Do transparent boundary conditions for the time-fractional heat equation preserve the diffusion sum rule?

The sum rule sets transparency, not flux conservation

A transparent boundary condition at a graph vertex is meant to let an incoming pulse pass into the outgoing bonds exactly as it would on an unbranched line, with no spurious reflection. On a three-bond star graph, the earlier classical result for β = 1 identified one condition that achieves this: the diffusion coefficients must satisfy κ1 = κ2 + κ3, where bond 1 is the incoming bond and bonds 2 and 3 are the outgoing ones.

It is worth separating two things that are easy to conflate. Kirchhoff flux balance at the vertex holds regardless of the sum rule; the paper confirms numerically that q1 = q2 + q3 to machine precision (max deviation around 10⁻¹⁷–10⁻¹⁸) whether or not the sum rule is satisfied. What the sum rule governs is different: how the incoming flux is apportioned between the two outgoing bonds, and whether bond 1 itself sees any reflected component. Breaking the sum rule leaves the flux identity intact while producing visible reflection on bond 1.

The vertex kernel changes order but not the condition

Solving the exterior branch problems with a Laplace transform, the authors show that the non-local vertex operator seen by the incoming bond is an Abel-type convolution kernel of order β/2. At β = 1 this reduces to the classical half-derivative kernel (t − τ)^(−1/2) that appeared in the earlier classical construction.

The kernel remains weakly singular but integrable: since 0 < β/2 ≤ 1/2 < 1, the integral of (t − τ)^(−β/2) over [0, t] is finite for every finite t. So the operator is a genuine convolution acting on the boundary trace, not a formal object. The key point for readers is that the fractional order enters both the branched and the free-space operators through the same kernel, which is why it cancels out of the transparency condition.

Why the sum rule survives the fractional generalization

The paper's main analytical result is that κ1 = κ2 + κ3 eliminates backflow for every β in (0,1], exactly as it does at β = 1. The authors prove this rigorously, together with uniqueness for the coupled vertex problem.

The mechanism is a cancellation rather than a coincidence. Because the fractional order appears in the same Abel-type kernel on both the branched and the unbranched sides, the transparency condition ends up depending only on the diffusivities. The paper states this directly: the transparency of the junction depends only on the network's diffusivities and not on the memory of the transport law. That is the result a reader should carry away, and it is a stronger statement than merely observing that the rule happens to work at a few tested β values.

How the numerical check was set up

The analytical result is validated at the level of the coupled partial differential equation system, not by discretizing the non-local vertex operator itself. Each bond is discretized with the standard implicit L1 scheme for the Caputo derivative, combined with a fully implicit spatial Laplacian. The L1 weights bj = (j+1)^(1−β) − j^(1−β) multiply the entire nodal history at every time step, so cost and memory scale as O(Nt²) in the number of time steps rather than O(Nt) as in a classical implicit scheme.

The vertex conditions involve only values at the same time level and are discretized as in the classical Crank-Nicolson scheme: a one-sided finite difference for the Kirchhoff flux and a single shared vertex unknown enforcing continuity. The fractional order affects only the bulk interior-node equations, not the vertex-coupling algebra. The authors note that a fast convolution technique would be needed for substantially longer simulations.

The validation compares the star-graph solution u1(x, t) on bond 1 against a reference solution obtained by solving the same time-fractional heat equation with diffusivity κ1 on a single unbranched line, truncated symmetrically and far enough that truncation is not visible on the time horizon considered. Both use the same scheme, the same Δx and Δt, and the same Gaussian initial datum on bond 1. If the junction is transparent, the two solutions should coincide on x ≤ 0; any deviation quantifies backflow.

Transparency error stays smallest at the sum rule

The paper measures a relative L2 transparency error E(κ1) at a fixed final time T, comparing the star-graph solution on bond 1 with the unbranched reference over x ≤ 0. Across β = 0.99, 0.7 and 0.5, with κ2 = 0.089 and κ3 = 0.08 fixed, E attains its minimum at or immediately next to κ1 = κ2 + κ3 = 0.169.

Table 2 reports the numbers for κ1 = 0.169 versus κ1 = 0.35 across five values of β. With the sum rule satisfied, E is 3.74% at β = 0.99, 3.03% at β = 0.90, 1.60% at β = 0.70, 0.68% at β = 0.50 and 0.23% at β = 0.30. With the sum rule broken, the corresponding values are 20.07%, 17.65%, 12.34%, 7.55% and 3.87%. The relative error is consistently three-to-five times smaller when the sum rule holds.

