Fractional Right Rectangular Rule in Numerical Methods

• The fractional right rectangular rule is a numerical quadrature method that approximates the Riemann–Liouville integral using right-endpoint values with kernel-adapted weights.
• It is integrated into finite element schemes for subdiffusion equations with delay, effectively handling low temporal regularity through both uniform and graded time meshes.
• The method achieves unconditional stability and optimal convergence, as confirmed by theoretical truncation error estimates and numerical validations.

Searching arXiv for the cited paper and closely related work on the fractional right rectangular rule. The fractional right rectangular rule is a time-discretization procedure for the Riemann–Liouville fractional integral that replaces the integrand on each temporal subinterval by its value at the right endpoint, with kernel-adapted weights derived from the weakly singular fractional convolution. In the context documented in "Finite element method for a constant time delay subdiffusion equation with Riemann-Liouville fractional derivative" (Bu et al., 16 Sep 2025), the rule is used within a fully discrete finite element scheme for a subdiffusion equation with constant time delay $\tau$, low temporal regularity at $t=0$ and $t=\tau$, and a Riemann–Liouville integral term coupled to a Caputo-type time discretization by the L1 formula. The term “right rectangular” refers to the use of right-node values on each interval, while “fractional” reflects the nonlocal kernel $\omega_\alpha$ governing the integral operator.

1. Definition and operator-theoretic setting

The underlying continuous operator is the Riemann–Liouville fractional integral of order $\alpha>0$, defined by

$${}_0I_t^\alpha u(x,t)=\int_0^t \omega_\alpha(t-s)\,u(x,s)\,ds,$$

where

$$\omega_\alpha(t):=\frac{t^{\alpha-1}}{\Gamma(\alpha)}.$$

This representation exhibits the essential feature that drives the numerical design: the kernel is weakly singular near $s=t$, so the operator is nonlocal and memory-bearing (Bu et al., 16 Sep 2025).

In the numerical setting of (Bu et al., 16 Sep 2025), the relevant integral order is $1-\alpha$, and the operator enters a delay subdiffusion equation through a delayed argument. The same paper formulates a fully discrete finite element method on a partition

$0=t_0<t_1<\cdots<t_n,$

with step sizes

$t=0$0

allowing both uniform and symmetric graded time meshes. This mesh flexibility is central because the exact solution is assumed to have low regularities at both the initial time and the delay time.

A different use of the phrase “rectangular fractional integral” appears in harmonic analysis, where a rectangular fractional integral operator is defined over axis-parallel rectangles rather than as a quadrature rule. In "The rectangular fractional integral operators" (Tanaka, 2023), the operator

$t=0$1

is studied under rectangular doubling weights. That work is analytically related by terminology and by its emphasis on rectangle-based fractional integration, but it does not define the numerical fractional right rectangular rule itself. This distinction prevents a common misconception: the quadrature rule in numerical fractional PDEs and the rectangular integral operator in weighted harmonic analysis are not the same construction.

2. Discrete formulation of the fractional right rectangular rule

In (Bu et al., 16 Sep 2025), the fractional right rectangular rule is introduced for the Riemann–Liouville integral on the above time grid. The discrete approximation at time level $t=0$2 is

$t=0$3

with weights

$t=0$4

where

$t=0$5

The rule therefore combines three ingredients: interval lengths $t=0$6, right-endpoint samples $t=0$7, and convolution-compatible weights $t=0$8 (Bu et al., 16 Sep 2025).

The shift by $t=0$9 is specific to the constant-delay formulation treated in the paper and encodes the delayed dependence of the solution. In the discrete weak formulation, the rule is inserted into the fully discrete finite element scheme as

$t=\tau$0

Here, $t=\tau$1 is the time-fractional derivative approximated by the L1 formula, while $t=\tau$2 is the delayed Riemann–Liouville integral approximated by the fractional right rectangular rule (Bu et al., 16 Sep 2025).

A concise summary of the discrete objects is as follows.

Quantity	Formula	Role
RL fractional integral	$t=\tau$3	Continuous nonlocal operator
Fractional right rectangular approximation	$t=\tau$4	Time discretization of delayed RL integral
Discrete weight	$t=\tau$5	Kernel-adapted coefficient

The use of right-node values distinguishes this rule from trapezoidal or midpoint-type fractional quadratures. The paper explicitly contrasts it with alternatives such as the fractional trapezoidal rule and emphasizes that the right rectangular version is efficient to implement and analyze on graded meshes (Bu et al., 16 Sep 2025).

