\overrightarrow{v}}_t+D\overrightarrow{v}& =0,t\ge 00\le x_i\le 1\hfill \\ \hfill v_0\left(t\right)& =g_l\left(t\right),\hfill \\ \hfill {\overrightarrow{v}}_i\left(0\right)& =f\left(x_i\right),\hfill \end{array}$$

(10)

We use the energy method to demonstrate the stability of the numerical solution of (10). This is equivalently performed by the following:

1.

Multiplying by the numerical solution ${\overrightarrow{v}}^{T}P$ from the left.

2.

Add the transpose of the resulting equation from step 1. above.

$$\begin{array}{cc}\hfill {\overrightarrow{v}}_t+D\overrightarrow{v}& =0\hfill \\ \hfill {\overrightarrow{v}}^{T}P{\overrightarrow{v}}_t+{\overrightarrow{v}}^{T}PD\overrightarrow{v}& =0\hfill \\ \hfill {\overrightarrow{v}}^{T}P{\overrightarrow{v}}_t& =-{\overrightarrow{v}}^{T}PD\overrightarrow{v}\hfill \\ \hfill {\overrightarrow{v}}^{T}P{\overrightarrow{v}}_t+{\overrightarrow{v}}_t^{T}P\overrightarrow{v}& =-{\overrightarrow{v}}^{T}PD\overrightarrow{v}-{\left(D\overrightarrow{v}\right)}^{T}P\overrightarrow{v}\hfill \\ \hfill {\displaystyle \frac{d}{dt}}\left|\right|\overrightarrow{v}{\left|\right|}_P^2& =-{\overrightarrow{v}}^{T}P\left(P^{-1}Q\right)\overrightarrow{v}-{\overrightarrow{v}}^T{\left(P^{-1}Q\right)}^{T}P\overrightarrow{v}\hfill \\ \hfill & =-({\overrightarrow{v}}^{T}Q\overrightarrow{v}+{\overrightarrow{v}}^T{Q}^T\overrightarrow{v})\hfill \\ \hfill & =-\left({\overrightarrow{v}}^T(Q+Q^T)\overrightarrow{v}\right)\hfill \\ \hfill & =-\left({\overrightarrow{v}}^T(E_N-E_0)\overrightarrow{v}\right)\hfill \\ \hfill & ={\overrightarrow{v}}^2{{|}_{x=0}-{\overrightarrow{v}}^2|}_{x=1}\hfill \end{array}$$

(11)

Equation (11) is the discrete analog of Equation (4), hence stability is established.

Simultaneous Approximation Terms

Thus far we have only considered numerical solutions without the imposition of boundary conditions. To be able to approximate the solutions numerically and still obey boundary conditions, we need to introduce Simultaneous Approximation Terms (SATs). The basic idea behind the SAT technique is to impose the boundary conditions weakly, such that the SBP property is preserved and an energy estimate can be obtained. The SAT acts as a penalty, pulling the boundary condition toward the initial conditions. When the boundary conditions are obeyed, the SAT has no contribution. We now impose boundary conditions through the introduction of S, the SAT terms given by $S=e_L(S_0,0,0,\dots ,0)$, with $e_L={[1,0,0,\dots ,0]}^T$ and $S_0=\mathrm{\Phi }(v_0-g{\left(t\right)}_l)$, $\mathrm{\Phi }$ is a parameter that we tune for stability as will be demonstrated below. Equation (10) is now given simultaneously as follows:

$$\begin{array}{cc}\hfill {\overrightarrow{v}}_t+D\overrightarrow{v}& =P^{-1}S\hfill \\ \hfill & =\mathrm{\Phi }{P}^{-1}{e}_L(v_0-g_l\left(t\right)).\hfill \end{array}$$

(12)

We conduct the energy estimates for (12) by multiplying through by $v^{T}P$.

$$\begin{array}{cc}\hfill {\overrightarrow{v}}^{T}P\overrightarrow{v}+{\overrightarrow{v}}^{T}PD\overrightarrow{v}& ={\overrightarrow{v}}^{T}P{P}^{-1}\mathrm{\Phi }{e}_L(v_0-g_l\left(t\right))\hfill \\ \hfill {\displaystyle \frac{d}{dt}}\left({\overrightarrow{v}}^{T}P\overrightarrow{v}\right)+{\overrightarrow{v}}^T(Q+Q^T)\overrightarrow{v}& =2{\overrightarrow{v}}^T\mathrm{\Phi }{e}_L(v_0-g_l\left(t\right))\hfill \\ \hfill {\displaystyle \frac{d}{dt}}\left({\overrightarrow{v}}^{T}P\overrightarrow{v}\right)& =-{\overrightarrow{v}}^T(E-E_0)\overrightarrow{v}+2\mathrm{\Phi }(v_0^2-v_0{g}_l\left(t\right))\hfill \\ \hfill {\displaystyle \frac{d}{dt}}\left|\right|\overrightarrow{v}{\left|\right|}_P^2& =v_0^2-v_n^2+2\mathrm{\Phi }(v_0^2-v_0{g}_l\left(t\right))\hfill \end{array}$$

(13)

Comparing (11) and (13) we see that stability is achieved when $2\mathrm{\Phi }\le -1$, which implies that $\mathrm{\Phi }\le -1/2$. $\mathrm{\Phi }$ determines the strength of the SAT and will be chosen below such that (12) is energy stable and consistent.