The residual error at κ1 = 0.169 is not exactly zero, and the paper attributes this to a discretization artifact of the L1/implicit scheme rather than to any failure of the sum rule. The authors also note that this comparison validates the sum rule's effect at the level of the coupled PDE system and its standard Kirchhoff condition; it is not an independent numerical confirmation of the specific operator identity beyond what the analytical argument already establishes.

Reflection shows up in the temperature profile

For β = 0.7 with the sum rule satisfied, the incoming pulse splits smoothly between bonds 2 and 3 with no visible distortion on bond 1 beyond ordinary sub-diffusive spreading, and the star-graph solution and unbranched reference are visually indistinguishable throughout the evolution.

When the sum rule is broken (κ1 = 0.35), bond 1 develops a visible secondary bulge near the vertex at t = 15, 30 and 60 that is absent in the reflectionless case. This is the reflected component of the pulse, visible directly in the spatial profile rather than only in the aggregate diagnostic E(κ1). The paper notes that the two panels of the vertex-flux figure look qualitatively similar in shape, so the reflection caused by breaking the sum rule is not directly visible in the vertex flux balance itself.

Convergence rates reveal where the vertex costs accuracy

On the uncoupled reference line, refining Δt with Δx = 0.025 fixed at β = 0.7 gives an observed temporal rate of about 1.03–1.05, below the asymptotic rate 2 − β = 1.3. The paper attributes this shortfall to limited time-regularity near t = 0: solutions of time-fractional equations typically retain only limited time-regularity there even for smooth initial data, and this initial-layer singularity governs the observed rate on a uniform mesh. Refining Δx with Δt = 0.02 fixed gives an observed rate of about 2.0–2.2, matching the expected second-order spatial accuracy.

Repeating the study on the coupled star-graph system with the sum rule satisfied, the rates away from the vertex (x = −2) match the uncoupled reference line closely: temporal ≈ 1.03–1.05 and spatial ≈ 2.04–2.22. At the vertex itself (x = 0), the spatial rate drops to ≈ 1.21–1.35, confirming that the one-sided finite difference used for the Kirchhoff flux is locally the dominant source of spatial error near the junction, while the bulk of each bond retains full second-order accuracy.

To confirm the reduced uniform-mesh temporal rate is attributable to the t = 0 initial layer, the authors repeat the temporal refinement on a graded mesh tk = (k/N)^r T with r = (2 − β)/β ≈ 1.86 for β = 0.7. The graded mesh gives an observed rate of about 1.28–1.29, closely matching the theoretical rate 2 − β = 1.3, against 1.03–1.06 on the uniform mesh with the same number of time steps. The authors also report that an earlier implementation contained a genuine indexing error in the L1 history pairing, which the sum-rule comparison alone would not have revealed because it degrades the reference and the star-graph solution consistently and largely cancels in their difference.

Where the invariance result stops applying

The sum-rule invariance is established for a three-bond metric star graph with a single fractional order β shared by all bonds. The paper states that the result generalizes to n-bond star graphs, but trees, loops and graphs with multiple vertices are not treated. Bond-dependent orders βj would break the common-kernel argument behind the invariance result; the authors relate this to, but do not resolve it with, existing edge-dependent-order and space-time fractional metric-graph frameworks.

The paper proves uniqueness for the coupled vertex problem but does not carry out a matching existence theory, leaving a full existence-and-uniqueness treatment for future work. A genuinely discrete fractional vertex condition is also left open. These are boundaries of the analytical result, not of the numerical demonstration, which is explicitly framed as validating the sum rule's effect at the level of the coupled PDE system rather than as an independent confirmation of the operator identity.

The practical priority this paper establishes is a comparison, not a new threshold: for a three-bond star graph with a shared fractional order, the transparency-optimal choice of κ1 is κ1 = κ2 + κ3 at every β tested, and the penalty for missing it is a three-to-five-fold larger relative transparency error plus a visible reflected bulge on the incoming bond. A reader deciding how to allocate diffusivities across a branched sub-diffusive network should treat the sum rule as the target and the fractional order as irrelevant to that target, while remembering that the evidence covers star graphs with one shared β.

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