3. Local truncation error and regularity assumptions

A defining theoretical property of the fractional right rectangular rule in (Bu et al., 16 Sep 2025) is the local truncation estimate stated in Lemma 2.1: $t=\tau$6 Here, $t=\tau$7 is the first time-step size and $t=\tau$8 is the grading parameter of the time mesh. The estimate identifies the method as locally consistent with order $t=\tau$9 under the solution regularity assumptions used in the paper (Bu et al., 16 Sep 2025).

The proof structure described in the source relies on Taylor expansion and on geometric properties of the graded mesh. The leading error stems from replacing the exact integrand value $\omega_\alpha$0 on each interval $\omega_\alpha$1 by the right-endpoint datum $\omega_\alpha$2 while controlling the weakly singular kernel $\omega_\alpha$3. Because the problem contains a constant time delay, the analysis must also track the effect of delayed arguments in the consistency bound.

The relevant temporal regularity assumption is

$\omega_\alpha$4

This condition reflects low regularity at the singular times and motivates the use of graded meshes. A plausible implication is that the right rectangular choice is not merely a low-order convenience; it is aligned with the regularity regime in which higher-order temporal quadratures may fail to realize their formal order.

A useful comparison comes from the stochastic-wave setting of "Difference methods for time discretization of stochastic wave equation" (Liu, 2021). There, a rectangle formula is used because the solution has only limited Hölder regularity in time, and the resulting convergence order is bounded by that regularity. Although the model, operator, and error norm differ from (Bu et al., 16 Sep 2025), both works support the broader principle that rectangle-based time quadrature is especially natural when temporal smoothness is weak.

4. Role in the fully discrete finite element scheme

The fractional right rectangular rule in (Bu et al., 16 Sep 2025) is not an isolated quadrature device; it is one component of a coupled space-time discretization. The paper develops a fully discrete finite element scheme for a subdiffusion equation with constant delay, using:

• the L1 formula for the Caputo fractional derivative,
• the fractional right rectangular rule for the Riemann–Liouville integral,
• a spatial finite element space $\omega_\alpha$5,
• and either a uniform mesh or a symmetric graded time mesh.

The mesh design is especially important. By setting the grading parameter $\omega_\alpha$6, the symmetric graded time mesh degenerates to a uniform mesh. This permits a two-track analysis: one for uniform stepping and another for graded stepping near nonsmooth points (Bu et al., 16 Sep 2025).

Within this framework, the right rectangular discretization approximates the delayed integral term $\omega_\alpha$7 or its discrete analog through the operator $\omega_\alpha$8. Since the delay induces nonlocality not only through the fractional kernel but also through the shifted time index, the numerical method simultaneously resolves fractional memory and fixed-delay memory.

This coupling has methodological significance. The L1 formula and the right rectangular rule are both first-order-type constructions in smooth regimes, yet the graded mesh is used to recover appropriate accuracy under low regularity. This suggests that the method is designed for compatibility between derivative discretization, integral discretization, and singular temporal behavior, rather than for isolated optimization of any single component.

5. Stability and convergence properties

The global analysis in (Bu et al., 16 Sep 2025) treats the right rectangular rule as part of the complete scheme and derives stability and convergence in two scenarios.

For the uniform time mesh, the paper introduces a discrete sequence $\omega_\alpha$9 and proves unconditional stability together with a local time error estimate. The temporal error is stated as

$\alpha>0$0

This result identifies a transition at the delay time: before $\alpha>0$1, the singular factor $\alpha>0$2 reflects the low-regularity regime; after $\alpha>0$3, the estimate becomes uniformly first order in $\alpha>0$4 (Bu et al., 16 Sep 2025).

For the symmetric graded time mesh, the analysis proceeds by introducing a discrete fractional Gronwall inequality, leading to stability and a globally optimal time error estimate

$\alpha>0$5

where $\alpha>0$6 is the number of subdivisions per delay interval and $\alpha>0$7 is the grading parameter (Bu et al., 16 Sep 2025).

The paper attributes an important role to the right rectangular rule in preserving these global rates. Because its local truncation error is $\alpha>0$8, mesh grading can be selected so that this error is compatible with, or not worse than, the error contributed by the L1 derivative approximation. This is a precise instance in which a low-complexity quadrature rule is analytically sufficient for optimal global behavior once the mesh is properly tuned.

A compact comparison of the reported convergence behavior is useful.