3. Riemann–Liouville SBP-SAT Operator

Riemann–Liouville derivatives are a type of fractional derivative that can be expressed as an integral. Finite differences can be used to approximate the Riemann–Liouville derivative by discretizing the integral. Since SBP operators are finite difference operators. They have been used to approximate both partial and ordinary derivatives. Here we will construct a numerical approximation of the time Riemann–Liouville derivative.

Let us consider the integral part of the Riemann–Liouville derivative. Following from [28], we use the product trapezoidal quadrature formula with respect to the weight function ${(t_{k+1}-\tau )}^{\beta -1}$

$${\int }_0^{t_{k+1}}{(t_{k+1}-\tau )}^{\beta -1}f\left(\tau \right)d\tau \approx {\int }_0^{t_{k+1}}{(t_{n+1}-\tau )}^{\beta -1}{\tilde{f}}_{k+1}\left(\tau \right)d\tau$$

(14)

where ${\tilde{f}}_{k+1}$ is the piecewise linear interpolant of f with notes and knots taken at $t_i$, $i=0,1,2,3,\dots ,k+1$. Applying standard techniques from quadrature theory [29], we can write the integral on the right side of Equation (14) as follows:

$${\int }_0^{t_{k+1}}{(t_{k+1}-\tau )}^{\beta -1}{\tilde{f}}_{k+1}\left(\tau \right)d\tau ={\displaystyle \frac{h^{\beta }}{\beta (\beta +1)}}\sum _{i=0}^{k+1}{a}_{i,k+1}f\left(t_i\right),$$

(15)

Consider the integral part of Equation (2) and allow $n=1$ and from Equation (15) let $\beta =1-\alpha$, written as follows:

$${\int }_0^{t_{k+1}}{(t_{k+1}-\tau )}^{-\alpha }{\tilde{f}}_{k+1}\left(\tau \right)d\tau ={\displaystyle \frac{h^{1-\alpha }}{(1-\alpha )(2-\alpha )}}\sum _{i=0}^{k+1}{a}_{i,k+1}f\left(t_i\right),$$

(16)

where

$$a_{i,k+1}=\left\{\begin{array}{cc}{k}^{2-\alpha }-(k-1+\alpha ){(k+1)}^{1-\alpha },\hfill & \hfill \mathrm{if}i=0.\\ {(k-i+2)}^{2-\alpha }+{(k-i)}^{2-\alpha }-2{(k-i+1)}^{2-\alpha },\hfill & \hfill \mathrm{if}1\le i\le k,\\ 1,\hfill & \hfill \mathrm{if}i=k+1.\end{array}\right.$$

(17)

Based on the analysis by [30] Unless very restrictive (and usually unrealistic) assumptions are made to ensure the smoothness of $f\left(t\right)$ also at the left endpoint of $[t_0,T]$, the rate by which the error reduces to 0 is therefore, away from $t_0$, proportional to $h^{1+\beta }$ when $0<\beta <1$. The same drop in the convergence order occurs also when higher-degree polynomials are employed; for this reason product trapezoidal quadratures of higher order are never taken into account for Riemann–Liouville derivatives when $0<\beta <1$. When $\beta >1$, order 2 of convergence is instead obtained.

Using the integral approximation in Equation (16) we can construct a matrix A. We approximate the differential part of the Riemann–Liouville derivative using the SBP operator defined above, and using the Gamma function property $\mathrm{\Gamma }(z+1)=z\mathrm{\Gamma }\left(z\right)$, we arrive at follows:

$$\begin{array}{cc}\hfill {(}_0^{RL}{D}_t^{\alpha }f\left(t\right))& \approx P^{-1}Q{\displaystyle \frac{h^{1-\alpha }}{\mathrm{\Gamma }(2-\alpha )}}({(k+1-i)}^{1-\alpha }-{(k-i)}^{1-\alpha })f\left(t_i\right)\hfill \\ \hfill & \approx P^{-1}QAf\left(t_i\right)\hfill \end{array}$$

(18)

Spectrum Analysis

Applying the new Riemann–Liouville numerical operator to (12) leads to the following:

$$P^{-1}QA\overrightarrow{v}+\lambda \overrightarrow{v}=\sigma P^{-1}{e}_0(v_0-f)$$

(19)