Time mesh	Reported temporal error	Analytical device
Uniform mesh	$\alpha>0$9 for $${}_0I_t^\alpha u(x,t)=\int_0^t \omega_\alpha(t-s)\,u(x,s)\,ds,$$0; $${}_0I_t^\alpha u(x,t)=\int_0^t \omega_\alpha(t-s)\,u(x,s)\,ds,$$1 for $${}_0I_t^\alpha u(x,t)=\int_0^t \omega_\alpha(t-s)\,u(x,s)\,ds,$$2	Discrete sequence $${}_0I_t^\alpha u(x,t)=\int_0^t \omega_\alpha(t-s)\,u(x,s)\,ds,$$3
Symmetric graded mesh	$${}_0I_t^\alpha u(x,t)=\int_0^t \omega_\alpha(t-s)\,u(x,s)\,ds,$$4	Discrete fractional Gronwall inequality

These results also clarify a potential misconception: the “right rectangular” qualifier does not imply that the method is restricted to uniform timesteps. In (Bu et al., 16 Sep 2025), the rule is explicitly formulated and analyzed on nonuniform graded meshes through the step-dependent weights $${}_0I_t^\alpha u(x,t)=\int_0^t \omega_\alpha(t-s)\,u(x,s)\,ds,$$5.

6. Numerical validation, interpretation, and related distinctions

Section 5 of (Bu et al., 16 Sep 2025) reports numerical tests that validate the theoretical predictions. On the uniform mesh, the observed convergence order approaches $${}_0I_t^\alpha u(x,t)=\int_0^t \omega_\alpha(t-s)\,u(x,s)\,ds,$$6 on $${}_0I_t^\alpha u(x,t)=\int_0^t \omega_\alpha(t-s)\,u(x,s)\,ds,$$7 and order $${}_0I_t^\alpha u(x,t)=\int_0^t \omega_\alpha(t-s)\,u(x,s)\,ds,$$8 for $${}_0I_t^\alpha u(x,t)=\int_0^t \omega_\alpha(t-s)\,u(x,s)\,ds,$$9. On the graded mesh, the convergence rates match $$\omega_\alpha(t):=\frac{t^{\alpha-1}}{\Gamma(\alpha)}.$$0. These experiments support the claim that the fractional right rectangular rule, when embedded in the proposed scheme, does not obstruct the expected temporal accuracy and behaves consistently with the truncation and stability analysis (Bu et al., 16 Sep 2025).

The numerical evidence also reinforces the interpretation of the rule as a method suited to low-regularity regimes. The improvement obtained on graded meshes is not due to a change in quadrature stencil, but to the interaction between mesh refinement near singular times and the kernel-aware right-endpoint weighting. A plausible implication is that, for subdiffusion-delay problems with startup and delay singularities, mesh design may be more consequential than replacing the right rectangular rule by a formally higher-order quadrature.

Two related distinctions are important for accurate terminology.

First, the fractional right rectangular rule in (Bu et al., 16 Sep 2025) is a numerical quadrature formula for a one-dimensional time-fractional convolution. By contrast, the rectangular fractional integral operators of (Tanaka, 2023) are analytic operators over geometric rectangles in $$\omega_\alpha(t):=\frac{t^{\alpha-1}}{\Gamma(\alpha)}.$$1, studied via weighted inequalities and dyadic-rectangle embeddings. The shared adjective “rectangular” refers to different structures.

Second, the broader literature on rectangle-based time discretizations includes non-fractional and stochastic settings. In (Liu, 2021), the rectangle formula for a stochastic spectral fractional wave equation yields a low-order method whose convergence is limited by temporal regularity, whereas a modified trapezoidal rule improves convergence when extra regularity is available. This comparison suggests an interpretive principle relevant to (Bu et al., 16 Sep 2025): right-rectangle-type rules are often selected not because they are universally most accurate, but because they remain robust under weak solution regularity.

Overall, within the evidence provided by (Bu et al., 16 Sep 2025), the fractional right rectangular rule is best understood as a kernel-consistent, delay-compatible, and graded-mesh-ready discretization of the Riemann–Liouville fractional integral. Its significance lies less in high formal order than in the fact that, under low regularity at $$\omega_\alpha(t):=\frac{t^{\alpha-1}}{\Gamma(\alpha)}.$$2 and $$\omega_\alpha(t):=\frac{t^{\alpha-1}}{\Gamma(\alpha)}.$$3, it supports unconditional stability, provable local consistency, and the reported global convergence bounds for the fully discrete finite element scheme.