Let us begin the analysis by re-arranging (19) as follows:

$$\begin{array}{cc}\hfill P^{-1}QA\overrightarrow{v}+\lambda \overrightarrow{v}& =\sigma P^{-1}{e}_0(v_0-f)\hfill \\ \hfill P^{-1}QA\overrightarrow{v}-\sigma P^{-1}{e}_0{v}_0+\lambda v& =-\sigma P^{-1}{e}_{0}f\hfill \\ \hfill P^{-1}(QA-\sigma e_0)\overrightarrow{v}+\lambda \overrightarrow{v}& =-\sigma P^{-1}{e}_{0}f\hfill \\ \hfill \mathrm{let}\tilde{Q}& =QA-\sigma e_0\mathrm{and}R=-\sigma P^{-1}{e}_{0}f\hfill \\ \hfill \mathrm{then}\\ \hfill (P^{-1}\tilde{Q}+\lambda I)\overrightarrow{v}& =R\hfill \end{array}$$

(20)

If $Re\left(\lambda \right)\ge 0$, then the following proposition holds.

Proposition 1. 

The discrete problem (20) leads to an invertible system if the matrix $P^{-1}\tilde{Q}$ has eigenvalues with strictly positive real parts.

In line with the above proposition, we demonstrate, in Figure 1, that for a discretization of 100 point and for $\sigma =-0.5$, the eigenvalues of $P^{-1}\tilde{Q}$ indeed have strictly positive real parts. This ensures the stability of the constructed SBP operator. In the next sections, we apply this SBP operator to a fractional differential equation and present the errors as a function of increasing resolution.

4. Numerical Example

In this section, we present a numerical example to illustrate the error of the fully discrete approximations developed for the Riemann–Liouville fractional derivative and show it in Figure 2. Consider the differential equation given by

$${}_a{}^{RL}D_t^{\alpha }f\left(t\right)+\lambda f\left(t\right)=\mathrm{\Gamma }(\alpha +1)+\lambda t^{\alpha }$$

(21)

with a with a know solution given by $f\left(t\right)=t^{\alpha }$. Allow $g\left(t\right)=\mathrm{\Gamma }(\alpha +1)+\lambda t^{\alpha }$. Following from (19) we discretize (21) as follows:

$$\begin{array}{cc}\hfill P^{-1}QA\overrightarrow{f}+\lambda \overrightarrow{f}& =\sigma P^{-1}{e}_0(f_0-g)\hfill \\ \hfill \overrightarrow{f}& =\sigma P^{-1}{e}_0{(P^{-1}QA+\lambda I)}^{-1}(f_0-g)with\hfill \\ \hfill \lambda & =2\mathrm{and}{f}_0\approx g\hfill \end{array}$$

(22)

Discussion

The positive eigenvalues, observed in Figure 1., lead not only to the uniqueness of the solution of (19) but also to null-space consistency of the SBP operator $D=P^{-1}QA$ [31]. The operator D is said to be null-space consistent when $D\overrightarrow{v}=0$ iff$\overrightarrow{v}\in Span\left\{\overrightarrow{1}\right\}$, where $\overrightarrow{1}={[1,\dots ,1]}^T$. If this property holds, the problem (19) can be reinterpreted as a Runge–Kutta time integration method satisfying various stability properties [32]. Looking at

Table 1. Comparison of numerical results at $t=2$ for different $\alpha$ values.

n	$\mathit{\alpha }=0.98$		$\mathit{\alpha }=0.5$
	Approx	Error	Approx	Error
15	2.26456	0.29209	2.79234	0.81987
30	2.12596	0.15349	2.49693	0.52446
60	2.05569	0.08322	2.28618	0.31371
120	2.02006	0.04759	2.13635	0.16388
240	2.00199	0.02952	2.03006	0.05759
480	1.99283	0.02036	1.95474	0.01773
Conv. Rate	$\mathbf{C}=\mathbf{0.78}$		$\mathbf{C}=\mathbf{1.09}$

data, we observe the convergence of the method for two fractional orders, $\alpha =0.98$ and $\alpha =0.5$. While the initial error for $\alpha =0.5$ is larger, its average convergence rate of $1.09$ is significantly higher than the $0.78$ rate for $\alpha =0.98$. This indicates that the operator converges more efficiently for $\alpha =0.5$, highlighting that a numerical scheme’s performance is critically dependent on the fractional order.

5. Conclusions

This paper successfully addresses a critical challenge in applied mathematics by establishing a provably stable and unique numerical solution for the Riemann–Liouville fractional derivative. We employed diagonal norm finite difference-based time integration within the summation-by-parts framework, developing second-order accurate discretizations. Our most significant finding is that these discretizations possess eigenvalues with strictly positive real parts for non-integer orders. This property is crucial as it ensures the invertibility of fully discrete approximations and implies a null-space consistency that connects our approach to established, stable Runge-Kutta time integration methods. While the overall convergence rate observed was first-order due to the integral quadrature, this work provides a robust and provably stable foundation for solving complex fractional differential equations. This advancement lays a significant foundation for future research to develop higher-order integral quadratures, which would further enhance the utility and precision of these methods for modeling real-world phenomena.